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Dan Prenzlow, Director, Colorado Parks and Wildlife • Parks and Wildlife Commission: Marvin McDaniel, Chair • Carrie Besnette Hauser, ViceChair
Marie Haskett, Secretary • Taishya Adams • Betsy Blecha • Charles Garcia • Dallas May • Duke Phillips, IV • Luke B. Schafer • James Jay Tutchton • Eden Vardy
�Abstract
Alldredge, Mathew W. Avian point count surveys: Estimating components of the
detection process. (Under the direction of Kenneth H. Pollock and Theodore R. Simons)
Point count surveys of birds are commonly used to provide indices of abundance
or, in some cases, estimates of true abundance. The most common use of point counts is
to provide an index of population abundance or relative abundance. To make spatial or
temporal comparisons valid using this type of count requires the very restrictive
assumption of equal detection probability for the comparisons being made.
We developed a multipleindependent observer approach to estimating abundance
for point count surveys as a modification of the primarysecondary observer approach.
This approach uses standard capturerecapture models, including models of inherent
individual heterogeneity in detection probabilities and models using individual covariates
to account for observable heterogeneity in detection probabilities. Twoobserver models
provided negatively biased estimates because they do not account for individual
heterogeneity in detection probabilities. Models accounting for individual heterogeneity
are always selected as the most parsimonious models for this data type.
We also developed a time of detection approach for estimating avian abundance
when birds are detected aurally, which is a modification of the time of removal approach.
This approach requires collecting detection histories of individual birds in consecutive
time intervals and modeling the detection process using a capturerecapture framework.
This approach incorporates both the probability a bird is available for detection and the
probability of detection given availability. Analyses presented demonstrate the
importance of models accounting for individual heterogeneity in detection probabilities.
�We recommend time of detection point count surveys be designed with four or more
equal intervals.
We also present a multiple species modeling strategy since many point count
surveys collect data on multiple species and present the approach for distance sampling,
multiple observer, and time of detection approaches. The purpose of using a multiple
species modeling approach is to obtain more parsimonious models by exploiting
similarities in the detection process among species. We present a method for defining
species groups which leads to an a priori set of species groups and associated candidate
models. Multiple species models worked well and in many cases gave more
parsimonious models than a species specific modeling approach, especially for the
multipleobserver and time of detection approaches. Parameter estimates for multiple
species models are more precise than single species models. We recommend this
approach for all situations where data on multiple species is collected.
Finally, we present a method for estimating the availability probability of birds
during a point count based on singing rate or detailed singing time data. This approach
requires data collected in conjunction with point count surveys that describe the singing
rates or singing time distribution of the bird population of interest. The singing rate
approach requires the assumption that an individual bird sings following a random
process but rates may vary between birds. We modeled this using a finitemixture
Poisson model. The singing time approach is a nonparametric approach and does not
require this restrictive assumption. Analysis of Ovenbird singing rate data demonstrates
the importance of accounting for availability bias when estimating abundance, especially
�as count lengths get short. We recommend this approach when “snapshot” type counts
are necessary.
Analyses presented throughout this thesis demonstrate the importance of
accurately modeling the detection process to estimate abundance. The importance of
accounting for individual heterogeneity in detection probabilities was evident in every
chapter. Using a point count method that accounts for individual heterogeneity is crucial
to estimating abundance effectively and making valid spatial, temporal and species
comparisons.
��ii
To my family:
I never would have been able to do this without their support and understanding. Not
only did my wife, Stacey, lovingly support me through this process, she also provided the
strength, encouragement and motivation to keep me moving forward. My son, Brennen,
was an inspiration, always there to remind me of the things that are really important. My
unborn son (May 6, 2004), reminding me of the miracles and joys that life offers. Special
thanks to my mother who also provided support and encouragement. Family is the one
truly important thing in life. I have been graciously blessed by my family, thank you.
�iii
Biography
I grew up in the mountains of Colorado and Wyoming. My free time was always
spent outside, skiing, hiking, fishing, hunting and otherwise goofing off in the mountains.
I learned to enjoy wildlife at an early age and am always at my best when interacting with
it and trying to learn and understand how all the pieces fit together. I also had the
opportunity of growing up and interacting with several wildlife biologists, including my
father, and gained many valuable experiences helping with their projects.
Seeing the stress and bureaucracy involved with wildlife research and
management I thought it might be wise to follow a different career path and let my
wildlife interests occupy my free time instead of my profession. I completed an
undergraduate degree in mechanical engineering in 1994 and then quickly decided that
office life was not for me. I started taking wildlife courses at Colorado State University
and worked on a few fish projects. Then I took a job working on a grizzly bear study in
Wyoming. While camped at the foot of the Tetons trapping and tracking bears I gained
the experience of a lifetime which solidified my career path.
I attended the University of Idaho for my Master’s of Science in Wildlife
Resources. My master’s project examined elk habitat and nutritional relationships on
industrial timber lands. My advisor, Jim Peek, taught me more about wildlife and
ecosystems than I could ever learn from books.
After completing my master’s I took a job as a wildlife biologist and continued
studying elk while my wife finished her degree. I really wanted to utilize my quantitative
skills and learn more about the population sampling techniques I had started studying at
CSU. Ken Pollock’s reputation and interest in wildlife sampling led me to North
�iv
Carolina State University. In 2002 I completed a master’s in biomathematics. Having an
interest in population dynamics and modeling as well, I decided to comajor in
Biomathematics and Zoology and threw in a Statistics minor for good measure.
�v
Acknowledgements
First I would like to thank my graduate committee for their support and advice
they provided me. Ken Pollock provided me many opportunities to work on a variety of
projects during my four years at NCSU and helped me develop a broad background in
sampling wildlife populations. Ted Simons brought me on board the “Bird Radio”
project and continuously advised me on the intricacies of point counts. Jim Nichols not
only was willing to fly down to NCSU for committee meetings but also helped me think
through the sampling and modeling issues I proposed and dealt with many questions
about model results (the ominous chat). Always remember to “think hard” about
something. Cavell Brownie helped and encouraged me on the statistical development of
my research and always had ideas on improving my techniques. Jim Gilliam, another
great thinker, always looked for applications to more general questions and challenged
my model assumptions. I owe the singing time model of Chapter 5 to this. I would also
like to thank Jaime Collazo for his assistance on this project.
This project was funded by the U.S. Geological Survey and the National Park
Service.
I also appreciate all of the field work that went before me. Without the
accessibility of these data sets for example analyses of my methods my thesis would have
been little more than statistical rambling. I would thank each field technician by name
but I know them only by the initials appearing on the data sheets.
Finally, I would like to thank my family once again for all of their support and
encouragement.
�vi
Table of Contents
Page
LIST OF TABLES
xii
LIST OF FIGURES
xvii
CHAPTER 1
INTRODUCTION
1
CHAPTER 2
ESTIMATING DETECTION PROBABILITIES FROM MULTIPLE
OBSERVER POINT COUNTS
12
ABSTRACT
13
INTRODUCTION
14
METHODS
17
Field Methods
17
PrimarySecondary Observer Model
19
Two independent observer model
20
Four or more independent observer models
23
Analyses
26
Comparison of Dependent and Independent Observer Models
26
Field data
27
Two independent observer examples
28
Four independent observer examples
29
Heterogeneity simulations
31
�vii
RESULTS
32
Comparison of dependent and independent observer models
32
Two independent observer examples
32
Four independent observer examples
32
Heterogeneity simulations
34
DISCUSSION
36
Field application
37
Example analyses
40
RECOMMENDATIONS
42
LITERATURE CITED
45
APPENDIX 1: Common and scientific names for the “Warbler” and “Vireo”
species groups used for analysis.
55
CHAPTER 3
TIME OF DETECTION METHOD FOR ESTIMATING ABUNDANCE FROM
POINT COUNT SURVEYS
56
ABSTRACT
57
INTRODUCTION
58
METHODS
61
Field Methods
61
Detection process
62
Approach and candidate models
64
General form of model
71
Models without heterogeneity
71
�viii
Heterogeneity models
76
Covariates
79
FIELD TRIALS
83
RESULTS
85
Three Interval Data Set
85
Four Interval Data Set
86
DISCUSSION
87
RECOMMENDATIONS
90
LITERATURE CITED
92
CHAPTER 4
MULTIPLE SPECIES ANALYSIS OF POINT COUNT DATA: A MORE
PARSIMONIOUS MODELING FRAMEWORK
99
ABSTRACT
100
INTRODUCTION
101
METHODS
104
Field Methods
104
Defining Species Groups
104
Model Selection
108
Multiple Species Modeling Strategy
109
ESTIMATION METHODS AND CANDIDATE MODELS
110
Distance
110
Time of Detection
113
Multiple Observers
119
�ix
FIELD TRIALS
123
Species Groups
124
Distance Analysis
125
Time of Detection Analysis
126
Multiple Observer Analysis
126
RESULTS
127
Species Groups
127
Distance
127
Time of Detection
129
Independent Observer
130
DISCUSSION
131
RECOMMENDATIONS
134
LITERATURE CITED
136
CHAPTER 5
MODELING THE AVAILABILITY PROCESS FOR POINT COUNT
SURVEYS USING AUXILIARY DATA
150
ABSTRACT
151
INTRODUCTION
153
METHODS
157
Field Methods
157
Availability Assuming Homogeneous Singing Rates
158
Availability Assuming Heterogeneous Singing Rates
160
Estimating Variance and Confidence Intervals
163
�x
Availability Using Singing Times
163
ANALYSIS OF FIELD DATA
167
RESULTS
169
Singing Rate Models
169
Singing Time Model
170
DISCUSSION
171
RECOMMENDATIONS
180
LITERATURE CITED
182
CHAPTER 6
EXECUTIVE SUMMARY
194
INTRODUCTION
195
Chapter 2: ESTIMATING DETECTION PROBABILITIES FROM MULTIPLE
OBSERVER POINT COUNTS
196
Objectives
196
Implications and Findings
197
Chapter 3: TIME OF DETECTION METHOD FOR ESTIMATING
ABUNDANCE FROM POINT COUNT SURVEYS
198
Objectives
198
Implications and Findings
198
Chapter 4: MULTIPLE SPECIES ANALYSIS OF POINT COUNT DATA: A
MORE PARSIMONIOUS MODELING FRAMEWORK
199
Objectives
199
Implications and Findings
199
Chapter 5: MODELING THE AVAILABILITY PROCESS FOR POINT
COUNT SURVEYS USING AUXILIARY DATA
200
�xi
Objectives
Implications and Findings
200
201
GENERAL CONCLUSIONS
202
APPENDIX 2: SURVIV CODE: Time of Detection Analysis
204
Single Species Code
205
Multiple Species Code (4 species example)
207
APPENDIX 3: SINGING TIME PROGRAMS
216
MATLAB Code for Singing Time Analysis
217
Screen Layout for Singing Time Data Collection
220
�xii
List of Tables
Chapter 2: ESTIMATING DETECTION PROBABILITIES FROM
MULTIPLE OBSERVER POINT COUNTS
Table 1: Comparison of the dependentobserver approach to the independentobserver approach using simulations of 1,000 data sets for each population
size, detectability and method. The unbiased scenario is the ratio of the
SE of the dependentobserver method to the SE of the independentobserver method. The biased scenario represents 10% and 20% of the
observations in the independentobserver data being dependent on the first
observer and compares the ratio of the SE of the dependentobserver
method to the MSE1/2 for the independent observer method.
49
Table 2: Model selection for the twoindependent observer examples giving the
∆AICc values for all 6 candidate models. The smaller ∆AICc values
indicate a more parsimonious model with 0 indicating the selected model.
AICc weights in parentheses.
50
Table 3: Abundance estimates (N) for the twoindependent observer examples.
Birds detected are the totals between the two observers. Model M0 was
selected as the most parsimonious for all data sets except the Ovenbird.
51
Table 4: Model selection for the fourindependent observer examples giving the
∆AICc values for all 8 candidate models. Model M 2h and model M obs,2h
are based on 2 point mixture models of heterogeneity. The smaller ∆AICc
values indicate a more parsimonious model with 0 indicating the selected
model. AICc weights are in parentheses.
52
Table 5: Abundance estimates (N) for the fourindependent observer examples.
Birds detected are the totals among the four observers. Detection
probabilities are given by p for model M 0 and pgroup1 and pgroup2 for model
M 2h . The proportion of the population in group 1 is given by pr(group1).
Standard errors are in parentheses.
53
�xiii
Table 6: Abundance estimates for fourobserver and twoobserver methods and a
single observer count from simulated heterogeneous data from three and
twopoint mixture distributions. For the threepoint mixture distribution
20% of the population had detection probability 1, 60% had detection
probability 0.75, and the remaining 20% had three different levels (low =
0.5, moderate = 0.3, and high =0.1) of detection probabilities. For the
twopoint mixture distribution half the population had high detection
probability (p=0.9) and the other half low detection probability (p=0.1 or
p=0.2), which gave capture histories similar to those observed in the
“Warbler” data set.
54
Chapter 3: TIME OF DETECTION METHOD FOR ESTIMATING
ABUNDANCE FROM POINT COUNT SURVEYS
Table 1: ∆AICc values for the 11 time of detection models fit to each data set.
∆AICc = 0.0 for the most parsimonious model for each data set. ∆AICc
weights (in parentheses) indicate the strength of the evidence for a given
model compared to the other models (the larger number indicates more
evidence for that model).
95
Table 2. Parameter estimates from the selected model for the 3 interval point
count data sets. λ1 is the proportion of the population that is in group 1.
Detection probability (pij) is the probability of detecting an individual
from group j in interval i. The detection probabilities for group 2 (pt2)
were fixed. Standard errors in parentheses.
96
Table 3: ∆AICc values for the four time interval Thrasher data set. A value of 0.0
indicates the most parsimonious model. ∆AICc weight is in parentheses
where weight is nonzero. NE indicates models that were not included
because of unreasonable parameter estimates.
97
Table 4: Estimated detection probabilities pij, probability of group occurrence λj
and standard errors for interval i and group j of the four time interval
Thrasher data set based on model Mth. Standard error for group two and
time interval two is not estimable.
98
�xiv
Chapter 4: MULTIPLE SPECIES ANALYSIS OF POINT COUNT DATA:
A MORE PARSIMONIOUS MODELING FRAMEWORK
Table 1: Number of parameters in candidate models for the time of detection
method with t time periods and s different species. Behavior models
assume a single behavioral response and heterogeneity models are based
on a 2 point mixture. Models with a superscript +spp indicate an additive
effect between observers and species and models with a superscript *spp
indicate an interaction effect between observers and species. Models with
superscript part indicate those that have similar detection probabilities
between species but different probabilities of being in the first group. An
additional parameter is required for each model to estimate abundance.
140
Table 2: Number of parameters in candidate model for the multiple observer
method using t observers, s species and heterogeneity models based on a 2
point mixture model. Models with superscript part indicate those that have
similar detection probabilities between species but different probabilities
of being in the first group. Models with a superscript +spp indicate an
additive effect between observers and species and models with a
superscript *spp indicate an interaction effect between observers and
species. An additional parameter is required for each model to estimate
abundance.
141
Table 3: Species groups for example analyses. Grouped first into three maximum
detection distance categories (≤100 m, > 100 m and ≤ 150 m, or > 150 m)
and then grouped by similarity in singing rates. Maximum detection
distance is from the actual data and is truncated by 10% of the largest
detection distances. All other categories are averages from rankings on a
scale of 1 to 5 from seven experienced birders familiar with the study area.
Higher ranks correspond to assumed higher values for each category.
142
Table 4: delta AICc for distance models using first 3 minute interval of time of
detection data.
143
Table 5: Distance analysis for first 3 minutes of 10 minute point count. Results
given for model with no species effect and for model with no species
effect. Observed count is after 10% truncation of largest observed
detection distances, EDR is the effective detection radius and density is
individuals per hectare. Standard errors are in parentheses.
144
�xv
Table 6: ∆AICc for time of detection multiple species models for unlimited radius
plots with 10% truncation of largest detection distances. Smaller values of
∆AICc indicate more parsimonious models. ∆AICc weights in
parentheses. Larger weights indicate more support for a given model.
Models with weights ≥ 0.20 are in bold for each species indicating
competing models.
145
Table 7: Parameter estimates from the time of detection method for each species.
The Probability that an individual is detected at least once during the count
pˆ T and the probability of being in the first heterogeneity group λ̂ and the
estimated abundance N̂ are given for the selected model and for the
selected model from a single species modeling approach. The
instantaneous rates formulation was used to estimate detection
probabilities. Standard errors are given in parentheses.
146
Table 8: ∆AICc for the four independent observer multiple species models for
unlimited radius plots with 10% truncation of largest detection distances.
Smaller values of ∆AICc indicate more parsimonious models. ∆AICc
weights in parentheses. Larger weights indicate more support for a given
model. Models with weights ≥ 0.20 are in bold for each species indicating
competing models. The number of observations for each species was
small for this data set so the groups have been modified for analysis.
148
Table 9: Independent Observer results for group B without Blackthroated Blue
Warbler, group C without Indigo Bunting and a combined group of one
species from groups D, E, and F. The probability of being in the low or
high detectability groups is given by πˆ and the probability of detection by
one of the 4 observers is given by p̂1 , p̂2 , p̂3 , and p̂4 These are reported
based on the selected model. The abundance estimate N̂ is given for the
selected model and for the selected model from a single species analysis.
149
�xvi
Chapter 5: MODELING THE AVAILABILITY PROCESS
FOR POINT COUNT SURVEYS USING AUXILIARY DATA
Table 1: Parameter estimates for the homogeneous Poisson model and the twopoint Poisson mixture models. Standard errors and percentile 95%
confidence intervals obtained with 1,000 bootstrap samples. ∆AIC value
of zero indicates the selected model. Availability probability estimates
p̂a (1), p̂a (2), and p̂a (3) are for 1, 2, and 3 minute point count surveys,
respectively.
186
Table 2: Availability probability estimates for one, two, and three minute point
count surveys using two simulated data sets for a five minute observation
period. One data set uses completely random singing times and the other
assumes birds sing in bouts of five songs. For one iteration a sample of
100 birds is drawn with replacement and 1,000 iterations are done for each
analysis. For each data set analyses are done for one, two, and three
minute point counts giving the availability probabilities p̂a (1) , p̂a (2) and
p̂a (3) , respectively. Percentile 95% confidence intervals are reported and
standard errors are in parentheses.
187
�xvii
List of Figures
Chapter 5: MODELING THE AVAILABILITY PROCESS
FOR POINT COUNT SURVEYS USING AUXILIARY DATA
Figure 1: Homogeneous Poisson model and twopoint Poisson mixture model fit
to Ovenbird singing rate data. Poisson mixture model is corrected for “size” bias
that occurs in this data, which is not a factor under the assumptions of the
homogeneous Poisson model.
189
Figure 2: Distribution of λ̂ from 1,000 bootstrap estimates for the homogeneous
Poisson model fit to the Ovenbird data set. Points within the percentiled 95%
confidence intervals are in black.
190
Figure 3: Distribution of λ̂1 , λ̂ 2 and δˆ from 1,000 bootstrap estimates for the
twopoint Poisson mixture model fit to the Ovenbird data set. Points within the
percentiled 95% confidence intervals are in black.
191
Figure 4: Distribution of the availability probability from simulated data with
random singing times using the singing time analysis approach for one, two and
three minute point counts. Data was simulated to be comparable to the Ovenbird
data set.
192
Figure 5: Distribution of the availability probability from simulated data
assuming birds sing in bouts of five songs. Analysis was based on the singing
time approach for one, two, and three minute point counts. Data was simulated to
be comparable to the Ovenbird data set.
193
�Chapter 1
Introduction
�2
There are a wide variety of field and statistical techniques for assessing animal
abundance, which include complete counts, partial counts, and capture methods (Seber
1982, Lancia et al. 1994, Williams et al. 2002). Rarely is it possible to conduct complete
counts as only portions of the area of interest can actually be counted and generally not
all animals in the sample areas will be observed. Such counts require that data are
collected in a manner that allows for the estimation of the fraction of the population that
is sampled. The actual sampling approach used is generally species and/or habitat
specific and may depend on the specific research question (Seber 1982, Lancia et al.
1994).
The interest in estimating animal abundance is that it is commonly used as a
measure of population health by ornithologists and other biologists (Lack 1954, 1966).
Abundance estimates over successive years can provide information on population
trends, which can be suggestive of population health (Ralph et al. 1995, Williams et al.
2002). Besides comparing abundance estimates between years it is also possible to
compare between spatially distinct areas, which can provide information on habitat
relationships or differences associated with management practices (Ralph et al. 1995).
Comparisons that may be of interest are between unmanaged areas, such as National
Parks, and actively managed areas, such as National Forests or state owned lands. These
comparisons can be important tools in adaptive management (Walters and Hilborn 1978)
and for understanding changes that occur in animal populations.
Although population estimates provide useful information about the state of
animal populations and are the focus of this dissertation, it is worth noting that it is also
necessary to obtain other demographic parameters to fully understand the dynamics of a
�3
population. This involves a complete understanding of the losses and gains occurring in
a population that are associated with birth, death and migration. Both abiotic and biotic
factors can affect population process and these factors must also be examined to fully
understand the dynamics of a population (Ricklefs and Miller 2000, Williams et al.
2002).
The lack of “good” quantitative measures of landbird (nongame bird species)
abundance in the past comes from two sources; lack of interest, and difficulty in
obtaining reasonable estimates (Nichols 1994). The lack of interest stems from the
historical concern for game birds and waterfowl which have been actively managed for
recreational use (Martin et al. 1979). Problems of obtaining reasonable estimates of
landbird abundance have slowed the development of valid statistical techniques but with
recent interest in landbird populations there has been a renewed interest in enhancing the
available methods. Of the available methods point count surveys are the most widely
used method for assessing abundance of landbirds (Ralph et al. 1995).
Recent interest in landbirds is due to concerns over possible declines of landbird
populations (Robbins et al. 1986, Askins et al. 1990). These declines have been the
motivation for programs such as Partners in Flight (Carter et al. 2000) that includes large
scale monitoring programs. The original objective of Partners in Flight was inventory
and monitoring of neotropical migrants, but this has been expanded to include other birds
of concern.
The number of people participating in national monitoring programs is also
evidence of the interest in bird populations. One such survey is the Breeding Bird Survey
(BBS), which has been conducted since 1966 (Sauer et al. 1997, 2003). Currently the
�4
BBS consists of about 3,700 active routes (nearly 2,900 surveyed annually) that are
distributed across the continental U.S. and Canada. Each route is 24.5 miles long and has
50 stops per route located at 0.5 mile intervals. Surveys are conducted during the
breeding season each year and are only done on days that satisfy a standardized protocol
to try and ensure that detection probabilities are constant over time. At each stop an
observer counts the number of birds detected of all species that are either heard or seen
during a three minute interval.
The BBS survey is representative of typical surveys for landbirds (Rosenstock et
al. 2002). A series of points are randomly placed over an area of interest. Then, using a
standardized approach, each point is surveyed for a set amount of time and all birds
detected are recorded. This can be done with either fixed radius plots or unlimited radius
plots. Such a count gives a measure of relative abundance of a population or provides an
index to abundance but does not provide an estimate of true abundance.
There have been a number of studies that have demonstrated that both observer
differences and environmental conditions affect the number of birds counted (Ralph and
Scott 1981). This led to the standardization of point counts as a means of providing
comparable counts both temporally and spatially. The general idea is that if counts are
conducted by the same observer or by observers with similar ability and are always done
on days with similar environmental conditions then the proportion of birds counted
should be similar between counts. Methods such as these have been highly criticized
throughout the literature on abundance estimation because it is now widely believed that
no amount of standardization can account for all of the variation associated with
�5
detection of animals (Burnham 1981, Wilson and Bart 1985, Johnson 1995, Barker and
Sauer 1995, Nichols et al. 2000, Rosenstock et al. 2002, Thompson 2002).
The general model for the relationship between a count statistic (Ci) and the true
abundance (Ni) is given by (Lancia et al. 1994, Williams et al. 2002)
E (C ) = N p
i
i
i
(1)
where i denotes the location or time of the count and pi is the probability of detection.
The premise behind standardizing counts and obtaining an index to abundance is that the
detection probability is constant across space and time because of the standardization.
With this assumption comparisons of abundance across space and time are made using
the count as an index to abundance.
An alternative approach is to collect count data so that the associated detection
probability can be estimated (Nichols et al. 2000, Farnsworth et al. 2002, Rosenstock et
al. 2002, Thompson 2002). With this additional information, direct estimation of
abundance is possible as
C
Nˆ = pˆ
i
i
(2)
i
Using this approach the resulting estimates of abundance can be used to draw inference
about population differences across space and/or time without the strict assumption of
constant probability of detection.
�6
The detection process consists of two components; the probability that an
individual is available for detection, and the probability that an individual is detected
given that it is available. The pi given in equation 2 is the product of these two
components (Marsh and Sinclair 1989) such that equation 2 becomes
Nˆ
i
=
C
pˆ pˆ
i
a
(3)
d
where pa is the probability that an individual is available for detection and pd is the
probability that an individual is detected given that it is available. Marsh and Sinclair
(1989) were concerned with aerial surveys of marine mammals where availability was
associated with an individual’s position in the water column and sea state. Most
estimation methods of animal abundance assume that all animals are available and thus
ignore this component. Examination of the original capturerecapture models (Otis et al.
1978, Seber 1982, Williams et al. 2002) shows that these models only estimate the
capture probability of animals available for capture, although recent temporary
emigration models (Kendall et al. 1997) do account for availability associated with
spatial location of an individual. In some situations the availability process may be very
important, including bird point counts which may require a bird producing a sound cue
before it can be detected.
The general methods for estimating abundance from point count data are distance
sampling, multiple observer approaches, the time of detection approach, double sampling
and repeated count methods. Distance sampling models the decline in the probability of
�7
detection associated with distance from the observer (Reynolds et al. 1980, Buckland et
al. 1993). The multiple observer method uses capturerecapture models to estimate the
detection probabilities of each observer (Nichols et al. 2000). The time of detection
method also uses capturerecapture models to estimate detection probabilities associated
with time intervals of a count (Farnsworth et al. 2002). Of these methods only the time
of detection method estimates the product of availability and detection given availability
while the other two approaches assume all animals are available. These methods will be
reviewed in more detail as this dissertation develops. The double sampling approach
(Bart and Earnst 2002) requires a “fast” method to obtain a count and then a “slow”
method to resample a portion of the area initially sampled and is assumed to be a census.
We do not believe that the double sampling method is appropriate in forested
environments. Repeated count methods (Royle and Nichols 2003, Royle 2004) estimate
both availability and detection given availability by repeated sampling of a set of points
over time. Repeated count methods also include the probability that an individual is in
the sample area as this may change between surveys.
The specific objectives of my dissertation are to examine the detection process
associated with auditory detection of birds, present some alternative methods for
estimating the detection process, and provide examples of these methods. In chapter 2, I
will present the multiple independentobserver approach, the relevant models and provide
an example analysis using this method. In chapter 3, I will present the time of detection
model, of which Farnsworth et al.’s (2002) removal model is a special case, develop
covariate models and heterogeneity models and present example analyses. In chapter 4, I
will develop multiple species models that exploit similarities in the detection process
�8
among similar species to model the detection probabilities which will give more
parsimonious models with better precision. In chapter 5, I will examine the availability
process more closely and present models that incorporate this directly from point count
data and an approach that uses auxiliary information on singing frequency to model the
availability process.
For the time of detection approach it is necessary to use program SURVIV (White
1983) to estimate model parameters when point count surveys are conducted with
unequal time intervals. In chapters 3 and 4 we present example analyses from data
collected with unequal time intervals. The SURVIV code used for these analyses is
given in appendix 1. The single species code is easily modified by changing the values
for new data. The multiple species code must be modified to fit the number of species in
the analysis. We give the SURVIV code for a four species analysis and the relevant
models.
To model the availability process using auxiliary data, it was also necessary to
develop computer programs for the analysis. We do not give the code for the bootstrap
since this is a standard analysis procedure. To analyze the singing time data we used
MATLAB and have included the code in appendix 2. We have also written a Windows
based computer program for collecting singing time data. This program can be used on
Windows based PDA’s. In appendix 3 we show the screen layout of the program and
provide the source code.
�9
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birds of eastern North America. Current Ornithology 7:1057
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Eds.). U.S. Department of Agriculture, Forest Service General Technical Report PSWGTR149.
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using capturerecapture data with Pollock’s robust design. Ecology 78:563578.
Lack, D. 1954. The Natural Regulation of Animal Numbers. Oxford University Press,
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Lack, D. 1966. Population Studies of Birds. Clarendon Press, Oxford.
Lancia, R.A., J.D. Nichols, and K.H. Pollock. 1994. Estimating the number of animals
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wildlife and habitats (T. Bookhout, Ed.). The Wildlife Society, Bethesda, Maryland.
�10
Marsh, H. and D.F. Sinclair. 1989. Correcting for visibility bias in strip transect aerial
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Martin, F.W., R.S. Pospahala, and J.D. Nichols. 1979. Assessment and population
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Related Quantitative Issues. Statistical Ecology, Vol. S11 (J. Cairns, G.P. Patil, and W.E.
Waters, eds.). International Coorperative Publication House, Fairland, MD.
Nichols, J.D. 1994. Capturerecapture methods for bird population studies. Proceedings
of Italian Ornithological Congress 6:3151.
Nichols, J.D., J.E. Hines, J.R. Sauer, F.W. Fallon, J.E. Fallon, and P.J. Heglund. 2000.
A doubleobserver approach for estimating detection probability and abundance from
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Otis, D.L., K.P. Burnham, G.C. White, and D.R. Anderson. 1978. Statistical inference
from capture data on closed animal populations. Wildlife Monographs, No. 62.
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point counts: Standards and applications. Pages 161168 in Monitoring bird populations
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replicated counts. Biometrics 60:108115.
�11
Sauer, J.R., J.E. Hines, G. Gough, I. Thomas, and B.G. Peterjohn. 1997. The North
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ed.). Edward Arnold, London.
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present but not detected. Auk. 119:1825.
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Annual Review of Ecology and Systematics 9:157188.
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biotelemetry data. Journal of Wildlife Management. 47:716728.
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phenology during the breeding season. Condor 87:6973.
�Chapter 2
ESTIMATING DETECTION PROBABILITIES FROM
MULTIPLE OBSERVER POINT COUNTS
�13
ESTIMATING DETECTION PROBABILITIES FROM MULTIPLE OBSERVER
POINT COUNTS
Mathew W. Alldredge, Biomathematics and Zoology, North Carolina State University,
Raleigh NC 27695.
Kenneth H. Pollock, Zoology, Biomathematics, and Statistics, North Carolina State
University, Raleigh, NC 27695.
Theodore R. Simons, USGS Cooperative Fish and Wildlife Research Unit, Dept. of
Zoology, North Carolina State University, Raleigh, NC 27695.
Abstract.—Point counts are commonly used to obtain indices of bird population
abundance. Recent methodological developments, including the dependentobserver
approach of Nichols et al. (2000) estimate detection probabilities which can reduce biases
associated with spatial and temporal variability in detection probability. We present an
independentobserver point count approach, which is a generalization of the dependentobserver approach. The independentobserver approach is essentially a closed population
capturerecapture method. Additional models can be parameterized using covariates,
such as detection distance, to account for heterogeneity associated with identified sources
of variation. By comparing abundance estimates from two and fourobserver point
counts we demonstrate a negative bias in twoobserver estimates. This negative bias,
caused by unobservable individual heterogeneity in detection probabilities, can be
accounted for when models with four independent observers are used. In four out of five
data sets examined heterogeneity models were selected, producing abundance estimates
15% to 26% higher than models that did not account for heterogeneity. The independentobserver approach is more efficient (smaller variance) than the dependentobserver
approach because it uses the full detection history of an individual and not just first
�14
detections. The method also allows the incorporation of detection distance estimates
which account for situations where detection probabilities decline as a function of
distance from the observer. Additionally, with four or more observers, the method
accounts for individual heterogeneity in detection probabilities which reduces the bias of
abundance estimates. Although independent observer methods are expensive and
possibly impractical for large scale applications, we believe they can provide important
insights into the sources and degree of perception bias (probability of detecting an
individual given that it is available for detection) in avian point count estimates, and that
they may be useful in a two stage sampling framework to calibrate single observer
estimates.
Introduction
Point counts are used extensively as indices of spatial and temporal differences in
bird abundance, and to assess habitat relationships, responses to environmental change or
management, and species diversity (Ralph et al. 1995a, Thompson 2002). They are used
across a spectrum of scales from long term continentalscale surveys such as the
Breeding Bird Survey (Robbins et al. 1986, Sauer et al. 1997, 2003) to short term site
specific studies (Ralph et al. 1995a).
There are fundamentally two approaches to abundance estimation using count
data. The first generates an abundance index using a standardized approach to control for
known sources of bias (e.g. weather, observer skill, time of day and season) (Conroy
1996, Sauer et al. 1997, Williams et al. 2002). The second uses statistical methods that
estimate how detection probabilities vary among observers and across space and time
�15
(Nichols et al. 2000, Williams et al. 2002). Estimated detection probabilities are used to
adjust the raw counts to reduce the bias of abundance estimates.
In a review of 224 papers reporting sampling techniques used to draw inference
about abundance, 95% relied on index counts (Rosenstock et al. 2002). Comparisons of
index counts across space or time require the strong assumption that the probability of
detection is constant for all locations and/or times. Assumptions of constant detection
probability have long been questioned (Burnham 1981, Wilson and Bart 1985, Johnson
1995, Barker and Sauer 1995). The weakness of these assumptions has motivated
considerable research into statistical approaches that estimate detection probabilities
directly for all study areas and time periods (Nichols et al. 2000, Bart and Earnst 2002,
Farnsworth et al 2002, Rosenstock et al. 2002, Thompson 2002). The general problem
with index counts is that no amount of standardization can account for the unobservable
or uncontrollable sources of variation that affect the raw count data (Burnham 1981,
Johnson 1995). Known sources that affect detection probabilities of birds are season
(Ralph 1981, Skirvin 1981), time of day (Robbins 1981, Skirvin 1981), stage of nesting
cycle (Wilson and Bart 1985), observer effects on singing frequency, habitat
characteristics and local species densities (McShea and Rappole 1997), and differences
among observers (Sauer et al. 1994). Conceptually, if 2 count statistics differ across
space or time it is not possible to distinguish if the difference is attributable to differences
in detection probability (observer differences, effects of habitat structure or other factors
affecting detection probability), or actual differences in abundance. For a thorough
comparison of these issues see Nichols et al. (2000) and Rosenstock et al. (2002).
The underlying model for estimating population size from count data is:
�16
Nˆ
i
=
,
Ci
pˆ
(1)
i
where the N̂ i is the estimated abundance, Ci is the count, p̂i is the detection probability,
and i denotes the time and location of the count (Lancia et al. 1994, Williams et al. 2002).
The probability of detection has two components (Marsh and Sinclair 1989); the
probability of being available for detection ( p̂a ) (i.e. if detections are auditory the
probability that the bird sings during the count interval), and the probability of detecting
( p̂d ) a bird given that it is available. There are currently five methods for estimating
detection probability for point count data. The methods employ; distance sampling,
multiple observers, time of detection, double sampling, and repeated counts. The point
transect distance or variable circular plot method (Reynolds et al. 1980, Buckland et al.
1993), and the dependent observer or primarysecondary observer approach (Nichols et
al. 2000) only estimate the probability of detection given availability. The time of
detection method (Farnsworth et al. 2002) estimates the product of availability and
detection given availability but it cannot separate the two components. The double
sampling approach requires a “rapid” sample and then a more intensive subsample of
plots to correct for observability bias (Bart and Earnst 2002). The repeated counts
method requires sampling the same plots over a period of time and estimates the product
of the probability of being available and the probability of detection given that an
individual is available but cannot separate these components (Royle and Nichols 2003).
�17
The repeated count approaches also include the probability that an individual is in the
sample area during the survey because individuals will move between successive surveys.
In this paper we focus on multiple observer methods of estimating detection
probability from point counts. Nichols et al. (2000) suggested that a completely
independent observer approach would provide more modeling flexibility and benefits
over the dependent observer approach if independence between observers was possible.
Our objectives are to; 1) present the two independent observer method and potential
models for estimating detection probability, including the use of detection distance
covariates, showing that the models are essentially closed capturerecapture models,
including the use of distance covariates, 2) present a more general model using four
independent observers showing that multiple observer models are essentially closed
capturerecapture models that allow for individual heterogeneity, 3) compare the
efficiency of the two independent observer approach to the primarysecondary observer
approach of Nichols et al. (2002), 4) present a two independent observer example to
demonstrate the procedure, 5) present a four independent observer example to model
inherent heterogeneity in bird detection probabilities and demonstrate potential bias in
twoobserver estimates, and 6) simulate data under a heterogeneous model to illustrate
the levels of heterogeneity typically present in data and the effect of heterogeneity on
twoobserver models and index counts.
Methods
Field methods.—The general sampling situation for the multiple observer
methods is a point count survey where multiple points are surveyed from an area of
interest. Point counts should be done using standardized guidelines that specify the time
�18
of year and time of day to conduct counts, suitable weather conditions, duration of
counts, spacing between points, etc. (Ralph et al. 1995b). This is a standard approach to
point counts used to maximize detection probabilities and reduce extraneous variability
among counts. For example the North American Breeding Bird Survey (BBS) requires a
requisite level of observer expertise, uses the same routes and stops each year, specifies
suitable weather conditions for counts, and uses a three minute count (Sauer et al. 1997).
When areas of interest are large, stratification by similar habitat is necessary to account
for differences in detection probabilities associated with habitat (Buckland et al. 1993,
Ralph et al. 1995b, Nichols et al. 2000).
The dependent observer method of Nichols et al. (2000) uses two observers, one
primary and one secondary, for each survey. The primary observer identifies all birds
seen or heard and communicates this to the secondary observer. The secondary observer
records birds identified by the primary observer and additional birds missed by the
primary observer. The role of the primary and secondary observer must be switched
during the survey, preferably so that one observer is primary for half the survey points.
For each point, the data for each species are; the number detected by the primary
observer, and the number missed by the primary but detected by the secondary observer.
The independent observer method uses essentially the same sampling design
except that observers conduct each point count simultaneously but independently of the
other observers. At the end of each point count observers combine their data and
determine the detection history for each bird identified during the count. For a twoobserver count the possible detection histories for each species are the number of birds
�19
seen in common by both observers, the number of birds seen only by the first observer,
and the number seen only by the second observer.
For both methods it is necessary to record the approximate direction and distance
of all detections and to track the movement of birds during the count. Tracking
movement avoids double counting of birds and minimizes matching errors with the
independent observer method. Recording the location of each detection is necessary to
match birds among observers using the independent observer method. Detection distance
estimates are use to determine the effective area sampled during the survey which is
necessary for making spatial or temporal comparisons (Ralph et al. 1995b). An
alternative to estimating detection distance is the use of fixed radius plots, where only
birds within a given radius are recorded.
Primarysecondary observer model.—The model proposed by Nichols et al.
(2000) is a modification of the model used by Cook and Jacobson (1979) to correct for
visibility bias in aerial surveys. The secondary observer only records detections not made
by the primary observer. The additional requirement that observers switch primary and
secondary roles creates two sets of data, which are equivalent to a generalization of a
removal study (Zippin 1858, Seber 1982) with two groups. If we let xij be the number of
individuals counted by observer i (i = 1, 2) for points when observer j (j = 1, 2) is
primary, then the probability that a bird in the sample area is detected by at least oneobserver is,
x x
pˆ =1− 12 21
x 22 x11
d
.
(2)
�20
Note that this method, like all multiple observer methods, is estimating the probability of
detection given availability of the animal. Abundance of the available portion of the
population is then estimated using equation 1.
The assumptions for this method are:
1. The probability of detection by a particular observer for a given species is the
same for all individuals of that species, regardless of whether the observer is
primary or secondary.
2. The population within the effective search radius is closed during the count.
3. Birds are identified correctly and not double counted.
4. All detections made by the primary observer are independent of the secondary
observer.
The first assumption will be violated if there is individual heterogeneity in detection
probability because those missed by the primary observer would likely have a lower
detection probability and also less likely to be detected by the secondary observer.
Two independent observer models.— If the same sampling technique is used but
the observer’s detections are independent, survey data are in the form of a LincolnPetersen closedpopulation capturerecapture model (Otis et al. 1978, Seber 1982).
pd1 – probability of detection by the first observer
pd2 – probability of detection by the second observer
x11 – number detected by both observers
x10 – number detected by the first observer only
x01 – number detected by the second observer only
�21
n1 – total number detected by the first observer (x10 + x11)
n2 – total number detected by the second observer (x01 + x11)
Using this notation the probability of detection by each observer is estimated by,
pˆ
d1
=
x
n
11
2
and
pˆ
d2
(3)
=
x ,
n
11
1
and the probability of detection by at least oneobserver is,
pˆ = 1 − (1 − pˆ )(1 − pˆ
d1
d
d2
)
.
(4)
The estimate of the probability of detection by at least oneobserver is then used with the
observed count in equation 1 to estimate population size.
The assumptions for the independent observer models are:
1. Independence of observations among observers.
2. If a fixed radius plot is used, then counts within the fixed radius circle are
accurate.
3. There are no matching errors among the observers so that assignments of
detection histories x11, x10, and x01 are accurate.
4. Detection probability for each species at all points is constant for each
observer.
5. There is no undetected movement into or out of the fixed radius plot.
Relevant capturerecapture models for the twoobserver case are model Mo (equal
detection probability between observers) and model Mobs (unequal detection probability
between observers; Mt of Otis et al. 1978) See Otis et al. (1978) and White et al. (1982)
�22
for a description of these capturerecapture models. Survey data can then be analyzed
using these models available in program CAPTURE (White et al. 1982) or program
MARK (White and Burnham 1999), which has the benefit of using information theoretic
model selection procedures. Additional models can also be used that incorporate
individual bird covariates, such as radial detection distance from observers, can be
developed using a generalized HorvitzThompson estimator of population size (Huggins
1989, 1991, Alho 1990). Using covariates in the models accounts for observable
heterogeneity in the detection probability of individual birds (Pollock 2002).
Modeling detection distance and other covariates requires conditioning the
probability of detecting a bird on availability (as before), and on the bird’s detection
distance from the observer. The probability of detection given availability for an
individual i, by observer j, can be represented as a function of detection distance as,
log ( p )= α + β δ (r ) ,
e
dji
j
j
i
(5)
where αj is the intercept, βj is the slope, and δ(ri) is a function of the detection distance
(such as detection distance squared). The detection distance function allows for four
additional models:
Model M d0 : equal intercept and slope terms between observers.
Model M*d
0 : equal intercept between observers but different slope.
Model M dobs : unequal intercept between observers but similar slope.
Model M*d
obs : unequal intercept and slope between observers.
�23
When covariates are included in the model it is necessary to use the generalized
HorvitzThompson (Horvitz and Thompson 1952) estimator of population size (Huggins,
1989, 1991, Alho 1990) instead of equation 1:
1
Nˆ = ∑ pˆ
n
i =1
;
(6)
i
where n is the number of birds detected and pi is the detection probability of an individual
bird. Program MARK provides the HorvitzThompson estimate of population size as a
‘derived parameter,’ when using the ‘Huggins closed captures’ data type.
Four or more independent observer models.—When four or more sampling
periods (in our case observers) are used in closed capturerecapture experiments, there
are conceptually eight models available for analysis (Otis et al. 1978, Pollock et al. 1990,
Williams et al. 2002). Only four of these models are reasonable models for independent
observer point count data:
M0
Equal capture probability.
Mobs
Observer variation in capture probability.
Mh
Individual capture heterogeneity.
Mobs,h Observer variation and individual capture heterogeneity.
Assumption four of the two independentobserver method is no longer necessary because
individual heterogeneity can be modeled with data from four or more observers. Using 3
observers may provide more precise estimates than two observers but it does not provide
the data necessary to fit the heterogeneity models.
�24
Behavioral response models are probably not relevant to analysis of independentobserver point count data. A behavioral response is a response by an individual to
capture (in our case, detection) that makes them either more or less likely to be captured
after first capture. Because observations on point counts are done simultaneously and
independently we assume that detections by one observer do not affect detections by the
other observers. This reduces the number of capturerecapture models that are relevant
for analysis of point count data to four, excluding covariate models.
The models available for analysis are based on assumptions about the sources of
variability in the data. Model M0 has the most restrictive assumptions by requiring that
the probability of detection is the same for all individuals in the population and that there
are no differences between observers in ability to detect individuals. Model Mobs is less
restrictive in that it allows for differences between observers but still requires that all
individuals in the population have equal detection probabilities for a given observer.
Probably the most important models in this group are those that incorporate
individual heterogeneity in capture probabilities. Individual heterogeneity indicates that
each individual in the population has a unique capture (detection) probability. All of the
other models identify the source of variation (temporal, behavioral) and model this
process. Accounting for individual heterogeneity is important because ignoring it will
cause a negative bias in population estimates caused by a positive bias in capture
estimates (White et al. 1982, Johnson et al. 1986, Williams et al. 2002). Because it is not
possible to account for unobservable or uncontrollable sources of variation that affect
count data (Burnham 1981, Johnson 1995), models that incorporate these unobservable
sources are potentially very useful.
�25
Because it is possible to identify and model some sources of individual
heterogeneity it is important to classify sources of heterogeneity as either observable or
unobservable. Observable heterogeneity includes differences due to factors like sex or
age that can be accounted for in a statistical model by stratification (Johnson et al. 1986)
or other factors which can be accounted for with covariates. Detection distance is a
covariate that could cause observable heterogeneity. Incorporating observable individual
heterogeneity into the independent observer models using covariates with four or more
observers is identical to that described previously for the twoobserver situation using the
generalized HorvitzThompson estimator.
Unobservable heterogeneity has been attributed to age, social status, innate levels
of activity, physical condition, and genetic variation (ie. covariates that are unknown for
individual birds) (White et al. 1982, Johnson et al. 1986). Heterogeneity in point counts
results from individual differences in age, social status, and singing rates, or from site
specific differences such as landscape structure, vegetative cover and background noise
that affect the detection of auditory or visual cues. Model Mh represents heterogeneity in
the detection probabilities of individual birds but no observer differences. Model Mobs,h
represents both observer differences and heterogeneity in the detection probabilities of
individual birds.
Three estimators are available to estimate abundance in the presence of individual
heterogeneity; the Jackknife estimator for model Mh, Chao’s estimator for model Mh and
model Mobs,h, and the finite mixture estimators for models Mh and Mobs,h. The Jackknife
estimator is based on linear functions of the capture frequencies (Burnham and Overton
1978, 1979) and is not a likelihood based approach. The Chao estimators are based on
�26
sample coverage (Chao et al. 1992, Chao and lee 1992) and also are not likelihood based.
Both the Jackknife estimator and the Chao estimators can be run using program
CAPTURE but not in program MARK. Program CAPTURE uses a multivariate
discriminate function procedure to select the appropriate model for the data (Otis et al.
1978), but covariate models cannot be included in the suite of models. An alternative
likelihood based approach to the heterogeneity models are finite mixture models of
heterogeneity (Norris and Pollock 1996, Pledger 2000), which can be parameterized in
program MARK and are likelihood based approaches. By employing Akaike’s
Information Criterion (AIC) model selection techniques (Burnham and Anderson 2002)
these approaches are applicable to an entire suite of models including the observable
heterogeneity and covariate models.
Analyses
Comparison of dependent and independent observer models.—Seber (1982)
compared the twosample removal method with the LincolnPetersen method (twoobserver Model Mt) and demonstrated the greater efficiency (smaller variance) of the
LincolnPetersen method, especially for low capture probabilities. We performed a
similar comparison between the dependent and independent observer methods using
simulations over a range of detection probabilities (0.6, 0.7, 0.8, and 0.9) and true
population sizes of 20, 50, 100, and 200. For each detection probability and population
size, 1,000 data sets were generated and analyzed with both methods. The standard error
(SE) was then calculated from the 1,000 estimated population sizes and the ratio of the
SE of the dependent observer method to the independent observer method was
�27
determined. The same set of simulations was also run for the scenario when the
assumption of independence was violated. This was done by allowing 90% or 80% of
the independent observer data to be independent but assuming for the remaining data that
if observer one detected a bird it was always detected by observer two. This assumption
violation caused a bias in the population estimate so we used meansquared error (MSE)
(MSE = Variance + Bias2) instead of the standard error for comparison purposes. Note
that for the dependent observer method there was no bias in the simulated data and thus
MSE is equivalent to the variance. These simulations were used to determine the ratio of
MSE1/2 of the dependent observer method to the independent observer method as a
measure of efficiency.
Field data.—Examples are provided for the independentobserver methods
presented in this paper using data collected in Great Smoky Mountains National Park
during June of 1999 (Simons unpublished data). Counts were conducted at 70 points
along low use hiking trails. All observers had been conducting point counts on the study
site for the previous six weeks during which their identification and distance estimation
skills were periodically validated. Before each count, observers estimated a 50m radius
circle by spotting landmarks using a laser range finder and began the count immediately
thereafter. Observers conducted variable circular plot 3 minute point counts (Reynolds et
al. 1980) between dawn and 10:15 am and only in good weather (no rain or excessive
wind) consistent with the recommendations for point count methodology detailed by
Ralph et al. (1995b). During each 3 minute count, observers recorded the number of
breeding pairs of each bird species seen or heard. Observers were separated and
�28
instructed to not look at the other observers during the count. At each point observers
recorded bird detections in all directions on an unlimited radius plot. Points were spaced
a minimum of 250 m apart and the location and movement of all individual birds detected
were mapped in order to avoid double counting. Following each count observers
compared their data sheets to determine the total number of birds detected and which
birds were seen in common.
We used the first two observers from the full fourobserver data set to construct a
twoobserver data set. Twoobserver analyses were also done for the other two observers
to confirm consistency but these will not be presented. For illustrative purposes we will
present analyses for three species; Ovenbird (Seiurus aurocpillus), Tufted Titmouse
(Parus bicolor), and Scarlet Tanager (Piranga olivacea), and two species complexes
(Warblers and Vireos see appendix for list of species) using both twoobserver and fourobserver methods.
Two independent observer examples.—The five twoobserver data sets were
analyzed using program MARK (White and Burnham 1999) with the ‘Huggins closed
captures’ data type. The a priori set of candidate models is:
1. equal detectability between observers, Model M 0
2. unequal detectability between observers, Model M obs
3. equal detectability between observers with distance function having the same
slope for both observers, Model M d0
4. equal detectability between observers at the point (equal intercept) with
distance function having a different slope for each observer, Model M∗0d
�29
5. unequal detectability between observers with distance function having the
same slope for both observers, Model M dobs
6. unequal intercept between observers with distance function having a different
d
slope for each observer, Model M∗obs
The most parsimonious models were selected using secondorder Aikaike’s Information
Criterion (AICc), an informationtheoretic approach with an adjustment for small sample
size (Burnham and Anderson 2002). Data were truncated following the recommendation
of Buckland et al. (1993) by discarding 10% of the largest detection distances for each
species.
Four independent observer examples.—Because this is a new approach and using
four observers at a point is not typical, we present this separately from the twoobserver
example. The fourobserver data set also allows for heterogeneity models of point count
data, which have not previously been investigated. We start by examining models
without heterogeneity and ones that model heterogeneity with detection distance
covariates. Comparisons to the twoobserver cases and examination of expected capture
histories from this analysis provide added detail about the presence of heterogeneity.
Using the six models discussed previously for the twoobserver examples, we
estimate detection probabilities and population sizes using the fourobserver data for the
same three species and two species complexes. These models either do not allow for
heterogeneity or model it only as a function of detection distance. Fourobserver models
were parameterized and run using program MARK with model selection based on AICc.
For a given species we then compared the population estimates from twoobserver and
�30
fourobserver data sets. Differences were interpreted as reflecting heterogeneity in the
data.
Examining the differences between the observed capture history and the predicted
capture history from a model based on estimates of assumed homogeneous parameters is
another method of detecting unobservable heterogeneity (Johnson et al. 1986). For
example, one could determine if the observed number of X1100 records was similar to that
predicted based on the selected model. The model used to generate predicted capture
histories was model M 0 for all data sets. We determined the expected value of the count
for the fifteen possible capture histories from all five data sets using the total number of
observations and the estimated probability of detection. We then compared the observed
counts for the fifteen possible capture histories using a Chisquare goodness of fit test
with fourteen degrees of freedom. If differences between observed and predicted counts
were evident we then looked at the standardized residuals to determine which capture
histories were different.
Because heterogeneity was evident, we ran the Jackknife and Chao version of the
capturerecapture heterogeneity model (Model Mh) using program CAPTURE (these
estimators cannot be obtained in program MARK) and we ran the finite mixture
heterogeneity models in program MARK. In all cases the three methods gave similar
estimates. The estimates from the twopoint mixture models are reported because model
selection is based on AIC criteria from the entire suite of models including those with
detection distance covariates. For consistency we denote the model describing
heterogeneity in detection probability as model M h and the model with observer
differences and heterogeneity as model M obs,h .
�31
Heterogeneity simulations.—We hypothesized that heterogeneity might be caused
by differences in calling/singing rates so we ran simulations to determine the effect of
various levels of heterogeneity on the other available models. We predicted that birds
with loud and frequent songs would have higher detection probabilities than birds with
quiet or infrequent songs. To examine this, we simulated heterogeneous data with a
threepoint mixture model for populations of 100 and 200 birds. We used 3 levels of
heterogeneity, low, moderate, and high, to give some understanding of the effect of
heterogeneity on the other candidate models. All simulations had 20% of the population
with probability of detection 1.0 and 60% of the population with detection probability
0.75. The remaining 20% of the population had detection probabilities of 0.5, 0.3, or 0.1
simulating low, moderate, or high heterogeneity, respectively. Using these
parameterizations 1,000 fourobserver data sets are generated and analyzed with
model M 0 and model M h . An additional 1,000 twoobserver data sets are generated with
this parameterization and analyzed with model M 0 . A single observer count is also
obtained for each simulation of the twoobserver data set to represent an index count.
Additional simulations were run using a twopoint mixture model for a population
of 200 birds and parameterized so that the expected capture histories approximated the
observer capture history for the “Warbler” group. The parameterizations that achieved
this were 100 birds with detection probability of 0.9 and 100 birds with detection
probability either 0.1 or 0.2. Again 1,000 simulations were run and analyzed for the fourobserver, twoobserver, and oneobserver counts. This was done to examine the
�32
performance of the nonheterogeneity models when heterogeneity is equivalent to that
evident in actual field data.
Results
Comparison of dependent and independent observer models.—Our simulations
showed that, when model assumptions were met, the independent observer method was
always more efficient than the dependent observer approach (Table 1). Detection
probability could not be estimated with the dependent observer method when detection
probabilities were low, and/or population sizes were small. Differences between the
methods decreased with increasing detection probability because when detection
probabilities are high almost all birds are detected by both methods. When the
assumption of independence was violated the efficiency of a particular method was a
function of the level of dependence, the detection probability, and the true population
size.
Two independent observer examples.—The total number of birds detected for the
twoobserver data sets (after 10% truncation of observations with the largest detection
distances) ranged from 31 (Titmouse) to 132 (Warbler) for the twoobserver data sets.
AICc scores selected either model M 0 or model M d from all data sets (Table 2). When
model M 0 was selected, model M d was always a reasonable alternative model based on
differences in AICc (∆AICc < 2). In contrast, when model M d was selected, model M 0
was not necessarily a reasonable alternative model (see for example the Ovenbird, Table
2). Models incorporating observer differences were never selected as the most
�33
parsimonious model, although for some data sets these models may provide reasonable
alternatives because ∆AICc is small.
Model M 0 detection probabilities for individual observers ranged from 0.77
(Tanager and Vireo) to 0.9 (Ovenbird) (Table 3). There were no differences in
population estimates for a given data set between model M 0 and model M d , but the
standard errors were different.
Four independent observer examples.—Excluding the heterogeneity models,
model M d0 was selected as the most parsimonious model of the fourobserver data sets
for all analyses except for the Tanager data set, for which model M 0 was selected.
Model M d0 was a reasonable alternative to model M0 for this case. All models not
incorporating individual heterogeneity gave similar population estimates (Table 4).
Species detection probabilities based on model M 0 (Table 5) were consistently lower for
the fourobserver analysis than for the twoobserver analysis (Table 3). Estimated
population size was > 10% higher for the fourobserver data sets (Table 5) than for the
corresponding twoobserver data sets (Table 3). It is also worth noting that the raw count
for the fourobserver data (Table 5) was > the twoobserver abundance estimates (Table
3), indicating that individual heterogeneity is present in the data for all species.
Observed capture histories were significantly different than expected under a null
hypothesis of homogeneity in detection probabilities for all five data sets (χ2≥ 34.8, p ≤
0.005, df=14). Examining the standardized residuals revealed that the number of
observations was greater than expected when only one observer detected a bird or when
�34
all four observers detected a bird, again, indicating that individual heterogeneity is
present in the data.
Model M h (Jackknife estimator) was selected as the most parsimonious model for
all five data sets using program CAPTURE. A twopoint mixture model for model M h
was selected as the most parsimonious model for all data sets except the Ovenbird data
set using program MARK (Table 4).
The estimate for the finite mixture for all five species was close to 0.5, indicating
that approximately half the population was in each detection group (Table 5). One group
of the finite mixture was always highly detectable (probability of detection > 0.90), while
the other group was generally much harder to detect (probability of detection ≤ 0.30) with
the exception of the Ovenbird (probability of detection = 0.51). Population estimates
(Table 5) from model M h (twopoint mixture) ranged from 15% (Titmouse) to 26%
(Warbler) higher than the estimates from the selected fourobserver model without
heterogeneity. The selected model for the Ovenbird using distance to model observable
heterogeneity gave the same population estimate as the twopoint mixture model, which
was similar to estimates from models that did not account for heterogeneity.
The difference in model selection for the Ovenbird reflects the lack of covariate
models in program CAPTURE. Model selection from CAPTURE ranked model M h first
and model M 0 as the next best. If we ignore the distance covariate models we see that
this ordering of models is the same using the AIC criterion in program MARK (Table 4).
It is encouraging that the two different methods of model selection and the three types of
heterogeneity estimators are in close agreement for these data.
�35
With the exception of the Ovenbird, comparison of abundance estimates from the
heterogeneity model to those obtained from models without heterogeneity demonstrates
the large (≥ 15%) negative bias in the estimates when heterogeneity is ignored. The
negative bias associated with the abundance estimate for the Ovenbird was only 1%. It
appears that the heterogeneity in Ovenbird detectability can be accounted for by using
distance covariates. Heterogeneity in detection probabilities for Ovenbirds had minimal
effects on abundance estimates, which is expected for this species because of its high
singing rates and loud vocalization. The detectability of Ovenbirds based on model M 0
was 0.82, while detectabilities of all other birds based on this same model were ≤ 0.72.
Heterogeneity simulations.—These simulations suggest that the estimation
accuracy of the available models is highest for the fourobserver model M h , followed by
the fourobserver model M 0 , the twoobserver model M 0 , and the oneobserver count
statistic (Table 6). True population size had no effect on the relative accuracy of various
models. Both fourobserver models were reasonably accurate (within 5% of the true
value), except under the most extreme heterogeneity when both models substantially
underestimated the true population. The twoobserver model M 0 was reasonably
accurate (within 4% of the true value) for the least heterogeneous data, but it did not
perform well as heterogeneity became more extreme. The oneobserver count statistic
underestimated the true population by ≥ 25% for all levels of heterogeneity.
Individual heterogeneity with a twopoint mixture distribution with extremely
different probabilities of detection between the two groups demonstrates greater
differences in the accuracy of the potential models (Table 6). For these simulations only
model M h performed well: all other estimators underestimated the true value by ≥ 20%.
�36
The estimates given by each estimator are similar to the estimates given by the
corresponding estimator for the Warbler data, which indicates how severe the
heterogeneity in detection probabilities may be in the actual data.
Discussion
We have presented an independent observer point count method as an alternative
to the primarysecondary observer approach of Nichols et al. (2000) and shown that the
method provides reasonable estimates of bird abundance. This is not surprising given the
similarities between the two methods and the additional information provided by
additional observers. We have also demonstrated that when model assumptions are met,
the method provides relatively more efficient estimates. The method also allows for
greater model flexibility and complexity. Examples include the incorporation of
individual covariates and models of individual heterogeneity when at least four observers
are used. Theoretically, primarysecondary observer approaches could also account for
individual heterogeneity. The method would require four observers with the added
complexity that the third observer would record birds that the first two observers missed,
and the fourth observer would record birds that the first three observers missed. While
possible we feel that this approach would not be logistically feasible to accomplish in the
field because the level of communication required between observers would become
unmanageable.
Incorporating detection distance into the independent observer approach has the
added benefit that a detection probability of one is not required at detection distance zero.
This is a restrictive assumption with the distance sampling method (Buckland et al. 1993)
and may not be reasonable for all surveys, especially when birds occur high in a forest
�37
canopy. It is important to consider other characteristics that may affect the detection
process, such as singing rate (Wilson and Bart 1985, McShea and Rappole 1997).
McShea and Rappole (1997) found that singing rates were lower for birds closer to an
observer. If singing rate does affect the probability of detection, then the distance
sampling assumption of a monotonic decline in detection probability would also be
violated.
Field application.—Nichols et al. (2000) thoroughly critiqued their primarysecondary observer approach and discussed potential problems with implementing their
method in the field. Given the similarities between our methods we will attempt to
recapitulate their main points.
The approach assumes that the population within the area being surveyed is
closed to movement into or out of the survey area during the count, and that birds are not
double counted. These issues are common to all point counts and they can be thought of
as a function of count duration. Double counting arises when the same bird is observed
at more than one location and is counted as two or more individuals. Reducing the count
duration limits the probability of birds moving into, out of, or within the survey area
during the count. Unfortunately reducing count duration reduces the total number of
detections on the count. On our study sites, detections are generally auditory and thus, if
birds have a low singing rate, short duration counts can significantly limit the proportion
of birds that are available to count. Optimal count durations should be long enough to
ensure that all birds are available for detection (sing at least once during the count) and
short enough to minimize movement.
�38
The independent observer approach requires that observations are independent
among observers. The primarysecondary approach of Nichols et al. (2000) required only
that the observations by the primary observer be independent of detections made by the
secondary observer. Many of the issues involved with independence between observers
are similar between the 2 methods. Independence may be violated if an observer obtains
cues from other observers. This could occur if oneobserver is writing down an
observation (Nichols et al. 2000), estimating a distance, or moving in a manner that
would draw the attention of other observers. Nichols et al. (2000) suggest that violation
of the independence assumption is most likely when there are few birds at a point or
when most observations are visual. In other words, it is harder to obtain cues from other
observers when detections are auditory. Our data are based almost exclusively (>95%)
on auditory detections.
Nichols et al. (2000) viewed differences between observers in their ability to
detect birds at different distances as the most serious source of error in their method. We
incorporate distance as covariates in the candidate models to account for this problem.
Note that the primarysecondary method can also incorporate detection distance
covariates, so this does not indicate that one method is better than the other. Because
these covariate models, which use all birds detected, are available, we do not recommend
using fixed radius plots. Modeling distance as a covariate using all detections provides
more information, and presumably better estimates than ignoring detections outside a
fixed radius. We do recommend 10% truncation of the farthest detections to remove
outliers, as recommended by Buckland et al. (1993). We must emphasize though that
�39
collecting distance data is essential to point count methods because even if it is not used
to estimate detection probabilities, it does provide an estimate of the sample area.
A final source of error results from the process of matching observer’s
observations. After each point count the observers must agree on which birds were
detected in common, and which birds were unique to an observer. This method requires
observers to determine which birds were seen by each observer. If surveys are conducted
during the breeding season when territories are fixed and birds are not in large flocks,
then matching errors should be minimal. If many birds of one species are present at a
point then it may be difficult to determine which birds are seen in common.
Both visual and auditory observations occur during point counts, and their
detection probabilities are usually not equal, especially if covariates such as distance are
used. Detection probabilities are also likely to vary among sex and age classes (Ralph et
al. 1995b). For example, during the breeding season auditory detections of adult
territorial males are likely to be much greater than auditory detections of females.
Determining the best point count study design will require careful consideration the
research objectives, and how species specific behavior and habitat characteristics might
affect detection probabilities.
Our examples were derived from data collected during the height of the breeding
season in Great Smokey Mountains National Park which is primarily composed of mature
deciduous forest habitats. Observations were primarily auditory detections of singing
males. Foraging females and nonterritorial males were observed on rare occasions. In
this context we recommend using only auditory detections of singing males for
estimating breeding bird abundance. Including visual observations or auditory detections
�40
of call notes that may include female and juvenile birds would overestimate breeding bird
abundance.
Example analyses.—Model selection for the twoobserver data sets suggests some
important characteristics of these data. First, we notice that models that incorporated
differences in detection probability between observers were not selected. This implies
that detection probability based on auditory cues was similar between two highly trained
observers. Models incorporating differences between observers were reasonable
alternative models for several species. This suggests that these models should be
included as candidates in most studies. These models may prove useful for species that
are hard to detect or on surveys where observer’s abilities vary more widely.
Incorporation of detection distance as a covariate also provided important improvements
to the model.
Model selection for the fourobserver data sets generally selected the same
models that were selected for the twoobserver data sets, excluding the heterogeneity
models. Again, estimates of detection probability and abundance were relatively precise
but accuracy was doubtful given the differences between the twoobserver and fourobserver results. The twoobserver abundance estimates were low and the detection
probability estimates were high compared to the fourobserver estimates, which was
consistent with the expected biases associated with individual animal heterogeneity.
Estimates based solely on a twoobserver approach may appear reasonable (as in our
examples) but caution is advised because no assessment of potential bias can be made
with this method.
�41
Heterogeneity models were generally selected as the “best” models. Potential
causes of heterogeneity in detection probabilities for point count surveys are; violation of
the independence assumption, combining types of detection (e.g. auditory and visual),
estimating detection probability for a group of species, and individual variation in singing
rates. Violation of the independence assumption caused by observer’s cueing off of each
other would affect the observed capture histories. This violation would result in more
capture histories where the majority of observers detected an individual bird and fewer
histories where only oneobserver detected an individual bird than expected, under
models of equal detection probability. This would result in a positive bias in detection
probabilities and an underestimation of abundance. Comparisons of observed to expected
capture histories, indicate that violations of the independence assumption are not the
main factor associated with heterogeneity in these data.
Combining multiple types of detections is a potential source of individual
heterogeneity that should also be considered carefully when studies are designed. For
example, the detection probabilities of songs may be very different from the detection
probabilities of calls or visual observations. Combined detection types may generate
results similar to the simulations presented for mixture models where singing birds had
high detection probabilities, calling birds had intermediate detection probabilities, and
visual observations had low detection probabilities.
Variability in the singing rates of individual birds is another potential source of
heterogeneity in detection probabilities (Wilson and Bart 1985, McShea and Rappole
1997). A bird that sings frequently is more likely to be detected than one that sings only
once during a survey period. Differences in singing rates may be associated with
�42
differences in pairing status and/or nest stage. For example, unpaired males may sing
more frequently than paired males and paired males that have not nested are likely to sing
at higher rates than a paired male that is incubating or caring for a brood (Wasserman
1977, Krebs et al. 1981, Lein 1981, Wilson and Bart 1985). This presents a situation
where the use of finite mixture models is biologically reasonable because individual
heterogeneity can be associated with biological groups based on an individual’s
reproductive stage.
So far we have described sources of heterogeneity and suggested possible ways to
account for it, but there are other unaccountable sources of heterogeneity. These sources
could arise from something as simple as whether the bird is facing toward or away from
the observer, or more complex interactions between sound attenuation of the auditory cue
and the surrounding landscape. Differences in song characteristics, vegetative
cover/density, topographic features, and background noise all affect the intensity and
attenuation of auditory cues (Richards 1981). Presence of conspecific territorial males,
competitors, predators, mating status, and nesting status could all affect the singing
behavior of individual birds. Stratification and careful survey design can provide partial
control of some of these sources of heterogeneity, but they will always be present in
survey data.
Recommendations
We believe that the independentobserver approach should be strongly considered
among the suite of statistical methods available for pointcount studies. The benefits of
this approach include; the number of candidate models available for analysis, the ease of
�43
implementation with existing software, the ability to run both general and covariate
models within program MARK, and the application of information theoretic approaches
to model selection. In particular, the incorporation of detection distance, a relatively well
studied methodology, as a covariate within the independentobserver framework may
provide more precise abundance estimates. We are encouraged that our three different
heterogeneity models gave similar results, suggesting that it may be useful to model
unobservable individual heterogeneity. Because the finitemixture models are likelihood
based models and can be compared to other models using AIC model selection, we
recommend their application when biologically meaningful phenomena that influence
detection probability, such as changes in singing rates during the breeding season, are
known.
We recommend conducting pilot studies involving multipleobservers collecting
multiple types of data prior to any implementation of a large scale survey. Such a study
will allow an assessment of the various sources of individual heterogeneity and selection
of the most efficient methodology. Pilot study results can provide guidance for the
allocation of singleobserver distance sampling, doubleobserver sampling, and/or more
than threeobserver sampling based on factors determined to be affecting the detection
process.
Finally, we feel that models combining the independentobserver and the time of
detection approach (Farnsworth et al. 2002) would be a useful extension of this work.
Combined models would allow separation of the probability of availability from the
probability of detection given availability. For species with high singing rates, this may
not provide much useful information because availability is often near one. For less
�44
vocal species this additional component of the detection process may have significant
effects on abundance estimates.
�45
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Table 1: Comparison of the dependentobserver approach to the independentobserver
approach using simulations of 1,000 data sets for each population size, detectability and
method. For the unbiased scenario, the ratio of the SE of the dependentobserver method
to the SE of the independentobserver method is presented. The biased scenario
represents 10% and 20% of the observations in the independentobserver data being
dependent on the first observer and compares the ratio of the SE of the dependentobserver method to the MSE1/2 for the independent observer method. NE represents
cases where the dependent observer estimate was frequently not estimable.
Detection Probability
True N
0.6
0.7
0.8
0.9
20
NE
NE
3.21
1.43
50
4.89
2.51
1.37
1.26
100
1.92
1.56
1.21
1.10
200
1.63
1.42
1.28
1.14
20
NE
NE
2.38
0.74
10%
50
3.58
1.77
0.90
0.68
Bias
100
1.43
1.03
0.80
0.63
200
1.10
0.87
0.69
0.54
20
NE
NE
1.93
0.63
20%
50
2.98
1.34
0.71
0.51
Bias
100
1.02
0.75
0.54
0.43
200
0.76
0.56
0.45
0.36
Unbiased
�50
Table 2: Model selection for the twoindependent observer examples giving the ∆AICc
values for all 6 candidate models. The smaller ∆AICc values indicate a more
parsimonious model with 0 indicating the selected model. AICc weights in parentheses.
Models
Date Set
Ovenbird
M0
M obs
5.93 (0.03) 7.91 (0.01)
M d0
0 (0.65)
M∗0d
M dobs
d
M∗obs
3.26 (0.13) 3.25 (0.13) 5.35 (0.05)
Tanager
0 (0.40)
2.03 (0.15) 1.06 (0.24) 2.76 (0.10) 3.15 (0.08) 4.80 (0.04)
Titmouse
0 (0.33)
2.14 (0.11) 0.10 (0.31) 2.31 (0.10) 2.31 (0.10) 4.60 (0.03)
Vireo
0 (0.33)
1.77 (0.14) 0.63 (0.24) 1.63 (0.15) 2.42 (0.01) 3.60 (0.05)
Warbler
0 (0.31)
1.31 (0.16) 0.53 (0.24) 2.17 (0.11) 1.86 (0.12) 3.26 (0.06)
�51
Table 3: Abundance estimates (N) for the twoindependent observer examples. Birds
detected are the totals between the two observers. Model M0 was selected as the most
parsimonious for all data sets except the Ovenbird.
Data set
Birds
Model M 0
N̂
N̂
Detected
Detectability
Model M 0
Model M d0
Ovenbird
72
0.90 (0.027)
73 (0.94)
73 (1.43)
Tanager
45
0.77 (0.055)
48 (2.09)
48 (2.28)
Titmouse
31
0.85 (0.052)
32 (0.98)
32 (1.22)
Vireo
89
0.77 (0.039)
94 (2.86)
94 (3.03)
Warbler
132
0.85 (0.026)
135 (2.10)
135 (2.19)
�52
Table 4: Model selection for the fourindependent observer examples giving the ∆AICc
values for all 8 candidate models. Model M 2h and model M obs,2h are based on 2 point
mixture models of heterogeneity. The smaller ∆AICc values indicate a more
parsimonious model with 0 indicating the selected model. AICc weights are in
parentheses.
Models
Data Set
M0
M obs
M d0
M∗0d
M dobs
d
M∗obs
M 2h
M obs,2h
Ovenbird
48.5
53.5
0.0
8.4
7.4
12.5
10.3
37.7
(0.0)
(0.0)
(0.95)
(0.01)
(0.02)
(0.0)
(0.01)
(0.0)
23.4
29.4
24.1
29.0
30.1
35.2
0.0
22.8
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(1.0)
(0.0)
28.9
34.1
26.0
30.3
31.2
35.8
0.0
6.2
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.96)
(0.04)
71.2
77.2
60.2
65.6
66.3
71.7
0.0
36.4
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(1.0)
(0.0)
99.0
102.4
83.9
88.5
87.3
92.1
0.0
4.2
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.0)
(0.89)
(0.11)
Tanager
Titmouse
Vireo
Warbler
�53
Table 5: Abundance estimates (N) for the fourindependent observer examples. Birds
detected are the totals among the four observers. Detection probabilities are given by p
for model M 0 and pgroup1 and pgroup2 for model M 2h . The proportion of the population in
group 1 is given by pr(group1). Standard errors are in parentheses.
Data Set
Ovenbird*
Tanager
Titmouse
Vireo
Warbler
Birds
Model M 0
Model M 2h
Detected
p
N
pr(group1)
pgroup1
pgroup2
N
81
0.82
81
0.41
0.51
1.0
82
(0.022)
(0.305)
(0.059)
(0.060)
(≈ 0.0)
(0.889)
0.66
56
0.54
0.30
0.92
63
(0.033)
(0.894)
(0.086)
(0.108)
(0.047)
(6.32)
0.70
39
0.53
0.28
0.97
45
(0.038)
(0.592)
(0.090)
(0.111)
(0.032)
(5.56)
0.67
112
0.55
0.25
0.93
134
(0.023)
(1.243)
(0.055)
(0.067)
(0.026)
(12.00)
0.72
155
0.47
0.18
0.91
196
(0.019)
(1.042)
(0.061)
(0.062)
(0.020)
(22.23)
55
39
111
154
*Selected model was model M d0 and the estimated population size was 82 (SE = 0.889).
�54
Table 6: Abundance estimates (averages over 1000 simulations) for fourobserver and
twoobserver methods and a single observer count from simulated heterogeneous data
from three and twopoint mixture distributions. For the threepoint mixture distribution
20% of the population had detection probability 1, 60% had detection probability 0.75,
and the remaining 20% had three different levels (low = 0.5, moderate = 0.3, and high =
0.1) of detection probabilities. For the twopoint mixture distribution half the population
had high detection probability (p = 0.9) and the other half low detection probability (p =
0.1 or p = 0.2), which gave capture histories similar to those observed in the “Warbler”
data set.
True
Level of
Population
Heterogeneity
4 Observer Estimate
2 Observer
1 Observer
Estimate
Count
Mh
M0
M0
N
3point MIXTURE
100
Low
100 (1.89)
98 (1.26)
96 (3.37)
75 (4.04)
100
Moderate
99 (3.52)
95 (2.00)
91 (3.49)
71 (3.89)
100
High
90 (3.99)
87 (2.21)
83 (3.11)
67 (3.63)
200
Low
200 (2.29)
197 (1.59)
192 (4.65)
150 (5.61)
200
Moderate
197 (4.51)
190 (2.83)
182 (5.16)
142 (5.64)
200
High
179 (5.20)
174 (3.05)
167 (4.43)
134 (5.14)
2point MIXTURE
200
Highest (p = 0.1)
188 (35.1)
135 (4.82)
122 (5.12)
100 (4.32)
200
High (p = 0.2)
205 (19.7)
160 (5.20)
142 (6.00)
110 (4.74)
�55
Appendix 1: Common and scientific names for the “Warbler” and “Vireo” species
groups used for analysis.
Vireo Group
Redeyed Vireo
Solitary Vireo
Vireo olivaceus
Vireo solitarius
Warbler Group
Blackthroated Blue Warbler
Blackandwhite Warbler
Blackthroated Green Warbler
Hooded Warbler
Wormeating Warbler
Yellowthroated Warbler
Chestnutsided Warbler
Canada Warbler
Dendroica caerulescens
Mniotilta varia
Dendroica virens
Wilsonia citrina
Helmitheros vermivorus
Dendroica dominica
Dendroica pensylvanica
Wilsonia canadensis
�Chapter 3
TIME OF DETECTION METHOD FOR ESTIMATING
ABUNDANCE
FROM POINT COUNT SURVEYS
�57
TIME OF DETECTION METHOD FOR ESTIMATING ABUNDANCE
FROM POINT COUNT SURVEYS
Mathew W. Alldredge, Biomathematics and Zoology, North Carolina State University,
Raleigh NC 27695.
Kenneth H. Pollock, Zoology, Biomathematics, and Statistics, North Carolina State
University, Raleigh, NC 27695.
Theodore R. Simons, USGS Cooperative Fish and Wildlife Research Unit, Dept. of
Zoology, North Carolina State University, Raleigh, NC 27695.
Jaime Collazo, USGS Cooperative Fish and Wildlife Research Unit, Dept. of Zoology,
North Carolina State University, Raleigh, NC 27695.
Abstract.—Point count surveys are often used to collect data on the abundance and
distribution of birds. These data are generally used as an index or a measure of relative
abundance. Valid spatial or temporal comparisons of these measures require assuming a
constant detection process. Eliminating this restrictive assumption requires information
about detection probabilities that allow for estimation of the detection process. The time
of removal approach for estimating detection probabilities is generalized to the time of
detection approach, which uses the complete detection history of an individual and has
fewer assumptions. We apply this model to point count surveys where detections are
aural and singing rates of birds are important components of the detection process. Our
model accounts for both availability bias and detection bias by modeling the combined
probability that a bird sings during the count and that it is detected given that it sings.
The model requires dividing the count into several intervals and recording detections of
individual birds in each interval. We develop maximum likelihood estimators for three
different forms of the detection process and provide a full suite of models based on
capturerecapture models, including covariate models. We present two examples of this
method; one for four species of songbirds surveyed in Great Smokey Mountains National
Park using three unequal time intervals, and one for the Pearlyeyed Thrasher
�58
(Margarops fuscatus) surveyed in Puerto Rico using four equal time intervals. Models
incorporating individual heterogeneity were selected for all data sets using informationtheoretic model selection techniques. Detection probabilities varied among time intervals
of the count, indicating a possible behavioral response to the observer. We recommend
applying this method to surveys with four or more equal intervals so that fewer
assumptions are necessary and analyses can take full advantage of standard capturerecapture software.
Introduction
Point count surveys are routinely used to determine animal abundance,
particularly breeding birds (Thompson 2002), but statistically valid methods for
collecting and analyzing such data are rarely used (Rosenstock et al. 2002). Most studies
do not correct for the detection process, relying instead on the raw count data as an index
to abundance (Ralph et al. 1995, Nichols et al. 2000). In general point count surveys
involve counting all individuals seen or heard at a set of points under standardized
conditions. For example, the North American Breeding Bird Survey (BBS), standardizes
the duration of counts, length of survey routes, distance between points, weather
conditions, time of year, etc. (Robbins et al. 1986, Sauer et al. 1997), but it does not
estimate detection probabilities directly. Current point count methods that account for
the detection process are; the distance sampling or variable circular plot method
(Reynolds et al. 1980, Buckland et al. 1993), multipleobserver methods (Nichols et al.
2000, chapter 2), the time of detection method (Farnsworth et al. 2002), double sampling
methods (Bart and Earnst 2002) and repeated counts (Royle and Nichols 2003). All of
�59
these methods are based on the general form for estimating abundance of species i as
(Lancia et al. 1994, Williams et al. 2002)
C
N̂ = p̂
i
i
(1)
i
where Ci is the observed count and p̂i is the estimated probability of detection for species
i.
The probability of detection p̂i is actually the product of two probabilities: the
probability that an individual is available for detection p̂ai and the probability that an
individual is detected given that it is available p̂ di . Ignoring these probabilities gives rise
to availability bias and perception bias (Marsh and Sinclair 1989). When point count
surveys are conducted in dense habitat, where most detections are auditory, availability is
the probability that a bird occurring within the survey area sings during the count. Both
distance sampling and multipleobserver methods ignore the availability process,
assuming that all birds are available for detection during the count interval.
Farnsworth et al. (2002) presented a method based on the time of first detection
for estimating detection probabilities that estimated the total detection probability (ie. the
product of the availability probability and the probability of detection given availability).
This method is based on dividing a count into multiple intervals and recording the
interval in which an individual is first detected. In collecting data on the interval of first
detection data are in the form of a typical removal experiment (Seber 1982) where
animals are removed from the population upon capture, or in this case detection.
Removal experiments presume that the number of animals caught on subsequent
occasions will decline as a linear function of the population size, and they use the decline
�60
to estimate initial population size. The removal method for point counts is similar in
concept presuming that the number of new detections declines with time, and using the
decline to estimate population size. The removal method is generalized to allow for
individual heterogeneity of detection probabilities.
Model assumptions for this method are (Farnsworth et al. 2002):
1. There is no change in the population of birds within the detection radius
during the point count (closed population).
2. There is no doublecounting of individuals.
3. All members of one group are detected in the first interval (applies to the
heterogeneity model for three time periods).
4. All members of the other group that have not yet been detected have a
constant per minute probability of being detected.
5. If counts with limitedradius are used, observers accurately assign birds to
within or beyond the radius used.
Assumption 2 requires observers to keep track of individual birds (usually by
mapping) during the count to avoid double counting. This requires observers to decide
whether a singing birds are new or whether they have been detected previously.
Capturerecapture methods (Otis et al. 1978, Williams et al. 2002) provide a
flexible and efficient framework for analyzing these data. A capturerecapture modeling
approach uses both first detections and all subsequent detections of an individual to
estimate the probability of detection. Seber (1982) demonstrated that capturerecapture
models are more efficient (smaller variance) than removal models. However, if
�61
subsequent detections are not tracked accurately, errors in the capture history will bias the
abundance estimates.
Our objective is to present a time of detection approach using a capturerecapture
framework based on all detections of an individual as a more general alternative to the
removal approach presented by Farnsworth et al. (2002). We first discuss field methods
required to collect data suitable for this method. We then discuss the detection process
and three ways of modeling it. A discussion of the candidate models and the general
form of the time of detection model is then presented. Following this development of the
general model, we present a finitemixture model for individual heterogeneity and a
covariate model for observable heterogeneity. Methods of analysis, including available
software, are also discussed. We then illustrate our methods with an example for three
unequal interval point counts and another for four equal interval point counts.
Methods
Field methods.—The sampling situation for the models we are proposing is the
same as that described by Farnsworth et al. (2002). Multiple locations are selected for
point counts within the area of interest, and at each point a count is conducted for a
specified amount of time. Field conditions in which point counts are conducted should
follow a standardized approach in order to maximize detection probabilities and
minimize other sources of variability (Ralph et al. 1995, Sauer et al. 1997). Depending
on the diversity of habitats and size of the area of interest it may also be necessary to
stratify by habitat type (Nichols et al. 2002).
The method presented by Farnsworth et al. (2002) was developed to estimate
detection probabilities for a single observer when a point count is divided into three or
�62
more intervals of variable length. This was done to present a model that would work
with existing types of point count data. A common approach, based on the
recommendation of Ralph et al. (1995), records birds detected in the first 3 minutes of a
count, the next 2 minutes and the final 5 minutes. The approach was recommended to
permit comparisons among studies using single 3, 5, or 10 minute counting periods.
The full detection history data for a point count survey will consist of the number
of birds observed with each of the 2n1 possible detection histories. For example, if three
time periods are used, then seven detection histories are possible (xw: x111, x110, x101, x011,
x100, x010, and x001). Mapping detections using multicolored pens (Simons and Shriner
2000), where each color represents a time interval, is an effective method for tracking the
time interval of detections.
Defining the sample area of a survey is necessary for making both temporal and
spatial comparisons. One approach involves collecting count data from limited radius
plots, so that the actual sample area is known (Nichols et al. 2000). An alternative
approach is to use unlimited radius plots and estimate the detection distance for each
individual observed during the count. This detection distance information can then be
used to estimate the effective sample area, similar to the approach used in distance
sampling (Buckland et al. 1993). Detection distance information can also be used as a
covariate in models to account for differences in detection probabilities among
individuals.
Detection process.—Estimating the detection process is key to obtaining reliable
estimates of abundance (Nichols et al. 2000, Farnsworth et al. 2002, Thompson 2002).
The detection process consists of both the probability that an individual is available for
�63
detection and the probability that it is detected given that it is available for detection
(Marsh and Sinclair 1989). Our method estimates the product of these two components.
Most birds in forested environments are detected by their songs or calls. In this situation
the availability process depends on the singing rates of individual birds and is thus a
function of the duration of each interval used in the point count. We will present the
availability process as the probability that a bird gives an auditory cue during the count,
but the method could be applicable to other situations affecting availability.
One approach to model the detection process is by assuming that singing rates of
a bird follow a Poisson process and that the probability of detecting all songs of a species
is equal. Under these assumptions the instantaneous detection rate (φi) or the “Poisson
detectability coefficient”, is used to model the probability that a bird sings during a
specified time period. The probability of detecting an individual in time interval i of
length ti using an instantaneous rates formulation (Seber 1982, pg. 3 and 296) is then,
p =1 − e ϕ it i
−
i
(2)
This formulation is consistent with that typically used for removal experiments where φi
corresponds to the “Poisson catchability coefficient” and ti corresponds to the effort on
the ith occasion (Otis et al. 1978, Seber 1982 pg 296). Farnsworth et al. (2002) used an
alternative formulation, the discrete rate formulation, which assumes a constant per
minute detection rate (γi) so that the probability of detecting an individual in time interval
i of length ti was given by
p =γt
i
i
i
(3)
The interval length ti that occurs in both equations 2 and 3 has important
consequences for the construction of candidate models and data analysis. Clearly if the
�64
instantaneous detection rate or per minute detection rate differs between time intervals
then the probability of detecting an individual during a given interval will be different for
each interval. Note that there is insufficient information in the data to model this
difference when using the removal approach. Equations 2 and 3 are particularly
important when the instantaneous or discrete detection rates are constant for the duration
of the count. If the time intervals are different then the probability of detecting an
individual during an interval will be different for each interval even when the detection
rates are constant. Using equations 2 or 3 can allow for unequal time intervals while
requiring only a single parameter for constant detection rate models. In contrast, if the
probability of detection is estimated for each interval directly, then a separate parameter
is required for the detection probability of each interval. When point counts consist of
equal intervals it is not necessary to use equations 2 or 3 because interval detection
probabilities are lo longer a function of the interval length and interval detection
probabilities are estimated directly.
Approach and candidate models.—A more general approach to the removal
models presented by Farnsworth et al. (2002) uses a capturerecapture framework which
tracks all time intervals in which individual birds are detected. The full time of detection
approach provides a complete set of models, of which the removal model is a special
case, for estimating detection probabilities. There are 8 general capturerecapture models
(Otis et al. 1978, White et al. 1982, Pollock et al. 1990, Borchers et al. 2002), all of which
may be applicable to the time of detection approach. Here we present each capturerecapture model and a description relevant to the time of detection method. We begin by
assuming that all intervals are of equal length.
�65
Model M0 represents a constant detectability coefficient for all individuals across
all time periods. This model is the simplest of all the models and requires estimation of
only two parameters, the detection coefficient and N. In general this model may
oversimplify the detection process because it assumes no individual heterogeneity and no
differences between the first detection and subsequent detections. This model is
generally not robust to heterogeneity or differences between first and subsequent
detection probabilities. This model may be applicable when a species is very easily
detected, for example, has very high rates of calling and very loud distinct calls. Such
situations reduce the affects of individual heterogeneity in detection probabilities caused
by variable singing rates because the probability of detecting any given call is high
regardless of whether or not the bird was detected previously.
Model Mt assumes equal detection coefficients for all individuals but different
coefficients among time periods. For t periods the number of parameters in this model is
t+1. The model does not account for individual heterogeneity in detection probabilities
or differences between first and subsequent detections. For example, this model is
applicable to situations where observer effects on singing rates (McShea and Rappole
1997) decline over time.
Model Mb assumes equal probability of first detection for all individuals across all
periods and a unique probability of subsequent detections that is equal for all individuals
across all subsequent periods. This model requires the estimation of the detection
coefficient for the first detection and one for all subsequent detections (plus one
parameter for N), for a total of 3 parameters in the model. The model is applicable to
situations in which the detection coefficient is constant over the duration of the count and
�66
individual heterogeneity does not affect the detection process but where the requirement
to track individuals detected during the count does affect the probability of subsequent
detections. We assume if observers are properly tracking individuals that detection
coefficients would be higher for subsequent detections than for first detections.
However, if observers have a tendency to ignore individuals previously detected then the
opposite would hold. It is important to note that under this model subsequent detections
do not provide any information for estimating the detection coefficient of first detections
or N. Model Mb is equivalent to the removal method (Otis et al. 1978) because only first
detections provide information about N. Model Mb is equivalent to Farnsworth et al.’s
(2002) model with no heterogeneity.
Model Mtb assumes an equal detection coefficient of first detection for all
individuals that differs between periods and a unique detection coefficient of subsequent
detection that is equal for all individuals but differs between subsequent time periods.
The full model contains more parameters than can actually be estimated so the
simplifying assumption is made that there is only a single effect of first capture. This
model estimates a detection coefficient of first capture for each time period and a single
coefficient for the change in detection probability for subsequent detections. The
probability of subsequent detections for a given time period is an additive effect to the
probability of first detection for all time periods. Again, only first detections provide
information about N. This is equivalent to the removal model, except that the removal
model requires setting 2 of the parameters equal so that N is identifiable. The model is
applicable in situations similar to those given for Model Mb but where the observer also
�67
has an effect on detections that diminishes over time. This model is overparameterized
and hard to fit to real data.
Model Mh assumes a unique detection coefficient for each individual which
remains constant across all time periods and is similar between first and all subsequent
detections. The number of parameters involved in all of the heterogeneity models
depends on how heterogeneity is modeled. This is discussed in the heterogeneity section
of this paper. In general this model is applicable when the detectability coefficient is
different among all members or groups of the population, but remains constant across
time periods, and is not affected by first detection. For example, if some members of the
population are unmated territorial males, some are mated territorial males, some are
mated and incubating territorial males, and some territorial males are actually caring for a
brood, we would expect singing rates and other behaviors to differ among groups. These
differences would result in different detectability coefficients among groups. Because it
is generally not possible to identify the status of individual birds differences in detection
probabilities are evaluated using mixture models (Norris and Pollock 1996, Pledger
2000).
Model Mth is similar to model Mh in that there is a unique detection coefficient for
each individual or group of individuals but this model also assumes that detection
probabilities change between time periods. Applicable situations include those discussed
for model Mh but where observer effects on singing behavior diminish over time.
Model Mbh assumes a unique detection coefficient of first capture for each
individual that remains constant across time periods and a unique coefficient of
subsequent detection that remains constant across time periods. Again, only first captures
�68
provide information about N so this model is equivalent to the removal approach with
heterogeneity. Farnsworth et al. (2002) present a 3 period removal model with a two
point mixture model of heterogeneity. Because there were only 3 periods, a very strong
assumption that all of the birds in one group are detected in the first interval is required.
The full two point mixture models of Norris and Pollock (1996) and Pledger (2000) are
available when four or more time periods are used. We believe that these models will be
very useful because individual heterogeneity and situations where subsequent detections
are affected by first detections are likely on avian point counts.
Model Mtbh is the most general model. It assumes a unique probability of first
capture for each individual that differs among time periods and a unique probability of
subsequent detection for each individual that differs among time periods. Fitting this
model requires several strong assumptions to reduce the number of parameters estimated.
We did not attempt to fit this model to our data.
The individual heterogeneity models have been the slowest to develop because of
their complexity, large number of parameters, and computing requirements. In fact only
recently have likelihood based methods been developed using finite mixtures (Norris and
Pollock 1996, Pledger 2000). The Jackknife (Burnham and Overton 1978, 1979) and
Chao (Chao et al. 1992) heterogeneity model estimators are not likelihood based
(Williams et al. 2002). The advantage to having maximum likelihood based estimators is
that models can be evaluated with likelihood ratio tests for evaluating sources of variation
and model selection techniques, such as Akaike’s Information Criterion (AIC), can be
used to select the most parsimonious model (Williams et al. 2002).
�69
The Jackknife estimator for model Mh is the original estimator for this model and
probably is the most commonly used (Williams et al. 2002). This estimator is based on a
generalized jackknife statistic using a linear function of capture frequencies for bias
reduction (Burnham and Overton 1978, 1979). Several tests of this estimator using
simulation have demonstrated that this estimator does work reasonably well under a
variety of situations (Otis et al. 1978, Pollock et al. 1990, Williams et al. 2002). This
estimator can be run using program CAPTURE (White et al. 1982).
Another approach commonly used for estimation under models Mh, Mth, and Mbh
is the sample coverage approach proposed by Chao et al. (1992), Chao and Lee (1992),
and Lee and Chao (1994). This approach utilizes the canonical form given in equation 1
by dividing the observed count by an estimate of the sample coverage. The estimators for
sample coverage are constructed using capture frequencies (Chao et al. 1992, Chao and
Lee 1992, Lee and Chao 1994). The sample coverage approach works well in some
situations but not in others (Williams et al. 2002). There are actually three estimators for
sample coverage under model Mth and Chao et al. (1992) provide guidelines on situations
in which each is applicable. Simulation results for model Mbh indicate that the jackknife
estimator of Pollock and Otto performs better than the sample coverage estimator (Lee
and Chao 1994). Both estimators Mh and Mth can be run using program CAPTURE.
The finite mixture heterogeneity estimators of Norris and Pollock (1996) and
Pledger (2000) are maximum likelihood estimators, which has the advantage of providing
a unified maximum likelihood approach of fitting and comparing the entire suite of
models. The idea behind this approach is that the entire population of interest can be
divided into a finite number of groups, each with unique probability of capture/detection.
�70
The nonparametric approach of Norris and Pollock (1996) first conditions on N to
determine the number of groups in the population and then N is varied to find the overall
maximum likelihood. Pledger (2000) does not condition on N but instead simply
specifies the number of groups at the outset and compares models with different numbers
of specified groups. Both methods often lead to the same results (Pledger 2000).
Simulation under the various heterogeneity models has demonstrated that the finite
mixture approach provides competitive models to the other approaches (Norris and
Pollock 1996, Pledger 2000, Williams et al. 2002). The finite mixture models can be fit
using program MARK (White and Burnham 1999), which requires that the number of
mixtures be specified.
It is worth noting that simulation studies have demonstrated that abundance
estimates will depend on the method used and the number of groups used. Norris and
Pollock (1996) provide estimates from their method along with the other commonly used
approaches to modeling heterogeneity and clearly demonstrate that the resulting estimates
of abundance can depend on the method used. Unfortunately there is no way of assessing
which method is correct. We would suggest that there may be some biological basis for
using finite mixtures for point count data since detections are likely to be a function of
singing rates which are known to vary in relation to the breeding state (mated, incubating,
brood rearing, etc.) of an individual bird. However, it is possible that this does not
account for all the variation in the data and these assumed groups may not be sufficient to
account for all the heterogeneity present in the data.
Other models relevant to estimating the detection process include those that
incorporate auxiliary information, such as detection distance, which can be incorporated
�71
as covariates. We develop these models in the “Covariate” section and demonstrate their
use with an equal time interval example.
General form of model.—Models of equal time interval data can be fit using
standard capturerecapture software (CAPTURE or MARK), but not for unequal time
interval data. Because much existing point count data have been collected using unequal
time intervals we present a general model and subsequent equations for both the
instantaneous rates and discrete rates formulations. This model is also applicable to
situations when equal time intervals are used, so we also present the equations for this
case, which simply models the interval probabilities of first detections (pi) and
subsequent detections (ci) directly.
Models without heterogeneity.—Model Mtb is the most general model without
heterogeneity. The remaining models without heterogeneity are just constrained forms of
model Mtb. Point counts with i intervals are comprised of 2i capture histories, one of
which is not observable because it represents individuals that are never detected. Model
Mtb requires 2i parameters but the minimal sufficient statistic for the distribution has
dimension 2i1 (Otis et al. 1978, Borchers et al. 2002). Because of this the model is
constrained by assuming that the change in detection probability after first capture is
constant across all subsequent captures (Borchers et al. 2002). We will also make this
assumption so that the probability of a subsequent detection (ci) during period i is,
c = p +δ
i
i
(4)
for equal intervals,
−
c =1 − e
i
(ϕ i +ν )t i
(5)
�72
for the instantaneous rates formulation, and
c = (γ i +η )
ti
i
(6)
for the discrete rates formulation, where δ, ν, and η are the change in the detectability
coefficients for all subsequent detections. Note that ci may change between periods even
though δ, ν, or η remain constant because either pi, φi or γi and ti can vary among time
periods.
We present the equations using the interval detection probabilities (pi’s and ci’s)
for clarity in the following equations. These are replaced by one of the appropriate rates
equations for unequal interval point counts. A count consisting of 4 intervals is
comprised of 15 observable detection histories (xw: x1111, x1110 , … , x0001). The set of
possible detection probabilities for the observable capture histories is denoted by πw. The
expected values of the counts for each detection history of the general model can be
written as,
E ( x1111 ) = Np1c2 c3c4 = N π 1111 ,
E ( x1110 ) = Np1c2 c3 (1 − c4 ) = N π 1110 ,
(7)
E ( x1010 ) = Np1 (1 − c2 )c3 (1 − c4 ) = N π 1010 ,
E ( x0001 ) = N (1 − p1 )(1 − p2 )(1 − p3 ) p4 = N π 0001
�73
where the pi’s and ci’s are from either the instantaneous or discrete rates formulation.
From this it follows that the probability of detecting an individual at least once during the
entire count (pT) is one minus the probability of never detecting an individual;
pT = 1 − (1 − p1 )(1 − p2 )(1 − p3 )(1 − p4 )
(8)
The full multinomial likelihood can be written as;
L( N , pi , ci , xw* ) =
(
N!
π wxw 1 − ∑ π w
∏
w
∏ xw ! N − ∑ π w ! w
w
(
w
)
)
N−
∑ xw
w
(9)
Where xw is the number of observations for each possible detection history, excluding the
history for the individuals never detected and πw (Equation 7) is the probability of
observing each capture history, excluding the probability of never seeing an individual.
Because N cannot be directly observed we condition on the total number of birds
counted (xT) to make the problem more amenable to numerical methods. The likelihood
is decomposed into a marginal distribution L1 and a conditional distribution L2 so that
detection probabilities can be estimated from the conditional, and abundance can be
estimated from the marginal distributions. The relationship between these likelihoods is
L=L1*L2. The marginal likelihood L1 can be written as,
⎛N⎞
L1 ( N  xT ) = ⎜ ⎟ (π i ) xT (1 − π i ) N − xT
⎝ xT ⎠
(10)
Where π. is the sum of the πw. The likelihood L2 is conditional on the observed capture
histories, given by,
xT
⎛
⎞ ⎛ πw ⎞
L2 ( pi , ci  xw ) = ⎜
⎟∏⎜ π ⎟
⎝ x1111...x0001 ⎠ w ⎝ i ⎠
xw
(11)
�74
From equation 11 the pi’s and c’s, φi’s and ν’s or γi’s and η’s (depending on which
formulation is used) can be estimated by maximizing the likelihood for the observed data.
From equation 8 the probability that an individual is detected at least once during the
count ( p
ˆ ) can be calculated. For models with time variation pˆ is given by;
T
T
pˆ T =1(1pˆ 1 )(1pˆ 2 )(1pˆ 3 )(1pˆ 4 )
(12)
for equal time intervals, by
p̂T =1eϕˆ1t1 eϕˆ2 t 2 eϕˆ3 t3 eϕˆ4 t 4
(13)
for the instantaneous rates formulation or by
pˆ T =1(1γˆ 1t1 )(1γˆ 2t 2 )(1γˆ 3t3 )(1γˆ 4t 4 )
(14)
for the discrete rate formulation. The variance of p̂T is obtained for each estimator by
reparameterizing to estimate p̂T directly. This can be done by solving one of the
equations of p̂T for the last interval parameter and using this in the likelihood. For
example, if we solve equation 13 for ϕ̂4 we get
ϕˆ4 =
1 ⎛ 1 − pˆ t ⎞
ln ⎜
⎟
t4 ⎝ eϕˆ1t1 eϕˆ2t2 eϕˆ3t3 ⎠
(15)
Replacing the final interval parameter with this equation will allow for direct estimation
of the probability that an individual is detected at least once and the associated variance.
An alternative approach is to use the delta method (Seber 1982) and obtain estimators for
�75
the approximate variances. Abundance is estimated from the general form (equation 1)
or the likelihood L1 as,
x
Nˆ = T
pˆ T
(16)
The observed count (xT) is one realization of a random variable and thus, has a variance
associated with it. Assuming that the observed count (xT) is from a binomial distribution
an estimate of the variance of abundance is (Nichols et al. 2000, Williams et al. 2002);
Var ( Nˆ ) =
xT2 Var ( pˆ T ) xT (1 − pˆ T )
+
pˆ T4
pˆ T2
(17)
With the framework of the model established, it is possible to determine the
number of parameters for each potential model. Capture history probabilities or the
expected values (equation 7) for four time periods indicate that there are potentially 8
parameters (p1, p2, p3, p4, c2, c3, c4, and N) estimated for the general model. Recall that
we must constrain the effect of subsequent capture to be equal across all time periods
(δ2=δ3=δ4). The 4 time period model Mtb thus, has 6 parameters. In general model Mtb
with t time periods has t+2 parameters to estimate. The remaining models are
constrained from this form. Surveys with t intervals are shown for the instantaneous rates
formulation.
Model M0: Set all rate parameters equal (φ1=φ2=…=φt, ν2=ν3=…=νt=0, and N).
Total parameters to estimate are 2.
Model Mt: Set rate parameters for subsequent detections equal to zero so that
subsequent detections have the same probability as initial detections (φ1,
φ2, …, φt, ν2=ν3=…=νt=0 and N). Total parameters to estimate are t+1.
�76
Model Mb: Set rate parameters for first detections equal and rate parameters for
subsequent detections equal (φ1=φ2=…=φt, ν2=ν3=…=νt, and N). Total
parameters to estimate are 3.
Analysis of point count data with these models will depend on whether the data
are from equal interval or unequal interval point counts. If data are collected with
unequal interval point counts then one of the rate formulations is required. This is not
possible using one of the standard capturerecapture programs. Analyis of unequal time
interval point counts requires maximizing the likelihood for each candidate model using a
program such as SURVIV (White 1983) that allows the user to input the likelihood for
each model.
Analysis of point count data from equal interval point counts is much easier
because standard capturerecapture analysis programs, such as CAPTURE and MARK,
can be used. Program CAPTURE is an older program that will run the full suite of
models automatically but it does not use current methods of model selection. Program
MARK requires the user to parameterize all candidate models and uses Akaike’s
information criterion (AIC), an informationtheoretic approach, (Burnham and Anderson
2002) for model selection. Program MARK also accommodates covariate models.
Heterogeneity models.—Many factors create unobservable individual
heterogeneity in detection probabilities. These factors often relate to individual behavior
or where the individuals are located (Burnham 1981, Johnson et al. 1986). Factors
inherent to an individual that may cause individual heterogeneity include age, breeding
status (mated, unmated, incubating, etc.), and singing rate. Other causes of heterogeneity
�77
are those associated with an individual’s location because detection probabilities are
being estimated from spatially distinct points. While stratification by habitat type can
eliminate some differences among points (Buckland et al. 1993, Nichols et al. 2000)
differences that cause heterogeneity exist in all data. These differences may include such
factors as foliage density, local background noise, topographic features and the presence
or absence of predators or competitors. Temporal differences in detection probabilities
between points sampled early in the day compared to those sampled later in the day
(Farnsworth et al. 2002) or between points done on consecutive days may also exist.
Some of these factors can be minimized by standardizing the conditions under which
point counts are conducted, but no amount of standardization will account for all of the
heterogeneity present in the population (Burnham 1981).
One source of heterogeneity that may be especially important given that these
models are based on singing rates, is individual variation in singing rate. Singing rates
generally change during the breeding season as birds pair and begin incubation or caring
for nestlings (Wasserman 1977, Lein 1981). Asynchronous breeding will cause
heterogeneity in the data. If singing rates decline over the course of the breeding season,
then sampling over this entire period would introduce heterogeneity into the data.
Singing rates are also affected by habitat, local abundance, and the proximity of
observers (McShea and Rappole 1997).
In general four or more time intervals are required to parameterize heterogeneity
models, unless very strong assumptions are made, as in Farnsworth et al. (2002). We will
develop the heterogeneity models based on four or more time intervals. The assumptions
�78
required for the restricted heterogeneity models are given with the presentation of the
three interval examples.
For point count surveys that use equal intervals it is possible to use several
different heterogeneity models. For unequal time intervals we must use the rates
equations and thus we will develop a finite mixture model of heterogeneity for this
situation because the likelihood equation must be parameterized and maximized in a
program such as SURVIV (White 1983). We use a finite mixture approach because it is
a maximum likelihood approach and we believe the benefits of a maximum likelihood
approach and informationtheoretic model selection warrant the use of this approach. We
present the twopoint mixture model because as Pledger (2000) suggested, twopoint
mixtures often provide the most parsimonious models and estimators with good
properties. This approach can be easily extended to more mixtures if appropriate. We
present this model for four sampling intervals, because it is doubtful that data would be
collected with more than four intervals.
Data from four interval point counts are summarized by the counts for the 15
observable detection histories xw. Assuming that only two groups (two mixtures)
comprise the population, the probabilities for each capture history are given by the sum
of the product of the proportion of animals in each group times the group specific capture
and recapture probabilities. For example, the expected value of the count for the 4
interval detection history for individuals detected in all intervals is
E ( x1111 ) = N (λ p11c21c31c41 + (1 − λ ) p12 c22 c32 c42 )
(18)
where λ is the proportion of animals in heterogeneity group one and the proportion of
animals in heterogeneity group two is 1λ as they must occur in one of the groups and the
�79
pij’s and cij’s are the probabilities of first detection and subsequent detection in the ith
interval for individuals in the jth group. There are similar expressions for the other xw.
The conditional likelihood L(pij, cij, λ  xw) is similar to equation 9,
xT
⎛
⎞ ⎛ π w1 + π w 2 ⎞
L2 ( pi , ci  xw ) = ⎜
⎟∏⎜
⎟
πi
⎠
⎝ x1111...x0001 ⎠ ∀w ⎝
xw
(19)
where π i is the probability of being detected at least once during the entire count, πw1 is
the probability of being in the first group and having capture history w, and πw2 is the
probability of being in the other group and having capture history w. Using this
likelihood, the pij’s and c’s, φij’s and ν’s, or γij’s and η’s (depending on the rate
formulation used) can be estimated by maximizing the likelihood for the observed data.
The probability that an individual is detected at least once during the count ( p̂T )
is calculated using the estimated detection coefficients,
pˆ T = 1 − ⎡⎣λˆ (1 − pˆ11 )...(1 − pˆ t1 ) + (1 − λˆ )(1 − pˆ12 )...(1 − pˆ t 2 ) ⎤⎦
(20)
for the equal interval formulation. For one of the other rates formulations (equations 2 or
3) the pij is replaced by the appropriate rate to estimate
pˆ
T
, as in equations 13 and 14.
The variance for pˆ T can again be calculated by reparameterizing the model estimators to
directly estimated pˆ T . Equations 16 and 17 are then used to estimate abundance and the
corresponding variance.
Covariates.—Previous examples describe cases in which sources of individual
variation were not identifiable. There are situations where sources of variation are
identifiable and can be quantified. Accounting for observable heterogeneity in capture
�80
probabilities when individual covariates that explain the differences in capture
probabilities are identified, is a special case of model Mh (Huggins 1989, 1991, Alho
1990). Many covariates might explain differences in detection probabilities from point
counts where detections are based on auditory cues. These include distance from the
observer, singing rate, direction of singing relative to the observer, foliage density,
pairing status, and so on. Unfortunately most of these are not quantifiable during a point
count. For example, it is not possible to assess the pairing status of a bird during a point
count and thus this source of heterogeneity cannot be accounted for with a covariate.
Singing rate is another factor that could be used as a covariate, although this should be
used as a categorical variable because not all songs are detected during a count.
Using detection distance as a covariate is very appealing because intuitively an
observer’s ability to detect a bird should decrease with increasing distance to the bird.
Because distance sampling on point counts is frequently used to estimate abundance,
using distance as a covariate seems even more reasonable. If observers can estimate the
location of birds from their songs with reasonable accuracy then models incorporating a
detection distance covariate will provide less biased estimates than those that do not. The
ability of observers to accurately estimate the distance to singing birds is a key
assumption and is one that is largely untested.
Covariates (including detection distance) are used to model detection probability
by making detection probability a linear function of individual covariates. An intercept
(β0) and a slope (β1) parameter are estimated to model the detection process. The original
suite of models described can be viewed as intercept only models. The effect of the
covariate term on the slope parameter is modeled as either a constant effect over time
�81
periods or a variable effect over time periods. This adds two additional models to each of
the original seven models described, or a total of 21 conceptual models.
Modeling detection distance as a covariate in model M0 will demonstrate the
approach. The original model M0 assumes a constant detection probability over all
periods and all detection distances. If we incorporate detection distance into the model
with a constant slope across periods, then the model assumes a constant detection
function over time periods. Alternatively we can model detection distance with a
different slope for each interval, which assumes that the detection probability at the point
(intercept) is constant across intervals but the effect of detection distance (slope) on the
detection probability is not constant between intervals. This may occur when the
observer’s arrival at the point affects a bird’s singing behavior, but the effect declines
over time. The other models can be constructed similarly from the original suite of
models by interpreting the original model parameters as reflecting the probability of
detection at the point, and interpreting the covariate terms as reflecting how detection
probabilities change with respect to covariates.
Covariate models of equal interval point counts can be parameterized in program
MARK, using the “Huggins Closed Captures” data type. The ability of MARK to
incorporate an entire set of candidate models into a single analysis using an informationtheoretic approach to model selection is a compelling reason to collect data using equal
time intervals.
Huggins (1989, 1991) and Alho (1990) used the linearlogistic function of
individual covariates conditional on the total number of animals detected, to model
detection probabilities as a function of the observed covariates. This model is used in
�82
program MARK for modeling equal time interval cases. Under this model, the
probability of detecting an individual, j, is given by,
pˆ
j
=
β +β x
e
1
0
1
j
(21)
+
+ eβ 0 β 1 x j
where β0 and β1 are estimated parameters, and xj is the measured covariate value for the
jth individual. We assume that movement during the point count is minimal so that xj is
constant for each individual.
If we assume that the probability of detecting the jth individual is constant across
time periods then the probability that this individual is detected at least once during a
count with t intervals is
pˆ = 1 − (1− pˆ j)
t
*
j
(22)
It is also possible to estimate intervalspecific detection probabilities for each individual
by modifying equation 16 to have interval specific estimates of β0 and β1 and using the
interval specific estimates to determine the probability of detecting each individual at
least once during the count. The abundance estimator for a count with a total of M
individuals detected is the HorvitzThompson estimator (Horvitz and Thompson 1952)
1
=
ˆ
∑
N
pˆ
M
j =1
*
(23)
j
Using time intervals of unequal length with covariate models becomes more
difficult because existing capturerecapture software cannot be used. In this situation
each individual in the sample has a unique likelihood that must be used to estimate the
model parameters. This is done by modifying one of the rates formulations (equations 2
�83
or 3) so that the detection probability is a function of the covariate. For example,
modifying the instantaneous rates formulation gives the detection probability as,
p = 1 − e (ϕ
−
i
+
β x )t
j
(24)
i
ij
such that the intercept (probability of detection at the point) is given by φi and β
represents the change in the probability of detection with respect to detection distance.
This form of the instantaneous rates equation is then used in the likelihood.
Field Trials
We present examples from two different field studies, one from point counts
using 3 unequal intervals, and the other from point counts using 4 equal time intervals.
The first example is based on point count data collected in Great Smokey Mountains
National Park from 1996 to 2000. We only used data from a single year (1998) and a
single observer to avoid temporal and observer effects. Point counts were conducted
along preestablished survey routes that followed trails in the park (Simons and Shriner
2000). All point counts were conducted in the first few hours of daylight on days
meeting acceptable environmental conditions for point counts. All point counts were
done during May and June of the breeding season. Point counts were divided into 3, 2,
and 5 minute intervals and the complete detection history was recorded for each
individual by using different colored pens for each time interval. Point counts were
unlimited radius plots and detection distance was recorded for all detections.
Analysis is restricted to the four most frequently detected species to avoid the
effects of sparse data on model selection. These species are the; Blackthroated Green
Warbler (Dendroica virens), Hooded Warbler (Wilsonia citrine), Ovenbird (Seiurus
�84
aurocapillus) and Redeyed Vireo (Vireo olivaceus), all of which have relatively loud
calls and high singing rates. All models (except the covariate models) were used and
analysis was done using program SURVIV. We truncated 10% of the data by omitting
observations with the largest detection distances, as recommended by Buckland et al.
(1993). Model selection was based on AIC (Burnham and Anderson 2002).
We modeled heterogeneity using constrained forms of the twopoint mixture
models. These models must be constrained so that all parameters are identifiable. We
constrained the models by fixing the detection probabilities for one of the groups in the
mixture. One model was to set all detection probabilities for one group equal to one,
similar to Farnsworth et al. (2002). This parameterization assumes that all individuals in
this group are detected in every interval. Alternatively, we set all interval detection
probabilities equal to 0.9 for one of the groups in the mixture. This value of 0.9 was
obtained from analysis of similar data from this same study area. A twopoint mixture
model of heterogeneity for fourindependent observer point count data showed that for
one of the mixtures the detection probability for Ovenbird was 1.0 and for Redeyed
Vireo was 0.89 (chapter 2). An analysis of singing rate data for the Ovenbird estimated
the availability probability at 0.93 (chapter 5). By modeling one of the heterogeneity
groups with a constant, model Mth has a slightly different interpretation as only one of the
groups will actually have time variation. The necessity of fixing parameters like this for
three time interval point count data is the primary reason for using four or more intervals
for point counts.
The other example is for the Pearlyeyed Thrasher based on point counts
conducted in the karst belt of northcentral Puerto Rico in 2003. Data were collected
�85
during the breeding season (midFebruary through May). Surveys were conducted using
three teams of two experienced and trained observers each. Forty nine point count routes
were used with one to 29 (average 9.7) point counts conducted on each route for a total of
477 points. Point count routes were located away from human habitation along “low
use” trails. The first point was located 500 m from the start of the trail and subsequent
points were located 200 m from the previous. Point locations were alternated from on the
trail to 50 m off the trail (randomly left or right).
Point counts were conducted from 0400 hours to 0800 hours on days with suitable
weather conditions. Each count was conducted for ten minutes. Counts were divided
into four equal time intervals and the complete detection history was recorded for each
individual by using different colored pens for each time interval. Point counts were fixed
100 m radius plots and detection distance within this plot was recorded for all detections.
During the previous year distances at each plot were flagged at 10 m intervals.
The Pearlyeyed Thrasher data were analyzed using the full suite of models,
including heterogeneity and detection distance covariate models. All analyses were done
using program Mark. Data were not truncated since a fixed radius plot was used. Model
selection was based on AIC (Burnham and Anderson 2002).
Results
Three interval data set.—Heterogeneity models were the most parsimonious for
the three unequal interval data sets from Great Smoky Mountains National Park (Table
1). The analysis clearly demonstrated the importance of heterogeneity models. The
∆AICc weights for all models without heterogeneity were always zero, indicating no
evidence supporting these models. Model Mth(0.9) was selected as the most parsimonious
�86
for three of the data sets and Model Mh(0.9) was selected for the other data set.
Heterogeneity models with interval detection probabilities set to 0.9 for one heterogeneity
group were selected for over models where a heterogeneity group had interval detection
probabilities set to 1. This indicates that the assumption of complete detection is too
restrictive for these data sets. All species except the Ovenbird showed evidence for time
variation in instantaneous rates of detection, indicating that detection probabilities do not
remain constant for the duration of counts. There was little support for the models
incorporating both heterogeneity and behavior, indicating that detection probabilities for
birds did not change after first detection.
Interval detection probabilities showed a consistent pattern; the shortest intervals
had the smallest detection probabilities and the longest intervals had the highest detection
probabilities (Table 2). This was not true for the Redeyed Vireo which had a detection
probability of 0.26 for the three minute interval and 0.29 for the two minute interval.
This may indicate an observer effect that made this species less detectable during the first
interval. Comparing the observed counts to the estimated abundance in the sample area
showed differences of 5% (Ovenbird) to 23% (Hooded Warbler).
Four interval data set.—Sixteen of the 21 conceptual models for the four interval
Pearlyeyed Thrasher data set gave reasonable results. The other 5 models gave results
indicating one or more parameters were not identifiable. The most parsimonious model
selected for this data set was model Mth (∆AICc weight = 0.70), with the remaining
support for model Mth(detection distance consant slope) (∆AICc weight = 0.30) (Table 3).
For all models, the general form of the model and it’s two covariate formulations were
always ordered together in model selection. Heterogeneity models were always “better”
�87
than models not accounting for heterogeneity. Of the models that did not account for
unobservable heterogeneity, models incorporating detection distance always had lower
AICc values.
Under the selected model, 29% of the population had low detection probabilities
while the remaining portion had high detection probabilities (Table 4). Low detection
probabilities ranged from 0.09 to 0.56 and the high detection probabilities ranged from
0.70 to ≈1.00. Since the detection probability was essentially one for one of the
heterogeneity groups during the second interval, abundance is equivalent to the count for
within this heterogeneity group. A detection probability of one indicates that all
individuals were seen. Note that the standard error is not estimable for parameter
estimates near the boundary. The total number of observations for this data set was 520
and the estimated abundance for the sampled area was 547 (SE=8.6), 5% higher than the
observed count.
Discussion
The time of detection method of modeling point count data collected from
consecutive time intervals is a less restrictive approach for estimating detection
probabilities than the removal method. Both approaches are promising because they
model the detection process and provide estimates of true abundance or density, and both
permit spatial and temporal comparisons of data without the unrealistic assumptions
necessary to compare index counts. These models can be applied to data sets with two or
more time intervals of equal or unequal length.
Our approach is a more general approach than the removal method of Farnsworth
et al. (2002). Recording the complete detection history of birds during a point count
�88
provides a much larger suite of models that can incorporate more sources of variation,
such as time variation, than the removal model. With the exception of the behavior
models which are equivalent to the removal models (Seber 1982) this approach is more
efficient (smaller variance) than the removal approach.
Differences between species in the estimated detection probabilities were
expected and are probably due to differences in singing rates and sound intensity. The
overall probability of detecting an individual at least once during a 10 minute count was
0.92 (SE = 0.006) for the Ovenbird, 0.79 (SE = 0.021) for the Blackthroated Green
Warbler, 0.71 (SE = 0.031) for the Redeyed Vireo, and 0.65 (SE = 0.037) for the
Hooded Warbler. Of these species Ovenbirds have the highest singing rates and thus, we
would expect them to have the highest detection probability. Comparing estimates of
detection probability to those given by Farnsworth et al. (2002) show very similar
estimates for all four of these species.
The time of detection approach also reduces the number of assumptions required
for the model. The removal method for three time intervals had five assumptions
(Farnsworth et al. 2002), which could be relaxed to four if more than 3 intervals were
used to collect data. The time of detection approach has only three assumptions for
counts conducted with more than three time intervals, which are generally required for all
analysis approaches to point count data. An additional assumption of both methods is
that species are identified correctly.
Assumption 1: there is no change in the population within the detection radius
during the point count (closed population). Violations of the closure assumptions are
more likely to occur for longer duration point counts and for wide ranging species. This
�89
method may not be applicable for wide ranging species, such as woodpeckers (Family
Picidae) or crows (Corvus spp.) where movement during the count may be significant.
Violations of the closure assumption are probably less of a problem for many small
breeding songbirds, such as the ones illustrated here, as they have relatively small fixed
territories during the breeding season (Farnsworth et al. 2002).
Short duration counts will reduce violations of the closure assumption. As the
total length of a point count decreases so does the probability that birds will either move
into or out of the sample area. This does not imply that all point counts should be
arbitrarily short because a point count must be divided into four intervals to fit the full
heterogeneity models. If the intervals are too short then interval detection probabilities
will be small and the variance on abundance or density estimates will be large. Careful
consideration must be given to the appropriate length of point counts for the species
surveyed. This may imply different survey protocols for different species groups.
Assumption 2: there is no doublecounting of individuals. Doublecounting
results in abundance or density estimates that are too large. Problems with doublecounting are likely to increase as the length of the count increases because undetected
movement of birds is more likely to occur. Our method requires observers to track
individuals during the count which should minimize violations of this assumption.
Proper training of observers to minimize doublecounting is also required (Farnsworth et
al. 2002).
Assumption 3: for unlimited radius plots, distance is measured accurately or for
limited radius plots, individuals are accurately assigned to within or beyond the
designated radius. We previously discussed the importance of knowing the sample area
�90
so that comparisons can be made between studies. Observers can be trained to estimate
detection distance or to assign birds to within a limited radius plot. The accuracy and
precision of observer ability to estimate distance has not been rigorously assessed. We
suspect that, even with training, observers tend to overcount individuals within fixed
radius plots, and that the accuracy of observers in estimating detection distances of songs
and calls may be poor.
The removal method also requires an assumption that the detection rate is
constant for the duration of the count (Farnsworth et al. 2002). This assumption is not
necessary with our method because there is sufficient information to model time variation
from the full detection history. Evidence for a time effect was found in four out of the
five data sets presented here. Therefore, it appears that the assumption of constant
detection rates under the removal model may not be valid. McShea and Rappole (1997)
found singing rates were affected by the presence of an observer. Movement of an
individual during the count would also affect the detection process as distance from the
observer may affect how likely an observer is to detect a call.
Recommendations
We developed a general model to analyze point count data collected for two or
more equal or unequal intervals. It can be used to analyze data in a variety of formats,
including those recommended by Ralph et al. (1995). We recommend that future studies
using this approach be designed with four or more equal intervals. The use of equal
intervals simplifies analysis and permits the use of standard capturerecapture software.
Use of four or more intervals allows for the application of full twopoint mixture models.
�91
Farnsworth et al. (2002) recommended combining the removal approach with the
distance sampling approach to provide better estimates. We have done this by
incorporating detection distance as a covariate in the models presented here. Further
development is needed to incorporate alternative forms of the detection function. One
obvious extension is to use distance squared, which is equivalent to the halfnormal
detection function applied in distance sampling. Given the number of factors affecting
the aural detections of birds in point count surveys distance may not be a useful
explanatory variable in all situations. For example, if singing rates of individual birds is
a significant factor affecting the detection of birds then detection probabilities may have
an initial increase with increasing detection distance before they decline at large
distances. This is because birds near observers may sing less than birds farther from
observers (McShea and Rappole 1997). Further investigation of the use of detection
distance as an explanatory variable for estimating aural detection probabilities is required.
Point counts based on auditory detections rely on an observer’s ability to
accurately estimate the detection distance to singing birds, even on fixed radius plots. A
rigorous assessment of the accuracy of detection distance estimation is needed before
those methods become widely adapted. Such an evaluation should also identify the bias
associated with fixed and unlimited radius plots.
For our examples we presented a single species modeling approach. Because
point count data usually consists of multiple species we recommend using a multiple
species modeling approach (chapter 4). If species with similar detection probabilities are
modeled together then more parsimonious models can be used. This is especially useful
for rare species or species that are hard to detect.
�92
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�Table 1: ∆AICc values for the 11 time of detection models fit to each data set. ∆AICc = 0.0 for the most parsimonious model for each
data set. ∆AICc weights (in parentheses) indicate the strength of the evidence for a given model compared to the other models (the
larger number indicates more evidence for that model).
Models
Species
M0
Mt
Mb
Mtb*
Mh(.9)
Mh(1)
Mbh(.9)
Mbh(1)
Mth(.9)
Mth(1)
Blackthroated Green Warbler
45.7
42.9
47.7
__
6.8
4.3
8.8
6.3
0.0
4.1
(0.00)
(0.00)
(0.00)
(0.03)
(0.09)
(0.01)
(0.03)
(0.75)
(0.10)
55.4
44.3
57.5
15.1
13.7
17.1
15.8
0.0
3.6
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.86)
(0.14)
56.6
40.7
58.6
0.0
12.2
2.0
14.3
15.2
14.9
(0.00)
(0.00)
(0.00)
(0.73)
(0.00)
(0.27)
(0.00)
(0.00)
(0.00)
78.0
58.2
77.7
0.1
10.4
2.2
12.4
0.0
2.7
(0.00)
(0.00)
(0.00)
(0.37)
(0.00)
(0.13)
(0.00)
(0.39)
(0.10)
Hooded Warbler
Ovenbird
Redeyed Vireo
__
__
__
*Parameter estimates for model Mtb were not reasonable. Standard errors were greater than 1.
95
�Table 2. Parameter estimates from the selected model for the 3 interval point count data sets. λ1 is the proportion of the population
that is in group 1. Detection probability (pij) is the probability of detecting an individual from group j in interval i. The detection
probabilities for group 2 (pt2) were fixed. Standard errors in parentheses.
Observed
λ1
p11
p21
p31
pt2
N
Blackthroated Green Warbler
377
0.53 (0.038)
0.40 (0.056)
0.24 (0.056)
0.54 (0.061)
0.9
425 (15.1)
Hooded Warbler
274
0.54 (0.050)
0.33 (0.062)
0.20 (0.056)
0.34 (0.065)
0.9
338 (17.4)
Ovenbird
444
0.54 (0.054)
0.53 (0.042)
0.39 (0.054)
0.71 (0.026)
0.9
465 (13.1)
Redeyed Vireo
397
0.45 (0.044)
0.26 (0.052)
0.29 (0.062)
0.44 (0.065)
0.9
457 (15.9)
Species
96
�97
Table 3: ∆AICc values for the four time interval Thrasher data set. A value of 0.0
indicates the most parsimonious model. ∆AICc weight is in parentheses where weight is
nonzero. NE indicates models that were not included because of unreasonable parameter
estimates.
Model
No Distance
Constant Distance
Time Distance
M0
170.2
165.2
164.2
Mt
148.1
143.1
139.7
Mb
102.3
99.5
97.4
Mtb
NE
NE
NE
Mh
72.2
70.3
53.4
Mbh
23.3
13.7
NE
Mth
0.0 (0.7)
1.7 (0.3)
NE
�98
Table 4: Estimated detection probabilities pij, probability of group occurrence λj and
standard errors for interval i and group j of the four time interval Thrasher data set based
on model Mth. Standard error for group two and time interval two is not estimable.
Group 1
Group 2
Parameter
Estimate pij
SE(pij)
Estimate pij
SE(pij)
p1j
0.21
0.036
0.78
0.030
p2j
0.09
0.090
≈ 1.0
NE
p3j
0.47
0.050
0.84
0.023
p4j
0.56
0.055
0.70
0.026
λj
0.29
0.037
0.71
0.037
�Chapter 4
MULTIPLE SPECIES ANALYSIS OF POINT COUNT
DATA: A MORE PARSIMONIOUS MODELING
FRAMEWORK
�100
MULTIPLE SPECIES ANALYSIS OF POINT COUNT DATA: A MORE
PARSIMONIOUS MODELING FRAMEWORK
Mathew W. Alldredge, Biomathematics and Zoology, North Carolina State University,
Raleigh NC 27695.
Kenneth H. Pollock, Zoology, Biomathematics, and Statistics, North Carolina State
University, Raleigh, NC 27695.
Theodore R. Simons, USGS Cooperative Fish and Wildlife Research Unit, Dept. of
Zoology, North Carolina State University, Raleigh, NC 27695.
Abstract.—Although many population survey techniques provide information on multiple
species, these data are rarely analyzed with multiple species models. We develop a
general multiple species modeling strategy based on exploiting similarities in
capture/detection processes among species in order to more parsimoniously model
multiple species data. Data are derived from point count surveys of breeding birds using
distance sampling, time of detection, and multiple observer methods and the set of
candidate models for each method. A method of grouping species together based on
similarities in detection probabilities is discussed and a set of characteristics important in
developing these species groups for aural detection of songbirds is given. Examples are
presented for each analysis method using nineteen species and six species groups
surveyed in Great Smoky Mountains National Park. Species effect models were
generally the most parsimonious models for the distance sampling method but group
models without species effects were generally “better” for the other two methods.
Population estimates were more precise for group models than for single species models,
demonstrating the benefits of exploiting similarities among species when modeling
multiple species data. Partial species effect models and additive models were also useful
because they modeled similarities among species while allowing for species differences.
�101
Although we present analyses for common species with relatively large observed counts,
we believe that multiple species modeling will be particularly beneficial for modeling
count data from rare species by “borrowing” information on the detection process from
more common species. Partial species effect and additive models may be particularly
useful in this aspect.
Introduction
Most available methods that are available for sampling animal abundance are not
species specific. Methods such as small mammal trapping (Webb 1965, Schwartz and
Whitson 1986, Mengak and Guynn 1987), mist netting birds (Nur et al. 1999), electro
fishing (Meador et al. 2003), and avian point counts (Ralph et al. 1997, Canterbury et al.
2000) are not species specific, and all provide information on multiple species.
Nevertheless, most analyses of abundance data are done for individual species. Because
the capture/detection process may be very similar among species, better estimate
precision and more parsimonious models are possible through analyses that exploit
species similarities.
The point count survey method is commonly used to estimate the relative
abundance or density of bird populations (Ralph et al. 1995, Thompson 2002). Several
recent papers have emphasized the necessity of understanding the detection process and
the limitations of using counts as indexes of abundance (Nichols et al. 2000, Rosenstock
et al. 2002, Farnsworth et al. 2002). The underlying estimator for estimating population
size and density from point counts is (Nichols 1992, Lancia et al. 1994, Williams et al.
2002):
�102
C
Nˆ = pˆ
(1)
i
i
and
C
Dˆ = apˆ
(2)
i
i
where N̂ and D̂ are the population estimates, Ci is the observed count for species i, pˆ i is
the estimated probability of detection for species i and a is the total area sampled.
There are currently three methods available for estimating abundance from
unrepeated point count surveys and modeling the detection process. The distance or
variable circular plot method models the probability of detection as a function of distance
from the observer, and typically provides an estimate of density (Ramsey and Scott 1979,
Reynolds et al. 1980, Buckland et al. 1993). Multipleobserver methods estimate the
probability of detection by each observer using a capturerecapture (independent
observers) (Chapter 2) or removal (primarysecondary observers) framework (Nichols et
al. 2000). Both methods assume all individuals in the sample area are available for
detection. The third approach, the time of detection/depletion method, estimates the
probability of detection over multiple time periods using a multiple detection capturerecapture or a first detection removal framework (Farnsworth et al. 2002, Chapter 3).
This method estimates the combined probability that an individual is available for
detection and that it is detected given that it is available.
The detection process modeled by all three methods is similar. It involves the
bird making itself available for detection, and the ability of observers to detect available
birds. In open habitats the detection of birds is primarily visual and thus availability
�103
depends only on the bird being present and not hidden from view. In forested habitats
detections are primarily auditory and availability depends on an individual bird singing or
calling during the count (Farnsworth et al. 2002) as well as being present. In either case,
the detection process is clearly based on an observer’s ability to detect birds by sight or
sound.
Because multiple species data are generally collected during point counts it is
reasonable to assume that the actual detection process for a given observer is similar for
similar bird species. Brightly colored birds that move regularly are more likely to be
detected visually than more cryptically colored birds with secretive behaviors. Observers
are more likely to detect birds with loud, frequent, high intensity songs then birds with
occasional, quiet, low intensity songs.
The standard approach to analyzing point count data is to estimate the detection
function for each species separately. This approach ignores any similarities among
species (see for example Ralph et al. 1995). When similarities in the detection process do
exist among species, then standard approaches will overparameterize models of the
detection process. Nichols et al. (2000) and Farnsworth et al. (2002) tested for species
effects in their analyses, but they did not develop a multiple species modeling framework.
Frameworks that exploit similarities among species in the detection process should
produce more parsimonious models. These models should also provide more precise
estimates for less common species that typically have small observed counts.
We present a multiple species modeling framework and explore its potential
applications to three methods of analyzing point count data. We begin by discussing the
importance of defining species groups and describe characteristics that can be used to
�104
define these groups. We then briefly review model selection and the principle of
parsimony, which provides the justification for using multiple species models. A general
multiple species modeling strategy is then developed. A brief review of each point count
estimation method is given along with a description of the multiple species candidate
models and the number of parameters required for each. Examples are then presented for
grouping species and analyzing grouped data using three current methods. We conclude
with a discussion of the applicability and benefits of this approach to multiple species
surveys.
METHODS
Field methods.—We assume that standard point count survey techniques (Ralph
et al. 1995) are used to sample multiple species populations. Specific point count
protocols will depend on survey objectives, survey design and the methods of analysis.
For details on field methods for distance sampling refer to Buckland et al. (1993), for
multiple observer techniques see Nichols et al. (2000) and chapter 2, and for the time of
detection approach see Farnsworth et al. (2002) and chapter 3.
Defining the area sampled is necessary for making spatial and temporal
comparisons. This is generally accomplished by using fixed radius plots or estimating
detection distances to all birds observed. Fixed radius plots may be difficult to apply in a
multiple species framework because the appropriate plot radius or maximum detection
distance will vary among species.
Defining species groups.—The objective of a multiple species modeling strategy
is exploit similarities among species to produce more parsimonious models of the
detection process. If only a few species are represented in the sample then analysis can
�105
be done for the entire data set. When sampling methods provide information on a large
number of species it will be beneficial to group species with similar characteristics
together for analysis. For example, species with very loud vocalization detectable at
large distances are likely to have a different detection process than species with quiet
vocalizations.
We suggest constructing a priori groups of species with similar characteristics
associated with the detection process. Using a priori groups will lead to biologically
meaningful hypotheses that can be tested. Using a posteriori groups or “data dredging”
for similar detection probabilities is not recommended because it does not necessarily
produce biologically meaningful groups and models characteristics of a particular data set
rather than the true underlying process. Species should be grouped together based on
similarity in characteristics thought to be important to the detection process. These
characteristics include sound intensity, singing rate, sound pitch, sound modulation,
plumage color, and movement. Their relevance will vary among species and habitats and
whether detections are primarily visual or auditory.
Detection distance, which is relatively easily measured may serve as a surrogate
for some of the characteristics listed above. Using this approach species groups are
constructed by combining species with similar maximum detection distance. In this case
truncating data by 10% of the maximum detection distances is advisable to remove
extreme observations (Buckland et al. 1993).
Sound intensity, or the energy content of a song (Gill 1995), is one of the most
intuitive song characteristics influencing the detection process. Very loud songs are
easily detected, whereas, quiet low intensity songs are less likely to be detected.
�106
Measuring sound intensity during a point count would be difficult because the effect of
distance would be confounded with any measurements taken. Actual measurement of
sound intensity should probably be done separate to the actual point count survey and
should be done at some standardized distance.
Sound pitch, or frequency of a song, can affect the detection process in two ways.
First, an observer may be able to hear sounds in certain frequency ranges better than
those in other ranges. In this situation sounds of similar pitch would have similar
detection probabilities for a given observer. The other factor associated with sound pitch
is how well a sound will travel in certain habitats. Lowfrequency sounds travel better or
are less subject to attenuation in dense vegetation than are highfrequency sounds
(Morton 1975, Wiley and Richards 1882). Grouping species by similarity in sound pitch
would create groups of species subject to similar sound attenuation. Sound pitch may
also be difficult to measure while conducting a point count survey.
Sound modulation, variation in either sound pitch or intensity (Gill 1995), may
also affect the detection process. The reasons sound modulation may affect the detection
process are the same as those discussed for sound intensity and sound pitch. Sound
modulation should also be considered for its affect on a listener’s ability to locate sound.
Songs that cover a broad frequency range are easier to locate than songs that cover a
narrow frequency range (Wiley and Richards 1982, Gill 1995). For example, alarm calls
cover a narrow frequency range and are difficult to locate in comparison to songs that are
used to attract mates. This has important implications to point count surveys as birds
must be located with some degree of accuracy for both fixed and unlimited radius point
counts.
�107
Singing rate is very important for modeling the detection process, especially when
there are a large number of birds singing simultaneously (McShea and Rappole 1997).
Because individual songs are detected with some probability, the more a bird sings the
greater the probability that an observer will detect it. For example, if the probability of
detecting any given song is 0.2 then the probability of detecting that bird at least once
during a count is 0.2 if it only sings once, 0.67 if it sings 5 times and 0.89 if it sings 10
times. If the probability of detecting a single song is low then grouping species by
similarity in singing rates may be useful because these groups will account for
differences among species in the probability of detecting an individual at least once
during the count. If the probability of detecting a single song is high, then this will not
matter because the probability of detecting an individual at least once during a count
approaches one, even with very low singing rates.
Plumage color and movement are important characteristics for point count
surveys when detections are visual, but they may also be important when detections are
primarily aural. Brightly colored birds, or birds that are actively moving during a count
may attract an observer’s attention to a specific area and increase the probability of
detection by ear. More cryptically colored birds or sedentary birds would not provide
this type of cue to an observer.
Using multiple characteristics to define species groups can be done by either
averaging the values across groups for each species or by using a hierarchical approach.
We prefer the hierarchical approach because it gives more weight to the characteristics
thought more important. A hierarchical approach is to first group by the most important
characteristic and then dividing groups further by using additional characteristics. These
�108
hierarchical levels can then be used for between group comparisons as well by starting at
the refined group levels for group comparisons.
Model selection.—Akaike’s Information Criterion (AIC) is an information
theoretic approach to selecting the most parsimonious models (Burnham and Anderson
2002). Models with a large number of parameters may have low bias but they have large
estimation variance associated with every parameter. Models with fewer parameters have
smaller variance associated with estimating parameters but have greater bias.
Parsimonious models provide the “best” balance between estimation variance and bias.
The two components of AIC are deviance from the saturated model, which
controls for bias, and the number of model parameters, which controls for estimation
variance associated with the number of parameters in the model. Using single species
models to analyze multiple species count data does not fully account for the number of
parameters and may lead to overparameterized models. For example, if a single species
analysis is used for 5 species, and a two parameter model is selected as the most
parsimonious model, then the total number of parameters estimated is ten. If the
detection process is similar among all of these species then the most parsimonious
multiple species model would not have species differences, it would only estimate two
parameters, and estimates would be more precise than the single species approach.
Comparing multiple species models to more familiar models that account for
animal level differences (e.g. comparing males versus females or juveniles versus adults)
or observable heterogeneity will illustrate the procedure from a model selection
framework. When capturerecapture experiments are used to estimate abundance of a
single species, models that account for observable heterogeneity, also known as animal
�109
level stratification are generally used (Chapman and Junge 1956, Seber 1982, Lebreton et
al. 1992, Borchers et al. 2002). For example, in some cases capture probabilities differ
between males and females or between adults and juveniles. In this situation one would
fit a model assuming males and females have the same capture probability and another
model assuming differences in capture probability. The most parsimonious model would
be selected based on AIC. Multiple species models are identical in concept to this
capturerecapture modeling approach. One candidate model assumes that all species in a
survey, or all species within a predefined group, have similar detection probabilities. The
alternative model assumes specieslevel differences and accounts for them as observable
differences in detection probabilities.
Multiple species modeling strategy.—The strategy we present is an a priori
hierarchical approach of first examining the within group structure and then, if warranted,
examining the between group structure. The first step in the modeling process is
examining within group structure, testing the assumption that species within defined
groups have similar detection probabilities. A set of candidate models for the within
group structure of the detection process is developed for each point count methodology.
If within group models suggest species differences then we do not proceed because the
number of model parameters will be similar to those for individual species. If no
differences in detection probability are found within a species group, we proceed to
between group models.
Multiple observer and the time of detection methods can model the species effect
as either an additive or an interaction effect. An interaction effect is equivalent to a
species specific analysis where observer or time differences in the detection probabilities
�110
vary among species. Using a linear models framework the interaction model has a
unique slope and intercept for all species in the analysis. The additive species effect
model assumes that there are differences between species but the differences in detection
probabilities between observers or time intervals are consistent between species. Only a
single parameter is necessary to model the difference between species using the additive
model, which is really just an adjustment to the intercept for each species in the analysis
but a common slope is used.
For consistency we will use similar notation to define the models for all of the
methods. Effects seen in the original single species models, such as time (t), observer
(obs), and difference in subsequent detection from first detection (b), will appear as
subscripts. Superscripts will be used to denote models with interaction species effects
(*spp), additive species effects (+spp) or without species affects (0).
Estimation methods and candidate models
Distance.—The distance sampling method (Ramsey and Scott 1979, Reynolds et
al. 1980, Buckland et al. 1993) is probably the most commonly used statistical method
for estimating abundance or density from point count data, primarily because the other
two methods have only recently been developed. Distance sampling involves observers
recording the radial distance to all species seen/heard. Distance sampling assumes that
the probability of detection for a given species is a monotonically decreasing function of
distance from the observer. If we let k be the number of points sampled with effective
search radius w, then equation 2 can be rewritten as (Buckland et al. 1993),
�111
C
Dˆ = kπ 2 pˆ
w
(3)
i
i
The probability of detection ( p
ˆ ) within a circle of radius w is,
i
pˆ =
i
2
∫
w
2 0
w
rg (r )dr ,
(4)
where g(r) is the detection function, which gives the probability of detecting an individual
given its radial distance (Buckland et al. 1993).
Distance methodology is based on the estimation of g(r), which determines the
number of parameters in the model. Buckland et al. (1993) recommend a class of reliable
models based on model robustness, shape criterion, and estimator efficiency. The
recommended model is based on a key function and possibly a series expansion to adjust
the key function, of the general form,
g (r ) = key(r )[1 + series(r )]
(5)
The recommended key functions for consideration are: the uniform, the halfnormal, and
the hazardrate, which have 0, 1, and 2, parameters respectively. Each series adjustment
term used in the model adds an additional parameter. Buckland et al. (1993) suggest that
the key function is often adequate for properly truncated data. In other cases one or two
adjustments may be necessary.
The assumptions for point count distance sampling are (Buckland et al. 1993):
1. g(0) = 1, certain detection at the point.
2. Objects are detected at their initial location.
3. Distances are measured accurately or placed in the correct distance categories.
4. Objects are detected independently.
�112
An additional underlying assumption is that all individuals within the sampled area are
available for detection visually or aurally or that inferences are only made about the
available portion of the population (Marsh and Sinclair 1989, Farnsworth et al. 2002).
This assumption is often overlooked but could be important for some species.
The distance sampling methodology has the most limited set of models for
examining multiple species data, because the method only estimates a detection function
relative to distance from the observer. Therefore, comparisons among species are
constrained to comparisons of the functions used to estimate abundance.
Following the general approach discussed above, the two candidate models for the
within group detection process are: 1) the same detection function for all species within
group j (no species effect) and 2) different detection functions for all species within group
j (species effect). If the “best” model for the groups indicates different detection
functions for all species, then among group comparisons are not necessary. If detection
functions are found to be the same for all species within a group, then among group
comparisons are recommended.
The number of parameters used when examining each species separately is just
the sum across all species of the number of parameters in each species specific detection
function. With a multiple species approach, the number of parameters used is the same if
the selected model represents different detection functions within groups. When a single
detection function is selected for all species within a group, then the number of
parameters is the sum of the number of parameters in each groupspecific detection
function. Parameters are further reduced if among group comparisons suggest that
common detection functions are appropriate.
�113
Time of detection.—The time of detection method was first proposed by
Farnsworth et al. (2002) as a removal method in which the data consists of the times of
first detection. A more general approach uses the full detection history of every
individual, applying the full set of capturerecapture models (Otis et al. 1978) as a
framework for analysis. Excluding the behavior models (Otis et al. 1978), the full time of
detection approach is generally more efficient (smaller variance) than the constrained
removal approach (Chapter 3 and see Seber 1982 pg.570). With this method, models
determine the probability of detection between time intervals of a point count. Ralph et
al. (1995) recommended using 3, 2, and 5 minute intervals so that future data are
comparable to other studies using these time intervals. Unlike other methods of
estimating detection probabilities from point count data, the time of detection method
actually estimates the product of the probability that an individual is available for
detection during a count, and the probability that an individual is actually detected given
that it was available (Farnsworth et al. 2002, chapter 3). This can be seen by examining
first detections in the second interval. These detections could either be of individuals that
were not available for detection in the first interval but are in the second, or of individuals
that were available in both the first and second intervals but were not detected in the first.
Note that both movement and singing rates can affect the availability process. The time
of detection approach assumes that movement is not a factor.
The assumptions for this approach are (Farnsworth et al. 2002, chapter 3):
1.
There is no change in the population of birds within the detection
radius during the point count (closed population).
2.
There is no doublecounting of individuals.
�114
3.
Detection distances are measured accurately, or if limited radius counts
are used, observers accurately assign birds to within or beyond the
radius used.
There are 8 general capturerecapture models (Otis et al. 1978, White et al. 1982,
Pollock et al. 1990), all of which may be applicable to the time of detection approach.
The capturerecapture models and a description relevant to the time of detection method
are (chapter 3):
Model M0: Equal detection probability for all individuals among all time periods.
Model Mt: Equal detection probability for all individuals but different detection
probabilities among time periods.
Model Mb: Equal probability of first detection for all individuals among all
periods and a unique probability of subsequent detections that is equal for
all individuals among all subsequent periods.
Model Mtb: Equal probability of first detection for all individuals but different
among periods and a unique probability of subsequent detection that is
equal for all individuals but different among subsequent time periods.
Model Mh: Unique probability of detection for each individual that remains
constant among all time periods.
Model Mth: Unique probability of detection for each individual that differs
among time periods.
Model Mbh: Unique probability of first capture for each individual that remains
constant among time periods and a unique probability of subsequent
detection that remains constant among time periods.
�115
Model Mtbh: Unique probability of first capture for each individual that differs
among time periods and a unique probability of subsequent detection that
differs among time periods.
Models Mb and Mbh are the removal models given by Farnsworth et al. (2002).
The full heterogeneity models (Burnham and Overton 1979, Chao et al. 1992,
Norris and Pollock 1996, Pledger 2000) are only applicable when four or more time
intervals are used. With three time periods it is possible to fit constrained twopoint
mixture models (Norris and Pollock 1996, Pledger 2000), by setting the detection
probability for one of the heterogeneity groups to a constant. The assumption here is that
some birds in the point count are very easily detected while other birds are harder to
detect. Farnsworth et al. (2002) made the assumption that the probability of detection for
one of the heterogeneity groups was one, and then estimated detection probabilities for
the other heterogeneity group. An alternative is to use information from other studies,
such as a four interval study, to set a more realistic value for the high detection
probability (chapter 3).
The structure of the time intervals used in the survey will determine how the
detection process is modeled with this approach (Chapter 3). If equal time intervals are
used, then the interval detection rate can be estimated directly using standard capturerecapture software for analysis, such as programs CAPTURE (White et al. 1982) or
MARK (White and Burnham 1999). When unequal time intervals are used then it is
necessary to model the detection rate as a function of time to allow for models with
constant detection rates. The detection rate can be modeled as a discrete per minute rate
(γi) (Farnsworth et al. 2002, chapter 3)
�116
p =γt
i
(6)
i
i
or as an instantaneous rate (φi) (chapter 3)
p =1 − e ϕ it i
−
(7).
i
For these situations the model likelihoods are not in the form required by the standard
capturerecapture programs. Therefore, it is necessary to parameterize the likelihood
using either discrete or instantaneous rates formulation that is maximized using a
program such as SURVIV (White 1983).
Models with no time effect assume a constant detection rate and interval
probabilities varying as a function of the interval length. The examples provided for this
section are from point counts with three unequal time intervals. We will also use the
instantaneous rates formulation (see chapter 3 for more details).
The most general model is Mtbh, and we present the likelihood for this model
using a twopoint mixture for heterogeneity. All other models are just constrained forms
of this general model. A point count with i intervals has 2i capture histories, one of
which is not observable because it represents those individuals never detected. For a four
interval point count there are 15 observable detection histories (w) with observed counts
(xw). The expected value for the number of birds detected in every interval is given by:
E ( x1111) =
N
(λ p
11
c c c
21
31
41
+ (1 − λ ) p
12
c c c
22
32
42
)
(8)
where λ is the proportion of animals in group one. The proportion of animals in group
two is 1λ, and the pij’s and cij’s are the probabilities of first detection and subsequent
detection in the ith interval for individuals in the jth group. The remaining expected values
are written similarly by modifying equation 8 to account for intervals where individuals
�117
are not detected. Using the rates formulation, the pij’s and cij’s are replaced by one of the
rate equations (equations 6 or 7). The conditional likelihood L(pij, cij, λ  xw) for the twopoint mixture model Mtbh is given by (chapter 3):
w ⎛
⎞ xw
⎛
+
xT ⎟⎞ ∏
π
w1 π w 2
=
(
,

)
⎜
⎟
⎜
p
L2 i ci xw ⎜ x … x ⎟ 1 ⎜
π • ⎟⎠
0001 ⎠
⎝ 1111
⎝
(9)
where xT is the total observed count, π. is the probability of being detected at least once
during the entire count, πw1 is the probability of being in the first group and having
capture history w, and πw2 is the probability of being in the other group and having
capture history w (chapter 3).
Maximizing the above likelihood for the observed data will give estimates of the
the detection rates (φij’s) and probability of being in one of the groups (λ), which can be
used to estimate the probability that an individual is detected at least once during a four
interval count as;
pˆ
T
(
)(
− ˆ
− ˆ
= 1 − ⎡λˆ 1 − e ϕ 11t 1… e ϕ 41t 4 + 1 − λˆ
⎢⎣
) (1 − e ϕˆ t …e ϕˆ t )⎤⎥⎦
−
12 1
−
42 4
(10)
The variance of pˆ T can be obtained by reparameterizing the model to include pˆ T as a
model parameter and thus, obtain an estimate of the variance for probability that an
individual is detected at least once. Abundance can then be estimated using the total
observed count (xT) as;
Nˆ =
x
pˆ
T
T
The variance is
(13)
�118
∧
Var ( Nˆ ) =
∧
xT Var ( pˆ )
2
T
pˆ
4
T
+
x
T
(1 −
pˆ
pˆ )
T
2
(14)
T
assuming that the observed count (xT) is from a binomial distribution (Nichols et al.
2000, Williams et al. 2002). All other models are obtained by constraining this general
form.
The set of candidate models using the time of detection method for multiple
species analysis has 14 models, although we propose 6 additional models. The 14
candidate models are obtained by using the seven single species models (chapter 3) and
analyzing them with or without a species effect. Three of the additional models are
variations of time and species effect models. One approach is allowing time specific
detection probabilities to vary independently among species so that there is no
relationship between the detection probabilities of different species. A second approach
provides for species differences, but the variations in detection probabilities with time are
consistent across species. If time specific detection probabilities are known for one
species, then a single species adjustment is added to obtain the detection probabilities for
the other species. In other words the first approach assumes a time and species
interaction, and the second approach is an additive effect related to species differences.
The other models we propose are individual heterogeneity models, which allow
for differences in detection probabilities among individuals. Model M 0h assumes that all
detection probabilities and probabilities of being in the first heterogeneity group are the
same for all species. Models M*spp
and M +spp
assumes that detection probabilities and
h
h
probabilities of being in the first heterogeneity group are different among species. We
�119
suggest that another reasonable model is M part
h , which assumes that detection
probabilities are the same for all species but that the probability of being in the first group
is not the same for all species. Differences in singing rates related to breeding stage are
one source of individual heterogeneity. Model M hpart describes a process in which the
probability of detection is similar among species at various breeding stages but that the
proportion of the population of each species in a given breeding stage varies.
The number of parameters for each of these models is given in Table 1.
Comparing the number of parameters for models without species effects to those with
species effects demonstrates the reduced number of parameters that can be achieved by
using multiple species models.
Multiple observers.—There are two approaches to the multiple observer method;
the dependent observer approach (Nichols et al. 2001) and the independent observer
approach (Chapter 2). The two approaches are very similar except that the dependent
observer approach treats the data as a removal method and the independent observer
approach utilizes the full capture history. The removal method (Nichols et al. 2000) uses
2 observers, one primary and one secondary. The primary observer records all
individuals detected and communicates this to the secondary observer. The secondary
observer records individualsnot detected by the primary observer. The independent
observer approach (chapter 2) differs in that all observers conducting the count record all
detections independent of the other observers. For a 2 observer count there are 3
potential capture histories (x11—seen by both, x10 and x01 seen by only one) instead of the
2 histories from the dependent observer approach.
�120
The multiple observer method of estimating abundance from point count data is
actually a direct application of capturerecapture models and can be analyzed with
programs, such as CAPTURE (White et al. 1982) or MARK (White and Burnham 1999).
Therefore, we will not present the likelihoods but simply describe the method and discuss
potential models. For more details on capturerecapture models refer to Otis et al.
(1978), Pollock et al. (1990), or Williams et al. (2002).
The assumptions for the multiple observer models are:
1. Counts within the fixed radius circle are measured accurately.
2. There are no matching errors between the observers so that the assignments to
x11, x10, and x01 are accurate.
3. Equal detection probability of each species at all points for each observer,
except with heterogeneity models.
4. There is no undetected movement into or out of the fixed radius circle.
An additional assumption for the dependent observer approach is that the primary
observer detects individuals independent of the secondary observer. The independent
observer approach assumes that observers detect individuals independent of one another.
We will only use the independent observer approach because this is the more
general method. The full suite of capturerecapture models presented in the previous
section are also parameterized for these types of data. However, the models with
behavioral effects are biologically unreasonable for this application (Chapter 2), because
all observations are done simultaneously, and observers make detections independently.
Therefore it is unreasonable to assume that detection by one observer affects detections
by other observers.
�121
For two observers the approach is in the form of a LincolnPetersen closedpopulation capturerecapture survey (chapter 2). If we let x11 be the number of
individuals detected by both observers, n1 be the number detected by observer 1, and n2
be the number detected by observer 2, we can write the probability of detection for each
observer as;
pˆ
d1
=
x
n
11
2
and
pˆ
d2
(15)
=
x ,
n
11
1
The probability of detection by at least oneobserver is,
pˆ = 1 − (1 − pˆ )(1 − pˆ
d
d1
d2
)
.
(16)
Using the canonical form (equation 1) we can then estimate abundance. This is done
using program MARK (White and Burnham 1999) with the Huggins Closed Captures
data type, which estimates the probability of detection for each observer and abundance
as a derived parameter. The number of parameters in a single species model for t
independent observers is 1 for model M0 and t for model Mt.
Individual heterogeneity in detection probabilities is presumably important in
point count data regardless of the analysis method used (Farnsworth et al. 2002, Chapter
2 and 3). It is possible to account for observable heterogeneity for two or more observers
using covariates, such as detection distance (Chapter 2). With four or more observers it
is possible to account for unobservable heterogeneity using heterogeneity estimators such
as the Jackknife (Burnham and Overton 1979), the Chao sample coverage estimator
(Chao et al. 1992) or the finite mixture estimators of Norris and Pollock (1996) and
�122
Pledger (2000). For illustrative purposes in this paper we will use a fourindependent
observer data set and twopoint mixture estimators of individual heterogeneity in
detection probabilities.
There are 12 candidate models for the multiple species independent observer
approach, excluding any covariate models (Table 2). These models are obtained by
adding a species effect to the four single species candidate models. Observer and species
effects are modeled two different ways. The first approach, which is equivalent to the
single species approach, assumes that observer effects differ across species and it models
the probability of detection for each observer separately for each species (denoted with
superscript *spp in the models). The second approach assumes that observer and species
effects are additive. It assumes the probability of detection varies among observers but
that the differences among observers is consistent across species (denoted with a
superscript +spp). This situation might arise when one observer is more skilled than the
other and consistently detects a larger proportion of the birds at a point. Detection
probabilities are estimated for each observer and a single parameter is used to adjust the
detection probabilities for each subsequent species. We also consider a partial individual
heterogeneity effect, similar to that used in the time of detection models. This partial
heterogeneity effect allows detection probabilities to remain constant among species but
the probability of being in the first heterogeneity group varies among species.
The number of parameters estimated for each model is provided in Table 2. Like
the time of detection method, the number of parameters required for models without a
species’ effect is less than the number required for models with a species effect.
�123
Field Data
To demonstrate the multiple species modeling approach we use point count data
for songbirds collected in Great Smoky Mountains National Park during the breeding
season. Survey points were located along low use hiking trails with a minimum 250 m
separation between points. Surrounding vegetation was closedcanopy deciduous
hardwood forest. Because of this detections were primarily aural (over 95%, Simons
unpub. data). All point counts were conducted between dawn and 10:15 in the morning
on days with good weather (no rain or excessive wind).
All observers were highly trained prior to conducting point counts. Training
included identification skills, familiarity with birds occurring in the area, and distance
estimation skills. Observers used laser range finders to delineated a 50 m radius circle
prior to starting each point count and to verify distance estimates during the count.
The time of detection data were extracted from point count surveys conducted
during May and June from 1996 to 1999. To avoid temporal and observer effects we
only used data from 1998 collected by one of the most experienced observers. This
restricted data set consisted of 323 survey points. We also omitted species with fewer
than 50 observations from analysis. Observation periods were delineated by birds
detected in the first three minutes, next two minutes, and final five minutes of a ten
minute count. Multicolor pens were used to denote the time of initial detection and
subsequent detections were recorded by underlining previous detections in the
appropriate color. The example data set for the distance method was obtained by using
only the observations recorded in the first 3 minutes of the time of detection data set.
�124
A fourindependent observer data set was collected in the same area during June
of 1999. All observers were highly trained and had been conducting point counts for six
weeks prior to this survey. Counts were conducted at 70 points and followed the same
protocol for weather conditions and distance measurements. Observers conducted
variable circular plot 3 minute point counts (Reynolds et al. 1980) and mapped the
location of each bird detected at the point. Attempts were made to track each bird during
the point count to avoid double counting and to simplify matching observations among
observers. Following each point count observers combined their data to determine the
total number of birds detected and identify birds seen in common.
Species groups.—Species groups were defined by obtaining rankings from 7
experts familiar with the species and study area used for our examples. Ranks were done
on a scale of 1 to 5 with 1 indicating the lowest probability of detection or smallest value
for a given characteristic and 5 indicating the highest probability of detection. These
ranks were then averaged for the 7 experts that provided ranks. Maximum detection
distance for each species was also used to define species groups.
We used the maximum detection distance, following 10% truncation of the largest
distances, as the first criterion for defining species groups. Because many characteristics
(sound intensity, pitch, modulation, etc.) can affect maximum detection distance, we
defined three groups: 1) species with maximum detection distance ≤ 100 m, 2) species
with maximum detection distance > 100 m and ≤ 150 m, and 3) species with maximum
detection distance > 150 m.
Because maximum detection distance may contain information about sound
intensity, sound pitch, and sound modulation, we chose to use singing rate to further
�125
refine these groups. Birds with similar singing rates within a distance group were used as
the final groups. Further refinement of groups was not necessary.
Distance analysis.—We used program DISTANCE (Thomas et al. 2002) to
analyze the distance data. Separate analyses were done for each group to determine if a
species effect was present within each group. Data format for DISTANCE was standard
except that the species identification had to be entered as an observation level variable,
which is the level at which detection distances are entered. This allowed for post
stratification by species (Rosenstock et al. 2002).
Data were truncated at 10% of the maximum detection distances for all analyses,
as recommended by Buckland et al. (1993). For each group an analysis was run for no
species effects and for species effects. For each of these effects the following key
functions and adjustments were run:
1. Halfnormal key function – Cosine adjustment
2. Halfnormal key function – Simple polynomial adjustment
3. Uniform key function – Cosine adjustment
4. Uniform key function – Simple polynomial adjustment
5. Hazard rate key function – Cosine adjustment
The appropriate key function and adjustment model are selected using AICc (AIC
corrected for small sample size) for both the species effect and no species effect models.
AICc was then used to choose between the species effect and no species effect models.
Between group comparisons were not warranted. We use the effective detection radius
(EDR) and the density estimate (D) to compare models with and without a species effect.
�126
Time of detection analysis.—Time of detection data were analyzed with program
SURVIV (White 1983) to estimate the detection parameters and twopoint mixture
heterogeneity parameters since the data are for three unequal time intervals and assumed
the instantaneous rates formulation (equation 7) to model the interval detection
probability. All 17 candidate models were initially run, but both of the models with time
and behavior were omitted from the final analysis because all parameters were not
identifiable. Selection of the most parsimonious model was done using AICc.
Population estimates and standard errors were derived from the estimated
detection probabilities and heterogeneity parameters (equations 1014). We report the
probability of detecting an individual at least once during the count (pT), the
heterogeneity parameter (λ), and the population estimates for the selected model and the
alternative species effect model.
Multiple observer analysis.—We used program MARK (White and Burnham
1999) with the “Huggins Closed Captures” and “Huggins Full Heterogeneity” data types
to analyze the fourindependent observer data. Twopoint mixture models were used for
the heterogeneity models because estimates from threepoint mixture models were not
reasonable. Model selection was based on AICc.
We report the heterogeneity parameter estimate, the observer specific detection
probabilities, and the population estimate for the selected model. We also report the
population estimate for “best” species effect model to demonstrate the improved
precision when models with no species effect were selected.
�127
Results
Species groups.—There were 19 species from the time of detection/distance data
sets selected for analysis. Three groups were defined in the ≤ 100 m category, two
groups in the 100 m to 150 m category, and one group in the over 150 m category, based
on similarities in singing rates (Table 3). Group sizes ranged from two to four species.
Because fewer points were sampled using the independent observer method, only
8 species had sufficient sample size for analysis and species groups were modified. No
analysis was done for group A, the Blackthroated Blue Warbler and Indigo Bunting had
to be omitted from groups B and C, respectively, and the Ovenbird, Scarlet Tanager, and
Tufted Titmouse were the only species used from groups D, E, and F to form the
combined group DEF.
Distance.—Species effects models were selected as the most parsimonious
models based on comparison of ∆AICc values for five of six species groups (Table 4).
These values reflect the difference between the AICc value of a particular model and the
model with the lowest AICc value. The model with ∆AICc of zero was the most
parsimonious model and competing models were those with ∆AICc <2 (Burnham and
Anderson 2002). When the species effect model was selected the no species effect model
had ∆AICc >2 indicating the species effect model was required to sufficiently explain the
variation in the data. The single instance when the no species effect model was selected
∆AICc=1.76 indicating that both the species effect and no species effect models had
similar ability to explain the variation in the data.
�128
All species in group A used a uniform key function, and three of the four had a
simple polynomial adjustment term. All species in group B used the hazard rate key
function, except the Darkeyed Junco, which used a halfnormal key function. In group
C the hazard rate key function was used for the Blackthroated Green Warbler and the
uniform key function was used for the other two species. The uniform key function was
used for all species in groups D and E, and the Hazard rate was used for all species in
group F. In general the shape of the detection function was similar among all species in a
group.
The effective detection radius did vary within groups except for group D which
selected the no species effect model (Table 5). Differences in the effective detection
radius were >10 for all other groups. These differences affected the shape of the
detection function when data were pooled across species. For example, for group F the
species specific detection function was a hazard rate with a cosine adjustment term for all
species, but the detection function for the group was a halfnormal with a cosine
adjustment term. In contrast, a uniform detection function with a cosine adjustment term
was selected for the species specific detection function and the group detection function
for group D.
Group D demonstrates the benefits of the multiple species approach. The
standard errors were generally smaller for the no species effect model for all groups but
these estimates were biased for all groups except group D. Density estimates for group D
were identical among species effect and no species effect models, but the standard errors
were smaller for the no species effect model (Table 5). The increase in precision is
especially obvious for the Veery, which had a smaller observed count.
�129
Time of detection.—Individual heterogeneity models explained the data more
parsimoniously than models that did not account for heterogeneity (Table 6). Individual
heterogeneity models were selected for groups A and D with no time effect and no
species effect on detection probabilities. Group A did not have a species effect on the
heterogeneity parameter, but group D did. Selected models for all other groups were
heterogeneity models with a time effect on the detection probabilities, groups B, C, E,
and F. Species effects on detection probability and the heterogeneity parameter were also
important in explaining the variation in the data for groups B and E. Species differences
in detection probabilities and the heterogeneity parameter do not appear in the selected
model but they may be important because the species effects model was very similar in
its ability to explain variations in the data (∆AICc weights 0.35 and 0.33, respectively).
The selected model for group F had a species effect on the heterogeneity parameter but
not on the detection probabilities. Models with both a time and behavior effect (change
in detection probability following initial detection) never gave reasonable parameter
estimates (standard errors >>1 for probabilities). Note that for some species groups, there
are alternative competing models to explain the data based on similar ∆AICc values and
similar ∆AICc weights. No models were analyzed to compare between groups because
similar models were not selected for similar groups. For example, groups A, B, and C
were all in a similar distance category but the models ranged from a heterogeneity model
with no species or time effects to one that incorporated both of these effects.
The estimated detection probabilities ranged from 0.81 (group F) to 0.92 (group
E) (Table 7). The estimated heterogeneity parameter ranged from 0.15 to 0.79, both of
these estimates occurred in group F. Models that used common parameters among
�130
species in a group showed increased precision for all parameter estimates. This is most
clearly seen in the increased precision for the abundance estimates, especially for species
with smaller observed counts.
Independent observer—Heterogeneity models explained the data in a more
parsimonious manner for all three independent observer species groups than models not
incorporating heterogeneity (Table 8). The heterogeneity model with no observer or
species effects was the selected model for group B and was a reasonable alternative
model for the other two groups (∆AICc weights ≥ 0.22). The heterogeneity model with
no observer or species effect of the probabilities of detection and a species effect on the
heterogeneity parameter was selected for group DEF and was also a reasonable
alternative model for group C (∆AICc weight = 0.26). An individual heterogeneity model
with an observer effect on the probability of detection and no species effect was selected
for group C. Selection for these simpler (fewerparameter models) may have been
partially because of the small observed counts in these data sets. Because the
heterogeneity model with no observer or species effects was a reasonable model for both
groups B and C, an additional model was run to determine if these groups could be
combined. The group effect model was selected (∆AICc = 13.8) suggesting that group
differences were important sources of variation in the data.
The heterogeneity parameter ranged from 0.34 to 0.58. The detection probability
was generally > 0.90 for the high detection probability heterogeneity group for all species
groups, with the exception of the detection probability of observer 3 in group C. The
detection probabilities for the other heterogeneity group were ≤ 0.36 for all species
groups. Comparing the abundance estimates for the selected model to the abundance
�131
estimates for the “best” species effect model shows very similar estimates but
considerably smaller standard errors for the selected models with no or partial species
effects.
Discussion
Application of multiple species models to population surveys offers a promising
approach to analyzing data when more than one species occurs in the sample. We have
demonstrated that following a multiple species modeling approach will give more
parsimonious models and better precision of estimates for species with similar detection
processes. Evidence from the analyses presented here clearly indicates that in many
cases single species analyses are overparameterized.
The multiple species analysis procedure allows for direct comparison between
models with and without species effects to determine if group based parameter estimates
are warranted. Another approach to multiple species analyses simply assumes
similarities among species and analyzes them as a group. This has been done for less
common species when observed counts are not sufficient for a single species approach
(Nichols et al. 2000). Testing this assumption in a multiple species analysis strengthens
inferences about the population.
It is important to consider the effect of small counts on model selection. When
observed counts are small, then estimation variance for model parameters will be larger
and model selection will tend to select models without species effects. This results in
greater bias but smaller estimation error. In other words, estimates may be precise but
not accurate.
�132
We feel that the additive species effect models are very important in situations
where observed counts are small for some species. This is done by “sharing” information
for estimation of parameters among species but still incorporates species effects.
Consider a simple case for a common and a rare species. When a multiple species
approach is used the precision of estimates will not be affected much for the common
species, but they will be improved for the rare species. As an example, assume that the
count was done using multiple observers and that the “true” underlying detection
processes are different among species. An additive model that assumes a constant
difference in the detection process among species accounts for this situation. Differences
in detection probabilities among observers are estimated using the full data set (both
species), which primarily is based on the species with a large count. In these cases only a
single parameter is needed to estimate the detection probabilities for the rare species.
The model “borrows” information from similar species to estimate the detection process
and it gives more precise estimates than a single species approach. Similar data
“sharing” was also demonstrated in models with a partial species effect where the
detection probabilities were constant among species but the heterogeneity parameters
were modeled with a species effect.
Individual heterogeneity models were always selected as the most parsimonious
models to describe the data. There are many sources of variation that cause heterogeneity
in animal surveys (Burnham 1981, Johnson et al. 1986). Temporal variation in singing
rates (Wasserman 1977, Lein 1981) is a potential source of heterogeneity. Habitat, local
abundance and proximity of observers has also been shown to affect singing rates of
breeding songbirds (McShea and Rappole 1997). Further investigation is needed to
�133
determine the effect of singing rate on detection probability and to identify and account
for other sources of heterogeneity.
Multiple species models were not useful in conjunction with the distance
sampling method. This is in part because these models have few parameters to begin
with. The species effect models had at most 3 parameters, so the reduction in model
parameters was not as great as in the other methods.
We feel that one of the problems with using the multiple species approach in
conjunction with distance sampling is related to species differences in the detection
radius. Pooling data for species with different effective detection radii changes the shape
of the detection function and can make it harder to fit. In our analyses, the only model
selected without a species effect was a group model where the effective detection radii
where almost identical among species. If singing rates significantly affect detection rates
then the assumption of declining detection probability with increasing detection radius
may also be violated. For forest songbirds, including Ovenbirds and Wood Thrushes,
singing rates have been shown to decrease with proximity of an observer (McShea and
Rappole 1997). If singing rates are important to the detection process then the use of
distance sampling may be problematic and multiple species models may not work if
observer effects differ among species.
The multiple species approach was beneficial and provided more precise
estimates of abundance for both the time of detection and multiple observer methods.
These methods worked well because they were exploiting similarities among species in
song structure and singing behavior and they were not as sensitive to detection distance.
Individual heterogeneity models proved to be the most applicable to these data. If
�134
singing rate is an important factor affecting aural detection of birds then mixture models
of individual heterogeneity are particularly applicable because heterogeneity groups will
reflect the proportions of the population in various breeding stages. Several studies have
documented changes in singing rates relative to the breeding stage of an individual
(Wasserman 1977, Lein 1981). Partial heterogeneity models may also be useful because
they model similarities among species detection probabilities but they do not restrict the
proportion of the population in each heterogeneity group.
We used both quantitative and qualitative information to define species groups.
The use of maximum detection distance worked well to categorize species into broad
groups but it did not allow complete classification of species groups. Singing rate
information created reasonable species groups within the distance categories. Using
assumed ranks is not the most desirable situation. People’s perception of species
detectability can be drastically different. For example, Farnsworth et al. (2002)
suggested that the Acadian Flycatcher was one of the more detectable species in their
surveys in Great Smoky Mountains National Park and attributed this to high singing
rates. Our average ranking from seven experts suggests that the Acadian Flycatcher
actually has one of the lower singing rates of the birds used in our analysis. A more
direct measure of characteristics used in defining species groups would be beneficial to
future studies, but would require extra field effort to collect the required data.
Recommendations
We believe that a multiple species modeling approach should be strongly
considered in the analysis of any data collected on multiple species. Because two of the
�135
methods presented here are based on the closed population capturerecapture models, the
benefits to analysis of similar data types and candidate models is clear. Development of
candidate models and gains in precision need to be investigated for other data types, such
as CormackJollySeber (Seber 1982, Williams et al. 2002) and tag return
models(Brownie et al. 1985, Williams et al. 2002).
Although the additive models were not selected in the analyses presented here
there were some cases where they were reasonable alternative models. The partial
heterogeneity models did demonstrate the benefit of “sharing” information to obtain more
precise parameter estimates. Application of these types of models for multiple species
analysis should be investigated further as they do not require the entire detection process
to be similar among species. The use of covariates, such as detection distance, as
additive effects in multiple species models should also be investigated.
�136
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�140
Table 1: Number of parameters in candidate models for the time of detection method
with t time periods and s different species. Behavior models assume a single behavioral
response and heterogeneity models are based on a 2 point mixture. Models with a
superscript +spp indicate an additive effect between observers and species and models with
a superscript *spp indicate an interaction effect between observers and species. Models
with superscript part indicate those that have similar detection probabilities between
species but different probabilities of being in the first group. An additional parameter is
required for each model to estimate abundance.
No Species Effect
Model
# Parameters
0
M0
1
M 0t
t
0
b
2
M
M 0tb
t+1
M 0h
3
M 0bh
5
M
0
th
2t+1
Species Effect
Model
# Parameters
spp
M0
s
M +spp
t
M
∗spp
t
spp
b
M
M
M
+spp
tb
∗spp
tb
part
h
M
M
M
M
spp
h
part
bh
spp
bh
part
th
+spp
th
∗spp
th
t+s1
ts
2s
t+s
s(t+1)
s+1
3s
2s+1
5s
M
ts+1
M
2t+2s1
M
s(2t+1)
�141
Table 2: Number of parameters in candidate model for the multiple observer method
using t observers, s species and heterogeneity models based on a 2 point mixture model.
Models with superscript part indicate those that have similar detection probabilities
between species but different probabilities of being in the first group. Models with a
superscript +spp indicate an additive effect between observers and species and models with
a superscript *spp indicate an interaction effect between observers and species. An
additional parameter is required for each model to estimate abundance.
No Species Effect
Model
# Parameters
0
M0
1
M
0
obs
M 0h
M
0
obs,h
t
3
2t+1
Model
M spp
0
+spp
M obs
Species Effect
# Parameters
s
t+s1
∗spp
obs
part
h
s+2
spp
h
3s
M
part
obs,h
s+t
M
+spp
obs,h
3s+2t2
M
∗spp
obs,h
s(t+1)
M
M
M
ts
�Table 3: Species groups for example analyses, grouped first into three maximum detection distance categories (≤ 100 m, > 100 m and
≤ 150 m, or > 150 m) and then grouped by similarity in singing rates. Maximum detection distance is from the actual data and is
truncated by 10% of the largest detection distances. All other categories are averages from rankings on a scale of 1 to 5 from seven
experienced birders familiar with the study area. Higher ranks correspond to assumed higher values for each category.
Maximum
Distance
75
80
75
80
Singing
Rate
2.6
2.6
2.6
2.0
Sound
Intensity
3.3
2.4
1.1
2.9
Sound
Pitch
3.4
3.7
3.9
2.4
Sound
Modulation
2.7
2.9
2.7
1.4
Plumage
Color
1.1
3.4
1.7
1.3
Movement
B
Blackthroated Blue Warbler
Darkeyed Junco
Hooded Warbler
Solitary Vireo
80
100
100
85
3.6
3.7
3.4
3.3
3.1
3.1
4.0
3.4
2.9
3.0
3.0
3.3
3.1
1.7
4.0
3.4
2.9
2.3
3.9
2.0
3.3
2.9
3.0
2.6
C
Blackthroated Green Warbler
Indigo Bunting
Redeyed Vireo
100
70
100
4.0
4.0
4.7
3.4
3.9
3.6
3.3
3.0
3.0
3.9
3.4
2.9
3.1
3.7
1.4
3.0
2.7
2.4
D
Scarlet Tanager
Veery
130
150
3.1
2.9
3.9
3.0
2.4
2.6
2.7
3.6
5.0
1.1
2.6
1.6
E
Ovenbird
Rufoussided Towhee
110
115
4.1
3.6
5.0
4.0
2.4
3.0
2.3
3.4
1.4
2.4
2.9
2.6
F
Redbreasted Nuthatch
Tufted Titmouse
Winter Wren
Wood Thrush
200
160
200
200
2.0
3.9
3.9
2.9
2.7
3.9
4.0
4.3
3.0
3.3
3.9
3.0
1.4
2.1
3.9
4.3
2.6
2.1
1.3
1.9
4.1
3.7
2.1
2.1
Group Species
Acadian Flycatcher
Blackandwhite Warbler
A
Goldencrowned Kinglet
Wormeating Warbler
1.9
3.7
4.4
2.7
142
�143
Table 4: delta AICc for distance models using first 3 minute interval of time of detection
data.
Species Group
A
B
C
D
E
F
No Species Effect
6.01
10.40
6.21
0.00
9.12
5.56
Species Effect
0.00
0.00
0.00
1.76
0.00
0.00
�Table 5: Distance analysis for first 3 minutes of 10 minute point count. Results given for model with species effect and for model
with no species effect. Observed count is after 10% truncation of largest observed detection distances, EDR is the effective detection
radius and density is individuals per hectare. Standard errors are in parentheses.
Group
A
Species
Acadian Flycatcher
Blackandwhite Warbler
Goldencrowned Kinglet
Wormeating Warbler
Obs. Count
58
90
64
56
Species Effect
EDR
Density
53.9 (6.86)
0.20 (0.041)
43.5 (1.63)
0.47 (0.072)
47.1 (2.34)
0.29 (0.061)
46.9 (3.14)
0.25 (0.056)
B
Blackthroated Blue Warbler
Darkeyed Junco
Hooded Warbler
Solitary Vireo
145
109
192
98
59.9 (2.16)
49.2 (2.75)
64.5 (4.41)
52.8 (4.88)
0.40 (0.060)
0.44 (0.080)
0.46 (0.078)
0.35 (0.079)
C
Blackthroated Green Warbler
Indigo Bunting
Redeyed Vireo
273
40
270
77.6 (2.98)
54.5 (2.03)
78.8 (6.11)
D
Scarlet Tanager
Veery
114
39
E
Ovenbird
Rufoussided Towhee
F
Redbreasted Nuthatch
Tufted Titmouse
Winter Wren
Wood Thrush
No Species Effect
EDR
Density
0.25 (0.022)
0.38 (0.035)
48.2 (1.22)
0.27 (0.025)
0.24 (0.021)
59.5 (6.57)
0.40 (0.094)
0.30 (0.070)
0.53 (0.124)
0.27 (0.063)
0.45 (0.049)
0.13 (0.027)
0.43 (0.075)
77.2 (1.96)
0.45 (0.036)
0.07 (0.005)
0.45 (0.036)
70.6 (1.85)
69.0 (2.54)
0.23 (0.029)
0.08 (0.018)
70.1 (1.51)
0.23 (0.025)
0.08 (0.009)
328
54
72.3 (1.69)
54.5 (1.75)
0.62 (0.061)
0.18 (0.037)
70.6 (1.40)
0.65 (0.057)
0.12 (0.009)
30
79
80
135
104.9 (15.86)
100.7 (9.11)
105.7 (7.58)
73.6 (10.84)
0.03 (0.011)
0.08 (0.018)
0.07 (0.014)
0.25 (0.077)
83.7 (6.22)
0.04 (0.007)
0.11 (0.018)
0.11 (0.019)
0.19 (0.031)
144
�145
Table 6: ∆AICc for time of detection multiple species models for unlimited radius plots
with 10% truncation of largest detection distances. Smaller values of ∆AICc indicate
more parsimonious models. ∆AICc weights in parentheses. Larger weights indicate
more support for a given model. Models with weights ≥ 0.20 are in bold for each species
indicating competing models. Models were omitted (NA) when parameter estimates
were not realistic.
Groups
Model
M 00
A
55.5 (0.00)
B
145 (0.00)
C
124 (0.00)
D
E
41.1 (0.00) 72.1 (0.00)
M spp
0
F
180 (0.00)
58.3 (0.00)
123 (0.00)
127 (0.00)
36.9 (0.00) 73.4 (0.00) 84.0 (0.00)
M
0
t
43.6 (0.00)
131 (0.00)
106 (0.00)
36.3 (0.00) 59.8 (0.00)
M
+spp
t
∗spp
t
0
b
spp
b
0
tb
+spp
tb
∗spp
tb
0
h
part
h
spp
h
0
bh
part
bh
spp
bh
0
th
part
th
+spp
th
∗spp
th
36.2 (0.00) 79.3 (0.00) 48.4 (0.00) 36.4 (0.00) 52.3 (0.00) 44.4 (0.00)
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
M
143 (0.00)
51.4 (0.00)
108 (0.00)
107 (0.00)
34.4 (0.00) 61.2 (0.00) 48.5 (0.00)
55.9 (0.00)
147 (0.00)
123 (0.00)
43.1 (0.00) 74.1 (0.00)
62.6 (0.00)
115 (0.00)
125 (0.00)
40.7 (0.00) 75.6 (0.00) 92.2 (0.00)
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
NA
0.0 (0.51)
20.3 (0.00)
3.6 (0.04)
3.2 (0.08)
5.8 (0.03)
75.2 (0.00)
4.5 (0.05)
4.3 (0.09)
5.0 (0.02)
0.0 (0.40)
6.6 (0.02)
5.9 (0.05)
9.6 (0.00)
8.4 (0.01)
7.2 (0.01)
1.9 (0.16)
7.7 (0.01)
8.9 (0.01)
2.0 (0.18)
22.3 (0.00)
5.6 (0.02)
5.2 (0.03)
7.9 (0.01)
77.3 (0.00)
6.6 (0.02)
6.3 (0.03)
7.0 (0.01)
2.1 (0.14)
8.6 (0.01)
8.0 (0.02)
18.1 (0.00) 15.7 (0.00) 11.1 (0.00)
6.1 (0.02)
11.8 (0.00) 17.3 (0.00)
182 (0.00)
1.7 (0.21)
21.0 (0.00)
0.0 (0.27)
15.1 (0.00) 22.3 (0.00) 69.4 (0.00)
6.4 (0.02)
5.3 (0.06)
1.3 (0.14)
2.0 (0.15)
0.8 (0.36)
0.0 (0.92)
23.7 (0.00) 19.2 (0.00) 0.23 (0.24) 16.8 (0.00) 12.9 (0.00) 10.8 (0.00)
17.3 (0.00)
0.0 (0.80)
0.14 (0.25)
6.5 (0.02)
0.0 (0.55)
10.5 (0.00)
�Table 7: Parameter estimates from the time of detection method for each species. The Probability that an individual is detected at least
once during the count pˆ T and the probability of being in the first heterogeneity group λ̂ and the estimated abundance N̂ are given for
the selected model and for the selected model from a single species modeling approach. The instantaneous rates formulation was used
to estimate detection probabilities. Standard errors are given in parentheses.
Selected Model
Group Species
A
B
Acadian Flycatcher
Obs.
Count
87
Blackandwhite Warbler
137
Goldencrowned Kinglet
82
Wormeating Warbler
96
Blackthroated Blue
Warbler
Darkeyed Junco
197
189
Hooded Warbler
274
Solitary Vireo
148
Alternative Single Species Model
λ̂
N̂
pˆ T
λ̂
N̂
pˆ T
0.89
(0.049)
0.89
(0.049)
0.89
(0.049)
0.89
(0.049)
0.58
(0.047)
0.58
(0.047)
0.58
(0.047)
0.58
(0.047)
98 (6.4)
0.91
(0.172)
0.86
(0.113)
0.92
(0.089)
0.92
(0.249)
0.90
(0.094)
0.83
(0.092)
0.84
(0.062)
0.91
(0.440)
0.48
(0.081)
0.72
(0.058)
0.65
(0.057)
0.42
(0.060)
220 (23.6)
154 (9.5)
92 (6.1)
108 (6.9)
0.51
(0.115)
0.58
(0.078)
0.67
(0.155)
0.54
(0.098)
95 (18.2)
160 (21.7)
89 (9.0)
104 (28.5)
227 (26.0)
326 (25.4)
Same model
163 (79.1)
146
�C
D
E
F
Blackthroated Green
Warbler
Indigo Bunting
377
64
Redeyed Vireo
397
Scarlet Tanager
161
Veery
67
Ovenbird
444
Rufoussided Towhee
79
Redbreasted Nuthatch
54
Tufted Titmouse
104
Winter Wren
106
Wood Thrush
153
0.89
(0.053)
0.89
(0.053)
0.89
(0.053)
0.53
(0.035)
0.53
(0.035)
0.53
(0.035)
424 (26.2)
0.9 (0.066)
72 (5.2)
0.85
(0.067)
0.85
(0.067)
0.52
(0.062)
0.75
(0.088)
189 (16.0)
0.92
(0.053)
0.89
(0.277)
0.51
(0.062)
0.50
(0.078)
483 (28.6)
0.81
(0.062)
0.81
(0.062)
0.81
(0.062)
0.81
(0.062)
0.79
(0.087)
0.78
(0.068)
0.50
(0.069)
0.15
(0.047)
67 (6.4)
446 (27.5)
79 (7.2)
419 (31.5)
0.93
(0.505)
0.87
(0.086)
0.55
(0.057)
0.55
(0.092)
0.51
(0.048)
0.86
(0.085)
0.82
(0.111)
0.53
(0.068)
0.74
(0.093)
186 (19.0)
131 (11.4)
188 (15.9)
454 (45.6)
82 (11.9)
Same model
89 (28.1)
128 (11.2)
69 (37.6)
0.79
(0.139)
0.77
(0.093)
0.87
(0.102)
0.85
(0.331)
0.80
(0.099)
0.78
(0.068)
0.56
(0.107)
0.13
(0.047)
68 (12.7)
135 (17.4)
121 (14.8)
180 (70.4)
147
�148
Table 8: ∆AICc for the four independent observer multiple species models for unlimited
radius plots with 10% truncation of largest detection distances. Smaller values of ∆AICc
indicate more parsimonious models. ∆AICc weights in parentheses. Larger weights
indicate more support for a given model. Models with weights ≥ 0.20 are in bold for
each species indicating competing models. The number of observations for each species
was small for this data set so the groups have been modified for analysis.
Models
M 00
B
56.33 (0.00)
Groups
C
103.52 (0.00)
DEF
114.68 (0.00)
M spp
0
55.94 (0.00)
103.22 (0.00)
107.66 (0.00)
M 0obs
61.04 (0.00)
105.35 (0.00)
116.26 (0.00)
60.69 (0.00)
105.07 (0.00)
109.22 (0.00)
70.55 (0.00)
109.37 (0.00)
111.91 (0.00)
0.00 (0.72)
0.76 (0.22)
1.18 (0.26)
M hpart
2.10 (0.25)
0.40 (0.26)
0.00 (0.47)
M
spp
h
7.73 (0.02)
4.47 (0.03)
4.61 (0.05)
M
0
obs,h
9.20 (0.01)
0.00 (0.32)
5.91 (0.02)
M
part
obs,h
11.34 (0.00)
1.30 (0.17)
4.29 (0.06)
M
+spp
obs,h
17.42 (0.00)
*
6.39 (0.02)
M
∗spp
obs,h
25.54 (0.00)
*
2.70 (0.12)
M
M
M
+spp
obs
∗spp
obs
0
h
*Models did not give realistic estimates. Standard errors for detection probabilities were
much larger than one.
�Table 9: Independent Observer results for group B without Blackthroated Blue Warbler, group C without Indigo Bunting and a
combined group of one species from groups D, E, and F. The probability of being in the low or high detectability groups is given by
πˆ and the probability of detection by one of the 4 observers is given by p̂1 , p̂2 , p̂3 , and p̂4 These are reported based on the selected
model. The abundance estimate N̂ is given for the selected model and for the selected model from a single species analysis.
Group Species
B
C
DEF
Obs.
Count
πˆ
Group Probability and Observer Detection Probabilities
p̂1
p̂2
p̂3
p̂4
Darkeyed Junco
36
Hooded Warbler
38
Solitary Vireo
51
Blackthroated
Green Warbler
47
Redeyed Vireo
72
Ovenbird
90
0.36 (0.066)
Scarlet Tanager
61
0.58 (0.094)
Tufted Titmouse
44
0.44 (0.097)
0.24 (0.083)
0.90 (0.027)
0.34 (0.065)
0.40 (0.057)
0.20 (0.091)
0.99 (0.016)
0.13 (0.070)
0.95 (0.030)
0.19 (0.072)
0.85 (0.054)
0.36 (0.062)
0.96 (0.021)
0.11 (0.051)
0.91 (0.050)
N̂ selected
N̂species
41 (2.9)
44 (5.7)
43 (3.0)
40 (2.0)
57 (3.7)
59 (5.9)
59 (5.4)
60 (7.2)
90 (7.6)
88 (7.9)
96 (3.3)
96 (3.7)
68 (3.6)
70 (5.6)
48 (2.4)
48 (3.3)
149
�Chapter 5
MODELING THE AVAILABILITY PROCESS
FOR POINT COUNT SURVEYS USING AUXILIARY DATA
�151
MODELING THE AVAILABILITY PROCESS FOR POINT COUNT SURVEYS
USING AUXILIARY DATA
Mathew W. Alldredge, Biomathematics and Zoology, North Carolina State University,
Raleigh NC 27695.
Kenneth H. Pollock, Zoology, Biomathematics, and Statistics, North Carolina State
University, Raleigh, NC 27695.
Theodore R. Simons, USGS Cooperative Fish and Wildlife Research Unit, Dept. of
Zoology, North Carolina State University, Raleigh, NC 27695.
Abstract.—Point count surveys based on aural detections of birds are commonly used to
estimate abundance of bird populations, to make inferences about population health, and
make spatial and temporal comparisons of populations. In order to make valid
comparisons it is necessary to estimate true abundance from count data by correcting for
the detection process. The detection process for point count surveys based on aurally
detected birds consists of availability (the probability that an individual sings during the
count) and perception (the probability an individual is detected given that it sings). Only
the time of detection method models both of these components from a single point count
survey but this method requires long survey periods which can be biased by movement of
birds. Short duration “snapshot” type methods (e.g. variable circular plot or distance
sampling and multiple observer counts) are not as subject to bias associated with birds
moving during the count but these “snapshot” methods do not account for availability.
Because availability bias increases as count length decreases it is important to include
this component when modeling the detection process. We propose two methods of
modeling the availability process; one that assumes an individual bird sings randomly
following a Poisson process and another nonparametric approach that makes no
�152
assumptions about the distribution of singing times for an individual bird. Data
collection for both approaches may be “size” biased or overrepresent birds with high
singing rates, which must be corrected for when estimating the availability probability.
For the first approach we develop likelihood based models to estimate the availability
probability for homogeneous singing rates and finitemixture Poisson models to account
for individual heterogeneity in singing rates. The nonparametric approach to estimating
the availability probability is based on resampling singing time data over the time
interval used for a point count survey and determining whether an individual sings during
the interval, an approach similar to doing a bootstrap. This approach does require that the
true distribution of singing rates can be obtained. We applied these approaches to singing
rate data collected for the Ovenbird (Seiurus aurocpillus) in Great Smoky Mountains
National Park, and to simulated data that were comparable to field data. The
homogeneous Poisson model did not fit the data well but a twopoint Poisson mixture
model did. The estimated availability probability for a three minute point count from the
twopoint Poisson mixture model was 0.87 (SE = 0.025), indicating that 13% of the
population did not sing during the count and could not be detected. Analysis of the
simulated singing time data indicates that if individuals sing following a random process,
results are comparable to the singing rate approach, but if individuals sing in bouts, the
availability probability will decrease or a smaller portion of the population actually sings
during a point count survey. The benefit of these methods is that they can be used to
correct for availability bias in conjunction with any point count method, including those
that use very short survey intervals. This approach does require a significant amount of
�153
field effort and may be most applicable to surveys intended to track spatial or temporal
changes in the abundance of a few target species.
Introduction
The importance of accurately modeling the detection process when estimating
animal abundance has long been recognized (Burnham 1981, Barker and Sauer 1995,
Johnson 1995, Williams et al. 2002). In spite of this, point count surveys for birds are
continually used as indices or measures of relative bird abundance rather than measures
of true abundance (Rosenstock et al. 2002). Recently methods for estimating the
detection process for point count surveys have been emphasized, and new methods have
been developed (Nichols et al. 2000, Bart and Earnst 2002, Farnsworth et al. 2002,
Thompson 2002, Rosenstock et al. 2002, Royle and Nichols 2003).
When the detection process is not measured, abundance estimates are subject to
both availability and perception biases (Marsh and Sinclair 1989). Given these two
components of the detection process the conceptual estimator of abundance is
C
Nˆ =
pˆ d pˆ a
(1)
where C is the observed count, pˆ d is the probability of detection given availability and
pˆ a is the availability probability. These parameters can be estimated jointly, as in some
current point count methods, or separately. Estimating these parameters separately gives
more flexibility in how the data are collected and the methods used for analysis.
Perception is the probability that an individual is detected given that it is available
for detection. Both the distance sampling or variable circular plot method (Reynolds et
al. 1980, Buckland et al. 1993) and the multiple observer methods (Nichols et al. 2000,
�154
chapter 2) model the probability of detection given availability but do not account for the
probability that an individual is available. These are “snapshot” type counts done over
short time periods to avoid confounding effects associated with movement of birds during
the count. Unless auxiliary information is collected to determine availability or it is
reasonable to assume the probability of being available is one, these methods only
estimate the available portion of the population,.
Availability refers to the portion of the population that has a nonzero probability
of detection. This is similar to the problem of temporary emigration in capturerecapture
studies where some portion of the population is in the sample area and can be observed
(available) and others are not. Estimation of temporary emigration uses the robust
capturerecapture design (Kendall et al. 1997) and was utilized for salamander surveys
where individuals are unavailable for capture when below the surface (Bailey et al.
2004a).
Nonavailability of birds in avian point counts is caused by birds not singing
during a point count or birds moving into or out of the sample area during the count. The
probability that a bird is available for detection during point counts based on auditory
detections is the probability that an individual sings at least once during the count.
Standard point count methods try to account for this by conducting counts early in the
morning when singing rates are highest (Ralph et al. 1995). Some counts, such as the
BBS (Sauer et al. 1997), also specify surveying early in the breeding season when singing
rates are assumed highest. It is doubtful, however, that such standardization validates the
assumption that all birds in the population sampled are available during a count. This is
especially true for species with low singing rates.
�155
The availability process is especially important in forested environments or dense
vegetation (Wilson and Bart 1985). When visual identification is limited, it is common to
detect most individuals aurally from calls and songs. Aural detections were made for
over 95% of birds detected in point count surveys of breeding birds in Great Smoky
Mountains National Park (Simons unpub. data). Therefore, the availability process in
these situations is directly related to the singing rates of the individuals encountered. The
availability process is of greatest importance for species with low singing rates relative to
the duration of the point count.
Current point count methods that incorporate both the availability process and the
detection process given availability are the time of detection method (Farnsworth et al.
2002, chapter 3) and repeated count methods (Bart and Earnst 2002, Royle and Nichols
2003, Royle 2004). All of these approaches estimate the joint probability of being
available and detection given availability. Availability using the repeated counts method
is related to both the probability of singing during a count and movement, because the
population within the sample area can change between successive counts.
The time of detection method divides a count into consecutive time intervals and
records all of the intervals in which an individual bird is detected. By constructing this
detection history, it is possible to use the entire suite of capturerecapture models (Otis et
al. 1978, Williams et al. 2002) to analyze these data (Chapter 3). A conceptual example
of this method for an individual that is missed in one interval but then detected in the next
interval will illustrate how availability is accounted for by the time of detection approach.
Two possibilities exist for why an individual is not detected in the first interval; one is
that the bird sang but the observer did not detect it, and the other is that the bird never
�156
sang so the observer could not detect it. This method estimates the total detection process
because both possibilities of not detecting a bird are represented.
The total survey length of each point count must be long (e.g. 10 minutes) to get
reasonable parameter estimates when using the time of detection method and therefore,
this method is subject to biases associated with movement. If movement occurs during
the sample period, then the “true” population of birds being sampled or the actual sample
area cannot be determined. Avoiding problems with movement requires the use of
shorter counts, which may suggest the use of distance or multiple observer methods for
estimating the probability of detection. Unfortunately no short count duration or
“snapshot” type count approach accounts for availability in the detection process.
The objective of this paper is to propose new methods of estimating the
probability that a bird sings during a point count. These methods are based on singing
data, collected separately from the point count that are then applied to any point count
method to correct for availability bias. One approach is to collect singing rate data and
model the availability process with a Poisson model which assumes a bird sings at
random times. The other approach is to record singing time data and estimate the
availability probability by resampling the data, which does not require any assumptions
about the distribution of songs beyond the data being representative of the population.
All models are based on estimating the probability that a randomly selected individual
from the population does not sing during a point count interval, the complement of which
estimates the availability probability.
We will first discuss the field methods to collect data for either the singing rate
approach or the singing time approach. A homogeneous Poisson singing rate model is
�157
then presented as a simple case for modeling the availability process. Finitemixture
Poisson models are then presented as more biologically reasonable models with
differences in singing rates associated with factors such as breeding phenology of an
individual. In this section we also discuss “size” bias (the tendency of over representing
birds with high singing rates in the sample) that may occur in the data and correct for this
in these models. The final model is a nonparametric approach that uses the actual singing
times of an individual to estimate the availability probability and variances by resampling the singing time data. We present an example of the singing rate models using
data collected on the Ovenbird and select between models using information theoretic
model selection. Examples for the singing time analysis method are given using
simulated data that are comparable to the Ovenbird data set. Two data sets are simulated;
one that has random singing times and one that assumes birds sing in bouts of five songs
each. We conclude with a discussion of this approach and compare this method with the
time of detection method of accounting for availability bias.
Methods
Field methods.—We assume that data collected to model the availability process
for point counts are collected at the same times and locations as the point count data.
This is important to ensure that the availability process being modeled from auxiliary
singing data are representative of the population of birds sampled with point counts.
These data could be collected at each point following the conclusion of a point count or
during travel to subsequent point counts. If point counts are stratified by habitat type
then singing data must be collected and analyzed for these same strata. Singing rates or
�158
times should be recorded over long time periods to accurately represent the singing
distribution for an individual bird (e.g. 10 to 20 minutes).
Ideally birds are sampled by visually locating birds in the absence of auditory
cues (possibly wearing headphones while locating birds). This will avoid issues of “size”
bias in the sample or overrepresenting birds with high singing rates. Recording singing
rates or singing times can begin immediately after a bird is identified visually.
In some situations and for some species it is only feasible to locate birds by
sound. In these cases it is important to ignore the song or singing bout that was used to
locate the bird. If this is not done then the singing rate of each individual is also biased
because the probability of a bird never singing during an observation period is zero.
Waiting a set period of time (e.g. one minute) after locating a bird is probably the best
approach to sampling when birds are identified by song.
Collecting data on singing rates only requires recording the number of songs or
calls given by an individual bird during the observation interval. Some effort is
necessary to ensure that a bird is present during the observation interval and has not
flown off. Singing time data require recording the exact start and stop times of each song
during an observation period. This can be done with either a stop watch or by using a
personal digital assistant (PDA) (we have developed a program to do this).
Availability assuming homogeneous singing rates.—Although we doubt that
average singing rates across the population of interest are homogeneous or identical
among individuals, this provides a reasonable starting place. The assumption for this
model is that all birds in the population of interest have the same singing rate parameter
(λ) so that on average every bird calls the same number of times during a time interval. If
�159
we let the random variable X represent the number of times an individual sings during a
time interval (t) and assume that singing rates are from a Poisson process then X is
distributed Poisson(λt). From a sample of n individuals we would have x1, x2, …, xn
realizations of the random variable X.
The likelihood for the set of n observations xi (i=1,…,n) under the model
assuming homogeneous singing rates from a Poisson process is:
n
L(λx i ,t)=∏ (λt) xi exp(λt)
(2)
i=1
Maximizing this likelihood or minimizing the negative natural log of the likelihood will
give the estimated Poisson parameter λ̂ given the data. This is equivalent to the moment
estimator for λ̂ or the mean of the observed singing rates. All models of singing rate
data are likelihood based and therefore, can be compared using informationtheoretic
approaches, such as Akaike’s Information Criterion (Burnham and Anderson 2002). The
AIC value for this homogeneous one parameter model is:
AIC=2ln(L(λx,t))+2
(3)
Modeling the singing rate parameter as a function of time (λt) is important
because it allows us to estimate the probability of availability for point counts of different
length. The probability of being available for detection (pˆ a ) is the probability that an
individual sings at least once during the count, p(X ≥ 0). The probability that an
individual sings at least once during a count of duration t* is estimated as the
complement of the probability that a bird never sings during a count such that:
ˆ
p̂ a =1p(X=0)=1exp(λt*)
(4)
�160
Availability assuming heterogeneous singing rates.—When the singing rate
parameter is not homogeneous across the entire population then singing rate data are
biased towards those individuals with higher singing rates. This is because individuals
with higher singing rates are more likely to be detected than individuals with lower
singing rates. This is known as a “size” biased sample (Patil and Ord 1976, Patil and Rao
1977, 1978) and results in a weighted distribution. This results in a positive bias or
overestimation of the availability probability.
When individual heterogeneity exists then the actual data being collected are for
the “size” biased distribution fw(x) and not the true distribution f(x). The relationship
between these two distributions is (Patil and Rao 1977, 1978):
f
w
(xθ)=
w(x)f(xθ)
w
(5)
where w(x) is the weighting function and w is a constant given by:
∞
w= ∫ w(x)f(xθ)dx
0
(6)
for X continuous or
w= ∑ w(x)p(xθ)
(7)
∀x
for X discrete. The constant w scales the cumulative distribution for the “size” biased
data Fw(x) to sum to one.
The weighting function w(x) is any function of the random variable that gives the
same chance of including any observation produced by the original distribution f(x). It is
reasonable to assume that including a bird in a sample is a function of singing rate. We
will let the weighting function w(x)=x, which gives a linear relationship such that the
probability of including a bird with a singing rate twice that of another is twice that of the
�161
bird with the lower singing rate. Patil and Rao (1977) found that this weighting function
was the most commonly used relationship, including wildlife surveys where group size
created a “size” bias.
Observations based on auditory cues often results in “size” biased data.
Therefore, we will present individual heterogeneity models of singing rate that
incorporate the correction for size bias. When data are collected so the data are not size
biased (e.g. visually locating birds or using a marked population of birds) or the true
singing rate distribution is represented by the data, then letting w(x)=1 for all singing
rates will give the appropriate model.
If a finite set of biological factors determine singing rates, then finite mixture
models (Lindsay 1983, 1986, Norris and Pollock 1996, Pledger 2000) of individual
heterogeneity are reasonable. For example, if singing rates are determined by pairing
status and nesting stage (Wilson and Bart 1985), then a finite set of factors affecting
singing rate is identifiable. Essentially this involves dividing the population into groups
based on an individual’s current breeding status and claiming that all individuals within a
group have on average the same singing rates. If a set of k groups are identified then this
would require estimating singing rate parameters λ1, λ2, …, λk, plus the proportion of the
k1
total population in each group δ1, δ2, …, δk1, 1∑ δ j . Using equation 5 and the
1
assumption that individual singing rates follow a Poisson process, the likelihood for a set
of n observations xi (i=1,…,n) under the finitemixture model of singing rates assuming a
kpoint mixture of Poisson processes is:
�162
L(λ1 ,...,λ k ,δ1 ,...,δ k1x,t)=
(8)
⎞
⎛ k1 ⎞
1 ⎛ k1
xi
xi
δ
x
(λ
t)
exp(λ
t)+
⎜⎜ ∑ j i j
⎜1∑ δ j ⎟ x i (λ k t) exp(λ k t) ⎟⎟
∏
j
i=1 wx i! ⎝ j=1
⎝ j=1 ⎠
⎠
n
Because the sum of the proportions across all groups must be one, the final term in this
probability represents the final group and it is equal to one minus the sum of the
proportions of the population that are in all the other groups.
Maximizing the likelihood or minimizing the negative natural log of the
likelihood of the observed singing rates gives the maximum likelihood estimates (MLE’s)
for λ1, λ2, …, λk and the proportion of the population in each group δ1, δ2, …, δk1. The
probability of being available for detection during a point count of length t* is given by
⎛ k1
⎞
⎛ k1 ⎞
pˆ a =1p(X=x)= ⎜⎜1∑ δ jexp(λ j t * ) ⎜1∑ δ j ⎟ exp(λ k t) ⎟⎟
⎝ j=1 ⎠
⎝ j=1
⎠
(9)
The AIC value is similar to the homogeneous AIC with a larger “penalty” for the number
of parameters. For a kpoint mixture model there are 2k1 estimated parameters, which
leads to the following equation for AIC:
AIC=2ln(L(λ1 ,...λ k ,δ1 ,...δ k1x,t))+4k2
(10)
Estimating variance and confidence intervals.—The estimated availability
probability is a function of random variables ( λˆ j ' s and δˆ j ' s ) that each have an
associated estimation variance. Because of this an exact derivation of variance can be
difficult to obtain (Williams et al. 2002), especially a general form for the heterogeneous
singing rate models when the number of mixtures varies. One approach is to use the
delta method (Seber 1982) to obtain the large sample approximation of the variance. We
�163
used the bootstrap (Efron 1979, Efron and Gong 1983, Manly 1998) to obtain empirical
estimates of variance and distribution free confidence intervals.
The bootstrap procedure treats the observed data of n observations as the
population and generates k new samples by selecting n observations from the data with
replacement. Model parameters are then estimated for each of the k samples. The mean
and variance of each estimate are estimated by the mean and variance of the set of
bootstrap estimates. Confidence intervals are constructed assuming that the estimates are
distributed normally and using the mean and variance to compute symmetric normal
theory confidence intervals. An alterative method uses the percentile approach which is
distribution free (Manly 1998). This approach involves ordering the k estimates for each
parameter and then selecting the endpoints of the confidence intervals as the values that
contain the central (1α)100% of the estimates.
It is also possible to use the bootstrap samples to calculate the variance of the
estimate of the availability probability, instead of deriving variance estimators based on
the variances of the parameter estimates. The probability of being available during a
count of length t is calculated (equations 4 or 9) for each of the k bootstrap samples using
the parameter estimates from that sample. The variance of this estimate across all
bootstrap samples is then used as the estimate of the variance for p̂a . Confidence
intervals are then constructed as discussed above.
Availability using singing times.—This nonparametric approach to estimating the
availability probability for detection during a point count uses the exact singing times of
individual birds and is not restricted by the assumption that the singing rate of an
individual bird is based on a Poisson process. This approach is much more robust to
�164
differences among individual birds and it is applicable in situations where birds sing in
singing bouts separated by periods of silence. This singing pattern clearly violates the
Poisson assumption. In many ways this approach is similar to the bootstrap just
discussed. Use of this approach requires that singing times are resampled from the
observed data based on the true distribution of singing rates.
As previously discussed, overrepresenting birds with high singing rates will lead
to a “size” biased sample. When sampling periods are sufficiently long so that all birds
in the sample sing at least once this bias can be corrected without assuming any
distribution of singing rates for the population. The data are corrected for “size” bias
based on the observed singing rates for each individual. Correcting the biased data is
done by rearranging equation 5 and using the probability mass function for the corrected
distribution of X, p(X=x) and the biased distribution of X, pw(X=x) as:
wp w ( X = x)
p ( X = x) =
w( x)
(11)
To solve for the weight, w, we use the fact that the sum of p(X=x) over all observable
singing rates, r, must equal one, such that:
w=
∑
∀r
1
p ( X = r)
w(r )
w
(12)
If not all of the birds in the sample actually sing during the sampling period then
the above formulas to correct for “size” bias do not work. One option is to sample birds
from the population differently so that an unbiased sample is obtained, possibly using
marked birds. If this is not possible then it will probably be necessary to assume a
�165
distributional form for the singing rates of the population (e.g. a Poisson distribution) or
use the singing rate method.
We assume that a sample of singing times from the population is representative
and treat it as the “population” of possible singing behaviors. In other words the sample
of singing times is assumed to represent what is observed during a point count, including
birds that do not sing during a specified interval, those that sing frequently, and those that
sing sporadically or in bouts. A single bird in the sample can provide much information
about what may occur in a point count. For example, if a bird is observed for 20 minutes
then examining any smaller portion of this may be a period when a bird does not sing,
sings constantly, or sings sporadically. This assumption implies that the singing rates
involved with this sample represent the true distribution of singing rates.
The number of observations to include in each sample replicate must be equal to
the population size estimate obtained from the point count survey. Sampling the data set
an infinite number of times will give a good estimate of the average availability
probability for the population but does not provide any information about the variance of
the estimate in relation to correcting for availability bias in a point count survey. Because
we are assuming that singing time data are representative of the population, then the
number of observations to include for each sample should correspond to the abundance of
birds estimated in the point count survey area. This estimated abundance in the survey
area corresponds to the portion of the population that is actually detected. In other words
each replicate sample is a potential realization of the availability probability for a
randomly selected sample of size N.
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To estimate the abundance in the sample area it is necessary to correct the
observed count for both availability and detection bias. Correction for detection bias is
based on the type of point count survey. To correct for availability bias the singing time
data is sampled sufficiently to obtain the average availability probability using the
algorithm described below. The variance of the availability probability is then estimated
by generating samples based on the estimated abundance.
Having estimated the abundance in the survey area it is now possible to estimate
the availability probability and its variance. Again, note that using the estimated
abundance does not affect the average availability probability but it does affect the
variance of this estimate. The following algorithm is used to generate the sample and
estimate the availability probability of each replicate (m) based on a point count survey
length of t* minutes from singing time data collected for t minutes on each individual.
1. Generate each observation, yi (i = 1,…,N), in the sample.
a. Randomly select a singing rate (x) for observation i based on the
probability mass function for p(X=x).
b. Randomly select an individual bird, i, (with replacement) with the
singing rate x.
c. Randomly select a starting time, ts from 0 to t.
d. Search the singing time data for bird i from ts to ts+t* and record if any
part of a song is within this interval. This is done by using starting and
ending times for each song.
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e. If the specified bird i sings during the interval then record a one,
otherwise record a zero. Let yi be the event that bird i sings, such that
if the bird sings yi = 1, otherwise yi = 0.
2. Repeat steps a through e for each observation in the sample of size N.
3. Estimate the availability probability for the current sample (replicate j) pˆ aj as,
N
p̂aj =
∑y
i
1
(13)
N
4. Repeat steps 1 through 3 to obtain m replicate estimates of pˆ aj .
The average availability probability is the sum of the availability probability
estimates p̂aj divided by the number of replicates m, and the variance of the sample
measurements p̂aj , is the sum of the square difference between the estimates and their
mean, divided by m1. Confidence intervals are generated by assuming that pˆ aj is
normally distributed or by using the percentile method, described for the bootstrap.
Analysis of Field Data
To demonstrate the use of auxiliary information to estimate the availability
probability we use singing rate data collected from 269 male Ovenbirds in Great Smoky
Mountains National Park during May of 2000 and 2001 (A. Podolsky unpub. data). Birds
included in the sample were located from auditory cues and the number of songs sung by
each bird were counted over a five minute interval. Observations were made in the first
few hours of the morning during the peak of daily singing activity.
�168
These data were used to estimate the availability probability using the
homogeneous singing rate model and two and threepoint Poisson mixture models of
heterogeneous singing rates. All estimates are maximum likelihood estimates (MLE’s)
and model comparison is based on AIC (Burnham and Anderson 2002). For each model
1,000 bootstraps were run to obtain a variance estimate and percentile confidence
intervals for all parameters and estimates of the availability probability. The availability
probability was estimated for one, two, and three minute point counts to demonstrate the
effects of the time interval used for point counts on availability.
We simulated comparable data to the Ovenbird data to demonstrate the singing
time method. This was possible because all birds in our sample sang at least once during
the sample period. Simulations assumed that songs were five seconds in length. Two
different data sets were simulated; one simulation assumed individuals sang at random
times and the other assumed individuals sang in singing bouts. The data set simulated
with random singing times meets the assumptions of the singing rate models, and should
have similar availability probabilities. The other data set was simulated assuming that
birds sing in bouts of five songs per bout. For birds with singing rates of five or less then
that individual would have a single singing bout corresponding to the specified singing
rate. Birds with singing rates greater than five would have multiple singing bouts but the
total number of songs would correspond to the specified singing rate. Songs within a
singing bout were simulated with no intersong interval. Multiple singing bouts were
simulated with random starting times for nonoverlapping singing bouts. We expect
lower availability probability when birds sing in bouts because there are larger time
periods when birds do not sing.
�169
Results
Singing rates for the Ovenbird data ranged from one to 23 songs during a five
minute interval. Plotting the frequency or probability of observing each singing rate
shows a bimodal distribution of the data (Figure 1). The average singing rate was 8.56
songs per five minute interval or 1.71 songs per minute.
Singing rate models.—Both the homogeneous Poisson and the twopoint Poisson
mixture models gave maximum likelihood estimates (MLE’s) for the model parameters.
The threepoint Poisson mixture model was problematic due to local minima, which
made finding the MLE’s for the model parameters difficult. Bootstrap estimates for the
threepoint Poisson model suffered similar convergence problems giving unrealistic
parameter estimates. Because of these unrealistic parameter estimates we will not present
the results of this model. The twopoint mixture Poisson model was the most
parsimonious model for these data based on AIC values (Table 1). The difference in AIC
for the homogeneous Poisson model was large (∆AIC = 182.7), indicating it is not a
reasonable alternative model for these data.
The homogeneous Poisson model did not fit the data well (Pearson χ2 = 719, df
=22, P < 0.0001), but the twopoint mixture Poisson model fit reasonably well (Pearson
χ2 = 29.6, df =20, P = 0.08)(Figure 1). The estimate of lambda for the homogeneous
Poisson model was 1.71, which corresponds to the average per minute singing rate. The
parameter estimates for the twopoint Poisson model indicate about 66% of the birds
were in a low singing rate group and the remainder sung at almost four times this rate
(Table 1). Parameter estimates for both the homogeneous and twopoint Poisson mixture
�170
models were reasonably precise based on the bootstrap variance estimates (Table 1). The
bootstrap distribution of λ̂ for the homogeneous Poisson model was approximately
normal (Figure 2) as were the distributions of λ̂1 , λ̂ 2 and δ̂ for the twopoint Poisson
model (Figure 3). The 95% confidence intervals using the percentile approach were
nearly symmetric for parameter estimates from both the homogeneous model and the
twopoint Poisson mixture model.
Estimates of the availability probability were 0.82 (SE = 0.009) for a one minute
count, 0.97 (SE = 0.003) for a two minute count, and 0.99 (SE = 0.001) for a three minute
count based on the homogeneous Poisson model. Availability probability estimates for
the twopoint Poisson mixture model were 0.57 (SE = 0.029) for a one minute count, 0.77
(SE = 0.030) for a two minute count, and 0.87 (SE = 0.025) for a three minute count
(Table 1).
Singing time model.—As expected the estimates of the availability probability
using the singing time approach were higher for the random singing data set than for the
singing bout data set (Table 2). All estimates were reasonably precise using a resample
size of 269 individuals. The difference between the availability probabilities for one and
two minute point counts was about 0.1 but it was only 0.06 for the three minute count.
This occurs as the length of the point count interval approaches the length of the singing
time interval.
Estimates of the availability probability for the random singing time data set
(Table 2) were similar to the estimates obtained from the twopoint Poisson mixture
model (Table 1). This is expected because the Poisson model assumes random singing
�171
times. This was just one realization of a random process, so we would not expect exact
correspondence between the two models.
The distribution of availability probability estimates is slightly skewed to the right
for both the random singing time data and the singing bout data (Figures 4 and 5). This
can also be seen in the confidence intervals that are slightly larger to the left of the mean
for most estimates (Table 2). We recommend the percentile confidence intervals in this
case. The overall shapes of the distributions are similar between the random singing time
analyses and the singing bout analyses.
Discussion
We have presented a method for accounting for availability bias in point count
surveys where individual birds are detected by song or call. This method allows for
correction of availability bias using auxiliary data, so it is applicable to any point count
method, even those that only correct for perception bias. We have provided two methods
of analysis, both of which incorporate individual heterogeneity in singing behaviors. One
method relies on the assumption that an individual bird’s singing times are randomly
distributed following a Poisson process. Individual heterogeneity in singing rates is
incorporated into this model by using a mixture of Poisson distributions. The other
approach is a nonparametric approach that does not require any assumptions about the
structure of an individual bird’s singing patterns. The approach is based on resampling
observed singing time data in relation to the time interval and abundance estimate from a
point count survey. It incorporates individual heterogeneity in singing behavior through
the resampling process.
�172
The field methods for these approaches are relatively straightforward but may
lead to a “size” biased sample. We have presented methods to account for this bias or
tendency to overrepresent individuals with high singing rates in the sample. Correcting
for “size” bias presents additional problems for the singing time approach if individuals
in the sample do not sing at least once during the observation period. If this occurs it
may be necessary to assume a distribution for singing rates or use the singing rate
method. Alternatively unbiased data could be collected by using birds identified by some
other means. If birds are found by visual detections or by randomly sampling a marked
population, then “size” bias is not a problem.
One concern with these approaches is censoring data when birds move undetected
during the sample period. If this occurs no songs will be recorded for the end of the
observation period. Unfortunately, in this situation it is not know if the bird is there and
not singing or has left the area. It is necessary to monitor birds visually or at close
distances to avoid these problems. Actually following birds as they move during the
sample period may be necessary for some species. If territory size is small for the species
of interest and movement distances are correspondingly small then this will be realistic.
However, if movement of birds is excessive and birds can make undetected movements
out of hearing range during the sample period, these methods will not work. In other
words, a key assumption is that the amount of time that a bird is monitored is known
without error. Radiotelemetry would be useful in testing this assumption.
For this paper the probability of being available for detection is related to whether
an individual actually sings during a point count survey. Point count data are primarily
collected for singing males. In fact, it may be advisable to ignore other types of
�173
detections because they will have different detection probabilities. If females or nonbreeding males also sing regularly and are likely to be included in the point count survey
then auxiliary singing data must also include these birds.
Singing rate data must be collected at times comparable to the point count
surveys. Factors such as the time of day (Robbins 1981a, Skirvin 1981), weather
conditions (Robbins 1981b), season (Best 1981, Skirvin 1981), breeding stage (Best
1981, Wilson and Bart 1985), local abundance of conspecifics (McShea and Rappole
1997) and local habitat characteristics (McShea and Rappole 1997) have been
documented to affect singing rates of birds. Because singing rates are so dependent on
these factors it is critical that singing rate data be collected across the same area as the
point count surveys, on the same days and over the same time period of the day. If this is
not done then the estimated probability of singing during a point count will be biased.
For example, collecting singing data in late morning following the conclusion of point
counts would lead to a negative bias in the availability probability if singing rates decline
later in the morning. This would then have a positive bias on the abundance estimate.
The best approach is to collect singing information using an additional observer,
following the conclusion of each point count or during travel between points to ensure
that the singing information is representative of the availability process involved with the
detection process in the point count.
Evidence that singing rates change during the breeding season highlights the
importance of correcting point count surveys for availability bias. Wilson and Bart
(1985) report that the probability that a House Wren (Trogolodytes aedon) sings at least
�174
once during a threeminute period was 0.7 before mating, 0.50.6 from mating to
completion of egglaying, 0.7 during incubation, and 0.5 or less following incubation.
Observation periods should be representative of the singing behavior of the
individual sampled. These observation periods should be longer than the length of the
point count survey that is being conducted, possibly more than twice as long. If birds
sing frequently during the entire period in which point counts are conducted and do not
have long intersong intervals then the duration of the sample can be shorter. As the
length of intersong intervals increases or becomes more variable then the duration of the
sample will also have to increase to accurately represent the singing characteristics of the
individual.
Collecting these data will require a large amount of time and may lead to attempts
to collect data on multiple birds at the same time. This should not be done for
conspecifics because they may counter sing and thus, data would not be independent.
This could be done for different species at the same time if movement of birds is
minimal. To collect this information may require an observer to follow a bird and this
may limit collection on multiple birds at one time.
The individual bird must be considered as well as the entire population of birds to
model the availability process using singing rate data. The individual must be considered
because observations during a point count are made at the individual level, which is clear
based on defining pa as the probability that an individual will sing during a count. We
make the assumption that the singing rate of an individual (i) follows a Poisson process,
with singing rate parameter λi. A Poisson process is a memoryless process (Tuckwell
1995), which means that the probability that a bird sings at some future time does not
�175
depend on the last time that it sang. In other words this assumes singing times are
completely random events. The average time between calls is given by the rate
parameter λi, which is the expected value of the Poisson process.
Because it is not possible to know the singing rate parameters of the individuals
detected during a point count, the probability of being available must be based on the
average availability of the population. We assume birds occurring in the sample area of a
point count survey represent a random sample from the population. In order to determine
the average probability of availability it is necessary to make assumptions about the
distribution of the singing rate parameter across the population. The simplest assumption
is that all individuals in the population have the same singing rate parameter (i.e. the
population is homogeneous with regard to singing rate). This assumption could be true
during certain periods of the breeding season (e.g. if all individuals have the same pairing
status or are in the same reproductive stage) but in general it is too restrictive for typical
applications.
It is more likely that there is individual heterogeneity in singing rates among
individuals of a population. Pairing status and breeding stage are major components that
effect singing behavior of males during the breeding season (Best 1981, Skirvin 1981,
Wilson and Bart 1985). Singing rates of unpaired territorial males are often higher
because these individuals are trying to attract mates (Best 1981). Singing rates generally
decline during the breeding season as males that are incubating or rearing broods sing
less (Wilson and Bart 1985).
If singing rates are determined primarily by pairing status and breeding stage then
it is reasonable to model individual heterogeneity with a finite mixture model (Lindsay
�176
1983, 1986), similar to the finite mixture models of heterogeneity for capturerecapture
data (Norris and Pollock 1996, Pledger 2000). If heterogeneity in singing rates is
associated with innate differences among birds, then biological interpretations of the
mixture groups is not possible.
The probability that an individual is available for detection during a count and the
probability of detection given availability are considered a nuisance parameter, because
these parameters only provide information used to estimate population abundance.
Because these probabilities are not of use to estimate abundance it is not necessary to
estimate the availability process separately if the time of detection method works
reasonably well. Unfortunately a second component of the availability process is
movement of individuals during the count. The time of detection method is biased if
movement that affects detectability occurs during the count.
Three types of movement can occur during a point count: movement in relation to
the observer at the point, small scale movements within the sample area, and larger scale
movements into and out of the sample area. Movement in relation to the observer is a
problem inherent with point counts and it can bias the count (Conant et al. 1981,
Buckland et al. 1993). If movement removes birds from the sample area, then observed
counts will always be too small. Alternatively, some species of birds may be attracted to
observers, in which case the counts will be too large. Minor responses by birds to an
observer will have less effect on the count but would bias distance estimates if this is
being used in the model of detection. One approach to account for observer related
movements is to wait at the point for a specified amount of time before beginning the
count, but the effectiveness of this has never been tested.
�177
Small scale random movements within the sample area are of less concern unless
they have a significant effect on the detection process. This type of movement may affect
detection distance estimates, but random movement should average out across the sample
of observed birds. Small scale movements may also result in double counting. This
occurs when a bird is counted, then moves and is recounted as a new individual. The
result is a positive bias in the count.
Large scale movements in and out of the count are also important. These
movements create a positive bias in the count because not all birds are in the sample area
at the same time. This makes inference about the population difficult because the sample
area or proportion of the population sampled cannot be determined (Scott and Ramsey
1981). These movements also affect the availability process when the time of detection
method is used (Chapter 3). Now a zero in the detection history could also represent an
individual that was not present in the sample area during that interval. The probability of
availability due to singing is confounded by movement. The best approach for avoiding
the confounding effects of movement is to shorten the count duration (Scott and Ramsey
1981, Buckland et al. 1993).
Shortening the duration of a point count minimizes movement problems (Scott
and Ramsey 1981), including those related to double counting, but increases the
importance of the availability component of the detection process. As the duration of the
count gets shorter the probability that an individual actually sings during the count also
gets smaller. The time of detection method does not work well with very short duration
counts. “Snapshot” type counts (e.g. distance sampling and multiple observer methods)
�178
minimize problems associated with movement. Availability probabilities on snapshot
counts must be estimated using auxiliary information.
The singing time model requires more detailed data and analysis is more complex
because it requires writing a computer program to resample the data. The overall field
effort between the two methods is the same. We have developed a computer program for
collecting this data which allows the user to input relevant details about the observation
period (date, observer, weather, etc.), the species, the clock time of the observation, and
the start and stop times of each song. Records of start and stop times are collected by
clicking on start and stop buttons. We have designed this so that it can be loaded on a PC
based pda for use in the field, and output is an excel file. The benefit of this is that
accuracy is maintained because the observer does not have to try and write down times
from a stop watch and data entry is not required following data collection.
The example presented was based on simulated singing time data and the true
singing rate distribution was known. In practice it may be necessary to correct singing
time data for “size” bias in singing rates, as seen for the Poisson models. Correcting data
for “size” bias is straightforward if all birds in the sample sing at least once. If not all
birds sing during the sampling period then correcting for “size” bias may require
assuming a distribution for singing rates.
The importance of correcting for availability bias is often overlooked in
abundance surveys or inference is only made about the portion of the population that is
available. Analysis of the Ovenbird data set demonstrates how important availability bias
can be in estimating animal abundance. The Ovenbird has a very loud song and a very
high singing rate so high detection probabilities are expected. High singing rates imply
�179
high availability probabilities for the Ovenbird. Probabilities (0.87 from the twopoint
Poisson mixture model) were high compared to species with lower singing rates. Based
on the estimate from the twopoint Poisson mixture model for a three minute count 13%
of the population was missed on field surveys because they were not available for
detection. This estimate is probably low because Ovenbirds sing in bouts. An estimate
of 20% (from the 3 minute singing time analysis) may be more realistic.
Correction of availability bias is as important as correction for perception bias or
the probability that an individual is detected given that it is available for detection.
Estimated probability of detection given availability of 0.80 for the Ovenbird was
determined from analysis of a fourindependent observer data set using a twopoint
heterogeneity model (Chapter 2). The magnitudes of both types of bias are comparable
and thus, estimates of animal abundance should account for both.
Such corrections for availability have been done for aerial surveys of marine
mammals (Marsh and Sinclair 1989), capturerecapture surveys of salamanders (Bailey et
al. 2004a, 2004b) and point count surveys of birds (Farnsworth et al. 2002, chapter 3). In
these surveys the availability process was related to individuals not being visible to
observers because of dive depths, not being susceptible to capture because individuals
remained underground to avoid desiccation, and individuals could not be detected
because they did not sing during the count. The importance of estimating the availability
process is seen in the salamander surveys where only 13% of the population was
available for capture and in surveys of dugongs where 30 to 100% of the population was
available for detection depending on water conditions and the position of an individual in
the water column.
�180
We recognize that accounting for availability bias in point count surveys using
auxiliary information on singing behavior of birds significantly increases the effort
required to obtain valid abundance estimates. Implementation of this technique should be
assessed with respect to the amount of availability bias thought to be present in the data
and the mobility of the species of interest. It may be a more efficient use of survey effort
to use the time of detection method when species are sedentary. Recall that this requires
a longer count period for the point count. It is better to use a very short point count for
more mobile species and spend the extra time collecting data to account for availability
bias. It is difficult to account for availability bias for broad scale monitoring studies
given the spatial range of the surveys and the multitude of species being monitored. In
these cases it may be necessary to select a few species of interest to examine the
availability process.
Data on the singing behavior of birds may be useful for understanding other
ecological questions as well. Spatial comparisons in changes in singing rates across the
breeding season may be indicative of the quality of breeding habitats. If better habitats
are occupied earlier in the year then changes in singing rates relative to breeding stage
could be used to assess differences in breeding habitats. Yearly differences in singing
rates may also indicate changes in populations or habitats. McShea and Rappole (1997)
found that singing rates varied within a species based on habitat characteristics and local
abundance of conspecifics.
Recommendations
The biological interpretation of mixture models in this application makes using
these models desirable. Modeling differences in singing rates that may be associated
�181
with breeding phenology lends support to these models and also has ecological
application for temporal and spatial comparison of singing rate differences. The
limitation of this approach is assuming that singing patterns of an individual are random
events. This method could be modified to model singing bouts as random events and the
number of songs for each bout as a random variable. Further investigation of the use of
mixture models and an evaluation of singing rate distributions is warranted.
The problems with the threepoint Poisson mixture model were discouraging but
expected because of the number of parameters in the model. Problems with convergence
of Poisson mixture models have been noted before and the use of truncated Poissons has
been recommended (Lindsay 1986). Further investigation of Poisson mixture models
with more than two mixtures is needed.
Requiring that data are either unbiased or that all birds in the sample sing is a
serious limitation to the singing time approach. Ideally the sample period would be
sufficient so that all birds sing at least once during the period. This was the case for the
Ovenbird data we presented but may not be realistic for all species. Further investigation
of the treatment of birds that do not sing during a sample period is necessary. This
includes an assessment of the weighting function used to correct for “size” biased data.
Comparing data based on auditory detections of birds, to data collected by other means
(e.g. visual detection of birds or sampling a marked population) is necessary to determine
the validity of the weighting function used here, and if it is reasonable to assume a
distributional form for singing rates.
�182
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�185
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�186
Table 1: Parameter estimates for the homogeneous Poisson model and the twopoint Poisson mixture models. Standard errors and
percentile 95% confidence intervals obtained with 1,000 bootstrap samples. ∆AIC value of zero indicates the selected model.
Availability probability estimates p̂a (1), p̂a (2), and p̂a (3) are for 1, 2, and 3 minute point count surveys, respectively.
Model
∆AIC
λ̂1
Homogeneous
95% CI
2Point
95% CI
182.7
1.71(0.052)
1.601.81
0.54(0.061)
(0.420.66)
0.0
λ̂ 2
2.02(0.069)
(1.882.16)
δ̂
p̂a (1)
p̂a (2)
p̂a (3)
0.66(0.036)
(0.590.73)
0.82(0.009)
0.800.84
0.57(0.029)
(0.510.63)
0.97(0.003)
0.960.97
0.77(0.030)
(0.700.82)
0.99(0.001)
0.991.00
0.87(0.025)
(0.810.91)
186
�187
Table 2: Availability probability estimates for one, two, and three minute point count
surveys using two simulated data sets for a five minute observation period. One data set
uses completely random singing times and the other assumes birds sing in bouts of five
songs. For one iteration a sample of 100 birds is drawn with replacement and 1,000
iterations are done for each analysis. For each data set analyses are done for one, two,
and three minute point counts giving the availability probabilities p̂a (1) , p̂a (2) and
p̂a (3) , respectively. Percentile 95% confidence intervals are reported and standard errors
are in parentheses.
Data Set
p̂a (1)
p̂a (2)
p̂a (3)
Random Singing
95% CI
Singing Bouts
95% CI
0.48 (0.049)
(0.380.57)
0.37 (0.051)
(0.270.47)
0.71 (0.046)
(0.620.79)
0.61 (0.051)
(0.510.70)
0.86 (0.036)
(0.790.92)
0.80 (0.040)
(0.720.88)
�188
List of Figures
Figure 1: Homogeneous Poisson model and twopoint Poisson mixture model fit to
Ovenbird singing rate data. Poisson mixture model is corrected for “size” bias that
occurs in this data, which is not a factor under the assumptions of the homogeneous
Poisson model.
Figure 2: Distribution of λ̂ from 1,000 bootstrap estimates for the homogeneous Poisson
model fit to the Ovenbird data set. Points within the percentiled 95% confidence
intervals are in black.
Figure 3: Distribution of λ̂1 , λ̂ 2 and δˆ from 1,000 bootstrap estimates for the twopoint
Poisson mixture model fit to the Ovenbird data set. Points within the percentiled 95%
confidence intervals are in black.
Figure 4: Distribution of the availability probability from simulated data with random
singing times using the singing time analysis approach for one, two and three minute
point counts. Data was simulated to be comparable to the Ovenbird data set.
Figure 5: Distribution of the availability probability from simulated data assuming birds
sing in bouts of five songs. Analysis was based on the singing time approach for one,
two, and three minute point counts. Data was simulated to be comparable to the
Ovenbird data set.
�189
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
3
5
7
9
observed
11
13
15
homog
17
19
2pt
21
23
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55
3
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58
4
1.
61
4
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5
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5
1.
70
6
1.
73
7
1.
76
7
1.
79
8
1.
82
8
1.
85
9
Frequency
190
100
90
80
70
60
50
40
30
20
10
0
lambda
�5
4
3
2
1
0
0.
67
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0.
69
6
0.
71
7
0.
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8
0.
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9
M
or
e
0.
65
0.
63
0.
61
0.
59
0.
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55
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1.
79
9
1.
84
0
1.
88
1
1.
92
2
1.
96
4
2.
00
5
2.
04
6
2.
08
7
2.
12
9
2.
17
0
2.
21
1
Frequency
Gamma
0.
74
0.
71
0.
67
0.
63
0.
60
0.
56
0.
52
0.
49
0.
45
0.
41
0.
38
0.
34
7
1
4
8
1
5
8
2
5
9
3
6
Frequency
191
100
90
80
70
60
50
40
30
20
10
0
Lambda 1
120
100
80
60
40
20
0
Lambda 2
100
90
80
70
60
50
40
30
20
10
0
�192
160
140
Frequency
120
100
80
60
40
20
0.
31
5
0.
35
4
0.
39
3
0.
43
2
0.
47
1
0.
51
0
0.
54
9
0.
58
3
0.
62
2
0
Availability Probability (t=1 min)
160
140
Frequency
120
100
80
60
40
20
0.
56
5
0.
59
8
0.
63
1
0.
66
4
0.
69
7
0.
73
0
0.
76
3
0.
80
1
0.
83
4
0
Availability Probability (t=2 min)
140
120
80
60
40
20
0.
95
6
0.
92
9
0.
89
8
0.
87
1
0.
84
4
0.
81
7
0.
79
0
0
0.
76
3
Frequency
100
Availability Probability (t=3 min)
�193
180
160
Frequency
140
120
100
80
60
40
20
0.
42
6
0.
47
0
0.
27
0
0.
30
9
0.
34
8
0.
38
7
0.
19
2
0.
23
1
0
Availability Probability (t=1 min)
160
140
Frequency
120
100
80
60
40
20
0.
44
5
0.
48
4
0.
52
3
0.
56
2
0.
60
1
0.
64
0
0.
67
9
0.
71
3
0.
75
2
0.
79
1
0
Availability Probability (t=2 min)
160
140
100
80
60
40
20
0
0.
63
2
0.
66
5
0.
69
8
0.
73
1
0.
76
4
0.
79
7
0.
83
0
0.
86
3
0.
89
1
Frequency
120
Availability Probability (t=3 min)
�Chapter 6
EXECUTIVE SUMMARY
�195
Introduction
Population abundance is one of the most common parameters used to assess the
current state or health of a wildlife population. Abundance is used to make spatial and
temporal comparisons of populations, to assess population trajectories and differences
between populations. This information is used to infer habitat relationships and judge the
effectiveness of management actions on wildlife populations of interest.
The benefit of using population abundance as a measure of population status and
for spatial and temporal comparisons is that this single variable contains information
about several population processes that contribute to population level changes. A series
of abundance estimates provide an estimate of population change which combines
information on birth, death, immigration and emigration. Population change also allows
for spatial comparison between areas of interest to provide information about habitat
quality or differences in management practices. For example, comparing abundance
estimates or rates of population change between managed (e.g. National forests) and
unmanaged areas (National Parks) provide a measure of management effectiveness.
Point count surveys are commonly used to provide indices of wildlife population
abundance or, in some cases, estimates of true abundance. The ease of application and
the limited effort required for these surveys compared to other methods of estimating
abundance makes them very practical to use in a wide variety of field situations. Point
count surveys are also used to collect information on multiple species which saves time
and effort for monitoring multiple populations of interest. This type of survey is used to
estimate local abundance of species of interest and for large scale monitoring of bird
populations in general.
�196
The most common use of point counts is to provide an index of population
abundance or relative abundance. To make spatial or temporal comparisons valid using
this type of count requires the very restrictive assumption of equal detection probability
for the comparisons being made. There are several statistical approaches available to
estimate the detection process from point count data and provide estimates of true
abundance. Each of these statistical approaches has a set of assumptions that must be met
and require additional data to be collected and/or extra field effort. The characteristics of
a particular field situation will dictate which point count method is applicable based on
the validity of particular assumptions.
In this thesis we have reviewed several of the available approaches to estimating
true abundance from point count data and provided more general modeling frameworks
for the multiple observer and time of detection approaches. We have discussed the
components of the detection process and factors that will affect it. We have presented a
multiple species modeling framework based on modeling similarities between species in
the detection process. We have also provided a method for estimating availability bias in
point counts using auxiliary data from song counts and distributions. We will review the
objectives of each chapter and the findings for each. We will then provide general
conclusions and implications about point count surveys.
Chapter 2: ESTIMATING DETECTION PROBABILITIES FROM MULTIPLE
OBSERVER POINT COUNTS
Objectives
1. Present and illustrate the two independent observer method and potential models
for estimating detection probability, including the use of detection distance
�197
covariates, showing that the models are essentially closed capturerecapture
models.
2. Present and illustrate a more general model using four independent observers
showing that multiple observer models are essentially closed capturerecapture
models that allow for individual heterogeneity.
3. Compare the efficiency of the two independent observer approach to the primarysecondary observer approach of Nichols et al. (2002).
4. Simulate data under a heterogeneous model to illustrate the levels of
heterogeneity typically present in data and the effect of heterogeneity on twoobserver models and index counts.
Implications and Findings
1. The independent observer approach gives more efficient (smaller variance)
estimates than the primarysecondary observer approach.
2. Twoindependent observer models appear to work well in practice and give
reasonably precise estimates.
3. Fourindependent observer model estimates indicate a negative bias in the twoindependent observer model estimates.
4. Individual heterogeneity in detection probabilities is important, can be extreme
and leads to negative bias in abundance estimates when models fail to account for
it.
�198
Chapter 3: TIME OF DETECTION METHOD FOR ESTIMATING ABUNDANCE
FROM POINT COUNT SURVEYS
Objectives
1. Present a time of detection approach using a capturerecapture framework based
on all detections of an individual as a more general alternative to the removal
approach presented by Farnsworth et al. (2002).
2. Discuss field methods required to collect data suitable for this method.
3. Present a finitemixture model for individual unobservable heterogeneity and a
covariate model for observable heterogeneity.
4. Illustrate our methods with an example for threeunequal interval point counts and
another for a fourequal interval point count survey.
Implications and Findings
1. The time of removal method (Farnsworth et al. 2002) is a special case of the time
of detection method.
2. Using the full detection history in a capturerecapture framework is more efficient
(smaller variance) than the time of removal approach using only first detections.
3. Time effects can be an important source of variation in point count surveys, which
cannot be modeled with the removal method.
4. Individual heterogeneity is important and leads to negative bias in abundance
estimates when models fail to account for it.
5. Modeling the detection process is important to estimate abundance but the
contribution of the availability component and detection given availability to the
overall probability of detection is unknown.
�199
Chapter 4: MULTIPLE SPECIES ANALYSIS OF POINT COUNT DATA: A MORE
PARSIMONIOUS MODELING FRAMEWORK
Objectives
1. Present a multiple species modeling framework to achieve more parsimonious
models and explore its potential applications to three methods (distance sampling,
time of detection and multiple observer) of analyzing point count data.
2. Discuss the importance of defining species groups and describe characteristics
that can be used to define these groups.
3. Provide examples of grouping species and an analysis for each point count
method using these species groups.
Implications and Findings
1. Defining species groups based on similarities in the detection process is critical to
multiple species modeling because this leads to a biologically reasonable set of a
priori candidate models.
2. Using categorical variables or subjective rankings to define species groups is
possible but may lead to poorly defined species groups as not all observers agree
on the rankings for certain species.
3. Multiple species modeling with the distance sampling approach did not provide
better estimates than a single species approach in my examples. Further
investigation of distance models of multiple species data is necessary.
4. Multiple species models for the multipleobserver and time of detection methods
did provide better estimates (more precise) than a single species approach.
�200
5. Accounting for individual heterogeneity in detection probabilities is important in
multiple species models. Failure to account for this leads to serious negative bias
in abundance estimates.
6. This approach can be extended to modeling rare or uncommon species that
typically cannot be modeled using a single species approach because of
insufficient data. Additive species effect models will be particular useful in this
respect.
Chapter 5: MODELING THE AVAILABILITY PROCESS FOR POINT COUNT
SURVEYS USING AUXILIARY DATA
Objectives
1. Investigate models for estimating the probability of a bird being available for
detection in relation to singing rates or times using data collected separately from
point count surveys. These estimates can be applied to any point count method to
adjust for the proportion of birds that are unavailable during the point count
survey.
2. Present a homogeneous Poisson singing rate model as a simple case for modeling
the availability process.
3. Present finitemixture Poisson models as biologically reasonable models for the
availability process associated with differences in singing rates relative to
breeding phenology of an individual.
4. Discuss “size” bias that may occur in the data and how to correct for this.
�201
5. Present a nonparametric approach that uses the actual singing times of an
individual, corrects for “size” bias in the data and then estimates the availability
probability and variances by resampling the singing time data.
6. Present an example of the singing rate models using data collect on the Ovenbird
and present an example of the singing time approach using simulated data.
Implications and Findings
1. Homogeneous Poisson and finite Poisson mixture models can be fit to singing rate
data using a maximum likelihood approach.
2. Obtaining maximum likelihood estimates for the threepoint Poisson mixture
model was problematic for our data.
3. The twopoint Poisson mixture model fits the data reasonably well and gives
reasonable estimates of availability probabilities. Changes in singing rate
associated with breeding/nesting stage provide biological justification for the use
of mixture models.
4. Individual heterogeneity in singing rates is important to model when estimating
the probability a bird sings during a point count. Failure to account for individual
heterogeneity in singing rates will lead to a positive bias in availability probability
estimates or a negative bias in abundance estimates.
5. The nonparametric approach using singing time data also seems to work well
based on simulation. The singing time approach is robust to nonrandom singing
behavior and will generally be more applicable to estimating the availability
process.
�202
6. This approach works with all point count survey methods which allows for
“snapshot” type approaches when movement is thought to be a problem in point
counts.
General Conclusions
Two of the chapters in this thesis provided generalizations of existing point count
models (Chapter 2 and 3). These generalizations give a more complete set of candidate
models that can account for potential sources of variation more completely. In general
the models presented in these two chapters are more efficient (smaller variance) than the
existing models.
Species differences in the detection process do exist, as seen in the examples
presented throughout this thesis, and failure to account for this will give invalid
conclusions about species differences. For example, species with low counts may
actually be more abundant than species with high counts but are just less detectable.
Exploiting species similarities in the detection process is also important and can lead to
more parsimonious models when done properly.
In every chapter of this thesis we found that individual heterogeneity in detection
probability or singing rates was a necessary component in the models to accurately
represent the detection and/or availability process. Given this we feel that it is extremely
important to use point count methods that will be able to account for individual
heterogeneity in detection probability in order to obtain realistic estimates of abundance.
Failure to account for individual heterogeneity will lead to negatively biased abundance
estimates and will give misleading spatial, temporal and species comparisons.
�203
It is possible that individual heterogeneity in detection probabilities is related to
differences in breeding stage among individuals. Singing rates or the number of times a
bird sings during a point count will influence the probability that an individual is
detected. There is sufficient evidence to support changes in singing rate of an individual
with respect to changes in breeding stage (Wilson and Bart 1985, Best 1981). The effect
of singing rate on the detection process is not well understood and the relationship of the
proportions of the population at various breeding stages to the finitemixture models of
heterogeneity is speculative. Further investigation of these relationships is necessary.
�Appendix 2
SURVIV Code:
Time of Detection Analysis
�205
SURVIV code for single species time of detection analysis of threeunequal interval
point count survey data.
proc title TIME OF DETECTION, POINT COUNT3 PERIODS;
proc model npar=7;
/* SPECIES 1, BTNW */
cohort=444;
/* COHORT = # OF SPECIES 1 SEEN DURING COUNT
ENTER THE NUMBER OF BIRDS THAT HAD EACH
HISTORY, THE ORDER IS X111, X100, X010, X001
X110, X101, AND X011*/
191: (1s(6))*s(7)*s(7)*s(7)+s(6)*(((1exp(3.*s(1)))*
(1exp(2.*(s(2)+s(4))))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
34: (1s(6))*s(7)*(1s(7))*(1s(7))+s(6)*((1exp(3.*s(1)))*
(exp(2.*(s(2)+s(4))))*
(exp(5.*(s(3)+s(5)))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
26: (1s(6))*(1s(7))*s(7)*(1s(7))+s(6)*(((exp(3.*s(1)))*
(1exp(2.*s(2)))*
(exp(5.*(s(2)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
46: (1s(6))*(1s(7))*(1s(7))*s(7)+s(6)*(((exp(3.*s(1)))*
(exp(2.*s(2)))*
(1exp(5.*s(3)))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
44: (1s(6))*s(7)*s(7)*(1s(7))+s(6)*(((1exp(3.*s(1)))*
(1exp(2.*(s(2)+s(4))))*
(exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
66: (1s(6))*s(7)*(1s(7))*s(7)+s(6)*(((1exp(3.*s(1)))*
(exp(2.*(s(2)+s(4))))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
�206
(exp(5.*s(3))))));
37: (1s(6))*s(7)*s(7)*(1s(7))+s(6)*(((exp(3.*s(1)))*
(1exp(2.*s(2)))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
Labels /* TO HELP IDENTIFY PARAMETERS */;
s(1)=p11 (p1 species 1);
s(2)=p21 (p2 species 1);
s(3)=p31 (p3 species 1);
s(4)=c21 (c2 species 1);
s(5)=c31 (c3 species 1);
proc estimate name=b;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=s(5);
s(6)=1.; s(7)=0.;
proc estimate name=t;
initial;
constraints; s(4)=0.; s(5)=0.;
s(6)=1.; s(7)=0.;
proc estimate name=dot;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=0.; s(5)=0.;
s(6)=1.; s(7)=0.;
proc estimate name=h1;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=0.; s(5)=0.;
S(7)=1.;
proc estimate name=h9;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=0.; s(5)=0.;
S(7)=0.9;
proc estimate name=bh1;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=s(5);
s(7)=1.;
proc estimate name=bh9;
initial;
constraints; s(1)=s(2); s(1)=s(3); s(4)=s(5);
s(7)=0.9;
proc test;
�207
SURVIV code for fourspecies time of detection analysis of threeunequal interval
point count survey data.
proc title TIME OF DETECTION, POINT COUNT3 PERIODS;
proc model npar=28;
/* SPECIES 1, RN */
cohort=54;
/* COHORT = # OF SPECIES 1 SEEN DURING COUNT
ENTER THE NUMBER OF BIRDS THAT HAD EACH
HISTORY, THE ORDER IS X111, X100, X010, X001
X110, X101, AND X011*/
13: (1s(6))*s(7)*s(7)*s(7)+s(6)*(((1exp(3.*s(1)))*
(1exp(2.*(s(2)+s(4))))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
8: (1s(6))*s(7)*(1s(7))*(1s(7))+s(6)*((1exp(3.*s(1)))*
(exp(2.*(s(2)+s(4))))*
(exp(5.*(s(3)+s(5)))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
5: (1s(6))*(1s(7))*s(7)*(1s(7))+s(6)*(((exp(3.*s(1)))*
(1exp(2.*s(2)))*
(exp(5.*(s(2)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
14: (1s(6))*(1s(7))*(1s(7))*s(7)+s(6)*(((exp(3.*s(1)))*
(exp(2.*s(2)))*
(1exp(5.*s(3)))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
6: (1s(6))*s(7)*s(7)*(1s(7))+s(6)*(((1exp(3.*s(1)))*
(1exp(2.*(s(2)+s(4))))*
(exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
3: (1s(6))*s(7)*(1s(7))*s(7)+s(6)*(((1exp(3.*s(1)))*
(exp(2.*(s(2)+s(4))))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
�208
(exp(5.*s(3))))));
5: (1s(6))*s(7)*s(7)*(1s(7))+s(6)*(((exp(3.*s(1)))*
(1exp(2.*s(2)))*
(1exp(5.*(s(3)+s(5))))))/
(1((1s(6))*(1s(7))*(1s(7))*(1s(7))+s(6)*
((exp(3.*s(1)))*(exp(2.*s(2)))*
(exp(5.*s(3))))));
/* SPECIES 2, TT */
cohort=104;
/* COHORT = # OF SPECIES 2 SEEN DURING COUNT
ENTER THE NUMBER OF BIRDS THAT HAD EACH
HISTORY, THE ORDER IS X111, X100, X010, X001
X110, X101, AND X011*/
28: (1s(13))*s(14)*s(14)*s(14)+s(13)*(((1exp(3.*s(8)))*
(1exp(2.*(s(9)+s(11))))*
(1exp(5.*(s(10)+s(12))))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
24: (1s(13))*s(14)*(1s(14))*(1s(14))+s(13)*((1exp(3.*s(8)))*
(exp(2.*(s(9)+s(11))))*
(exp(5.*(s(10)+s(12)))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
8: (1s(13))*(1s(14))*s(14)*(1s(14))+s(13)*(((exp(3.*s(8)))*
(1exp(2.*s(9)))*
(exp(5.*(s(9)+s(12))))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
22: (1s(13))*(1s(14))*(1s(14))*s(14)+s(13)*(((exp(3.*s(8)))*
(exp(2.*s(9)))*
(1exp(5.*s(10)))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
10: (1s(13))*s(14)*s(14)*(1s(14))+s(13)*(((1exp(3.*s(8)))*
(1exp(2.*(s(9)+s(11))))*
(exp(5.*(s(10)+s(12))))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
9: (1s(13))*s(14)*(1s(14))*s(14)+s(13)*(((1exp(3.*s(8)))*
(exp(2.*(s(9)+s(11))))*
(1exp(5.*(s(10)+s(12))))))/
�209
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
3: (1s(13))*s(14)*s(14)*(1s(14))+s(13)*(((exp(3.*s(8)))*
(1exp(2.*s(9)))*
(1exp(5.*(s(10)+s(12))))))/
(1((1s(13))*(1s(14))*(1s(14))*(1s(14))+s(13)*
((exp(3.*s(8)))*(exp(2.*s(9)))*
(exp(5.*s(10))))));
/* SPECIES 3, WO */
cohort=106;
/* COHORT = # OF SPECIES 3 SEEN DURING COUNT
ENTER THE NUMBER OF BIRDS THAT HAD EACH
HISTORY, THE ORDER IS X111, X100, X010, X001
X110, X101, AND X011*/
43: (1s(20))*s(21)*s(21)*s(21)+s(20)*(((1exp(3.*s(15)))*
(1exp(2.*(s(16)+s(18))))*
(1exp(5.*(s(17)+s(19))))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
14: (1s(20))*s(21)*(1s(21))*(1s(21))+s(20)*((1exp(3.*s(15)))*
(exp(2.*(s(16)+s(18))))*
(exp(5.*(s(17)+s(19)))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
0: (1s(20))*(1s(21))*s(21)*(1s(21))+s(20)*(((exp(3.*s(15)))*
(1exp(2.*s(16)))*
(exp(5.*(s(16)+s(19))))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
19: (1s(20))*(1s(21))*(1s(21))*s(21)+s(20)*(((exp(3.*s(15)))*
(exp(2.*s(16)))*
(1exp(5.*s(17)))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
15: (1s(20))*s(21)*s(21)*(1s(21))+s(20)*(((1exp(3.*s(15)))*
(1exp(2.*(s(16)+s(18))))*
(exp(5.*(s(17)+s(19))))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
8: (1s(20))*s(21)*(1s(21))*s(21)+s(20)*(((1exp(3.*s(15)))*
�210
(exp(2.*(s(16)+s(18))))*
(1exp(5.*(s(17)+s(19))))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
7: (1s(20))*s(21)*s(21)*(1s(21))+s(20)*(((exp(3.*s(15)))*
(1exp(2.*s(16)))*
(1exp(5.*(s(17)+s(19))))))/
(1((1s(20))*(1s(21))*(1s(21))*(1s(21))+s(20)*
((exp(3.*s(15)))*(exp(2.*s(16)))*
(exp(5.*s(17))))));
/* SPECIES 4, WW */
cohort=153;
/* COHORT = # OF SPECIES 4 SEEN DURING COUNT
ENTER THE NUMBER OF BIRDS THAT HAD EACH
HISTORY, THE ORDER IS X111, X100, X010, X001
X110, X101, AND X011*/
103: (1s(27))*s(28)*s(28)*s(28)+s(27)*(((1exp(3.*s(22)))*
(1exp(2.*(s(23)+s(25))))*
(1exp(5.*(s(24)+s(26))))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
8: (1s(27))*s(28)*(1s(28))*(1s(28))+s(27)*((1exp(3.*s(22)))*
(exp(2.*(s(23)+s(25))))*
(exp(5.*(s(24)+s(26)))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
1: (1s(27))*(1s(28))*s(28)*(1s(28))+s(27)*(((exp(3.*s(22)))*
(1exp(2.*s(23)))*
(exp(5.*(s(23)+s(26))))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
11: (1s(27))*(1s(28))*(1s(28))*s(28)+s(27)*(((exp(3.*s(22)))*
(exp(2.*s(23)))*
(1exp(5.*s(24)))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
11: (1s(27))*s(28)*s(28)*(1s(28))+s(27)*(((1exp(3.*s(22)))*
(1exp(2.*(s(23)+s(25))))*
(exp(5.*(s(24)+s(26))))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
�211
(exp(5.*s(24))))));
9: (1s(27))*s(28)*(1s(28))*s(28)+s(27)*(((1exp(3.*s(22)))*
(exp(2.*(s(23)+s(25))))*
(1exp(5.*(s(24)+s(26))))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
10: (1s(27))*s(28)*s(28)*(1s(28))+s(27)*(((exp(3.*s(22)))*
(1exp(2.*s(23)))*
(1exp(5.*(s(24)+s(26))))))/
(1((1s(27))*(1s(28))*(1s(28))*(1s(28))+s(27)*
((exp(3.*s(22)))*(exp(2.*s(23)))*
(exp(5.*s(24))))));
Labels /* TO HELP IDENTIFY PARAMETERS */;
s(1)=p11 (p1 species 1);
s(2)=p21 (p2 species 1);
s(3)=p31 (p3 species 1);
s(4)=c21 (c2 species 1);
s(5)=c31 (c3 species 1);
s(6)=pi1 (pi species 1);
s(7)=gc1 (p grp spp 1);
s(8)=p12 (p1 species 2);
s(9)=p22 (p2 species 2);
s(10)=p32 (p3 species 2);
s(11)=c22 (c2 species 2);
s(12)=c32 (c3 species 2);
s(13)=pi2 (pi species 2);
s(14)=gc2 (p grp spp 2);
s(15)=p13 (p1 species 3);
s(16)=p23 (p2 species 3);
s(17)=p33 (p3 species 3);
s(18)=c23 (c2 species 3);
s(19)=c33 (c3 species 3);
s(20)=pi3 (pi species 3);
s(21)=gc3 (p grp spp 3);
s(22)=p14 (p1 species 4);
s(23)=p24 (p2 species 4);
s(24)=p34 (p3 species 4);
s(25)=c24 (c2 species 4);
s(26)=c34 (c3 species 4);
s(27)=pi4 (pi species 4);
s(28)=gc4 (p grp spp 4);
proc estimate name=M();
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
�212
s(1)=s(15); s(1)=s(16); s(1)=s(17);
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M()grp;
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(8)=s(9); s(8)=s(10);
s(15)=s(16); s(15)=s(17);
s(22)=s(23); s(22)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M(t);
initial;
constraints;
s(1)=s(8); s(2)=s(9); s(3)=s(10);
s(1)=s(15); s(2)=s(16); s(3)=s(17);
s(1)=s(22); s(2)=s(23); s(3)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M(t)grp;
initial;
constraints;
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M(b);
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
s(1)=s(15); s(1)=s(16); s(1)=s(17);
�213
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=s(5);
s(4)=s(11); S(4)=s(12);
s(4)=s(18); s(4)=s(19);
s(4)=s(25); s(4)=s(26);
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M(b)grp;
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(8)=s(9); s(8)=s(10);
s(15)=s(16); s(15)=s(17);
s(22)=s(23); s(22)=s(24);
s(4)=s(5);
S(11)=s(12);
s(18)=s(19);
s(25)=s(26);
s(6)=1.; s(13)=1.; s(20)=1.; s(27)=1.;
s(7)=0.; s(14)=0.; s(21)=0.; s(28)=0.;
proc estimate name=M(h);
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
s(1)=s(15); s(1)=s(16); s(1)=s(17);
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(h)grp;
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(8)=s(9); s(8)=s(10);
s(15)=s(16); s(15)=s(17);
s(22)=s(23); s(22)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(bh);
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
s(1)=s(15); s(1)=s(16); s(1)=s(17);
�214
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=s(5);
s(4)=s(11); S(4)=s(12);
s(4)=s(18); s(4)=s(19);
s(4)=s(25); s(4)=s(26);
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(bh)grp;
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(8)=s(9); s(8)=s(10);
s(15)=s(16); s(15)=s(17);
s(22)=s(23); s(22)=s(24);
s(4)=s(5);
S(11)=s(12);
s(18)=s(19);
s(25)=s(26);
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(th);
initial;
constraints;
s(1)=s(8); s(2)=s(9); s(3)=s(10);
s(1)=s(15); s(2)=s(16); s(3)=s(17);
s(1)=s(22); s(2)=s(23); s(3)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(th)grp;
initial;
constraints;
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(h*);
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
s(1)=s(15); s(1)=s(16); s(1)=s(17);
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
�215
s(6)=s(13); s(6)=s(20); s(6)=s(27);
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(bh*);
initial;
constraints; s(1)=s(2); s(1)=s(3);
s(1)=s(8); s(1)=s(9); s(1)=s(10);
s(1)=s(15); s(1)=s(16); s(1)=s(17);
s(1)=s(22); s(1)=s(23); s(1)=s(24);
s(4)=s(5);
s(4)=s(11); S(4)=s(12);
s(4)=s(18); s(4)=s(19);
s(4)=s(25); s(4)=s(26);
s(6)=s(13); s(6)=s(20); s(6)=s(27);
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc estimate name=M(th*);
initial;
constraints;
s(1)=s(8); s(2)=s(9); s(3)=s(10);
s(1)=s(15); s(2)=s(16); s(3)=s(17);
s(1)=s(22); s(2)=s(23); s(3)=s(24);
s(4)=0.; s(5)=0.;
s(11)=0.; s(12)=0.;
s(18)=0.; s(19)=0.;
s(25)=0.; s(26)=0.;
s(6)=s(13); s(6)=s(20); s(6)=s(27);
s(7)=0.9; s(14)=0.9; s(21)=0.9; s(28)=0.9;
proc test;
�Appendix 3
Singing Time Programs
�217
MATLAB code for singing time analysis.
cdf = [0.229381331, 0.353629552, 0.474691922, 0.565488698, 0.645772164,
0.696745794, 0.729514555, ...
0.784470499, 0.814205116, 0.85625836, 0.903177269, 0.928664083,
0.953660767, 0.972775878, 0.981696263, ...
0.987669735, 0.99104299, 0.995290792, 0.997302909, 0.998258665,
0.999168908, 0.999168908,1]; % cdf is : 1*23
sample = 3*60; % sampling interval
interval = 5*60; % total duration for experiment data
rate_freq = zeros(23,2); rate_freq(1:23,1) = (1:1:23)'; % data entry numbers for
each rate from 1 to 23
rate_freq(1:23,2) = [12, 13, 19, 19, 21, 16, 12, 23, 14, 22, 27, 16, 17, 14, 7, 5, 3, 4,
2, 1, 1, 0, 1]'; % need to change for new data
rate_details =
[1,293,298,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0;
1,288,293,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0;
1,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0; … Continue data matrix];
rand('state',sum(100*clock));
loop1 = 1000; % outer loop
loop2 = 100; % inner loop
for i = 1:1:loop1,
counts = 0; % total number of the overlapping counts
for j = 1:1:loop2,
seed = rand(1); % determine direction
if seed <=0.5,
direction = 1;
else
direction = 1;
end
ratef = 0;
while (ratef ==0),
seed = rand(1); % determine rate
for k=1:1:22, % pick a rate
if seed <=cdf(1),
rate = 1;
break;
�218
elseif (seed > cdf(k) & seed <= cdf(k+1))
rate = k+1;
break;
end
end
% pick a record, need the information of how many records for each rate
ratef = rate_freq(rate,2);
end
if rate > 1,
rate_end = sum(rate_freq(1:rate1,2));
else
rate_end = 0;
end
rate_start = rate_end + 1; % locate rate_details
new_rate_end = rate_start + ratef  1; % now we need to pick a record from
rate_details(rate_start:new_rate_end,:)
dh = 1.0/ratef;
new_cdf = 0:dh:1;
seed = rand(1);
for k=1:1:ratef,
if seed <=new_cdf(1)  ratef ==1,
record = 1;
break;
elseif (seed > new_cdf(k) & seed <= new_cdf(k+1))
record = k+1;
break;
end
end
% now the record is the number inside the subset of the choosen rate
index1 = rate_start + record  1;
if direction ==1,
start1 = (intervalsample)*rand(1); % generate start point from [0,intervalsample]
end1 = start1 + sample;
m = 2;
while rate_details(index1,m) ~=0 & m <=11
if rate_details(index1,m)>=start1 & rate_details(index1,m)<=end1,
counts = counts + 1;
break
end
m = m + 1;
end
else
start1 = sample + (intervalsample)*rand(1); % generate start point from
[sample,interval]
�219
end1 = start1  sample;
m = 2;
while rate_details(index1,m) ~=0 & m <=11
if rate_details(index1,m)>=end1 & rate_details(index1,m)<=start1,
counts = counts + 1;
break
end
m = m + 1;
end
end
end
est_p(i) = counts/loop2;
end
out=est_p';
save sing_rand3.out out ascii double
mean(est_p)
var(est_p)
�220
Singing Time Program for a Pocket PC.
�
Dublin Core
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Title
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Dissertations/Theses
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Text
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Dublin Core
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Title
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Avian point count surveys: estimating components of the detection process
Description
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Point count surveys of birds are commonly used to provide indices of abundance or, in some cases, estimates of true abundance. The most common use of point counts is to provide an index of population abundance or relative abundance. To make spatial or temporal comparisons valid using this type of count requires the very restrictive assumption of equal detection probability for the comparisons being made. We developed a multipleindependent observer approach to estimating abundance for point count surveys as a modification of the primarysecondary observer approach. This approach uses standard capturerecapture models, including models of inherent individual heterogeneity in detection probabilities and models using individual covariates to account for observable heterogeneity in detection probabilities. We also developed a time of detection approach for estimating avian abundance when birds are detected aurally, which is a modification of the time of removal approach. This approach requires collecting detection histories of individual birds in consecutive time intervals and modeling the detection process using a capturerecapture framework. This approach incorporates both the probability a bird is available for detection and the probability of detection given availability. We also present a multiple species modeling strategy since many point count surveys collect data on multiple species and present the approach for distance sampling, multiple observer, and time of detection approaches. The purpose of using a multiple species modeling approach is to obtain more parsimonious models by exploiting similarities in the detection process among species. We present a method for defining species groups which leads to an a priori set of species groups and associated candidate models. Finally, we present a method for estimating the availability probability of birds during a point count based on singing rate or detailed singing time data. This approach requires data collected in conjunction with point count surveys that describe the singing rates or singing time distribution of the bird population of interest. The singing rate approach requires the assumption that an individual bird sings following a random process but rates may vary between birds. Analyses presented throughout this thesis demonstrate the importance of accurately modeling the detection process to estimate abundance. The importance of accounting for individual heterogeneity in detection probabilities was evident in every chapter. Using a point count method that accounts for individual heterogeneity is crucial to estimating abundance effectively and making valid spatial, temporal and species comparisons.
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<p>Alldredge, M. W. 2004. Avian point count surveys: estimating components of the detection process. Dissertation, North Carolina State University, Raleigh, USA. <a href="https://repository.lib.ncsu.edu/bitstream/handle/1840.16/4576/etd.pdf?sequence=1&isAllowed=y" target="_blank" rel="noreferrer noopener">https://repository.lib.ncsu.edu/bitstream/handle/1840.16/4576/etd.pdf</a></p>
Creator
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Alldredge, Mathew W.
Subject
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Ecology
Zoology
Biostatistics
Extent
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240 pages
Date Created
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2004
Rights
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<a href="http://rightsstatements.org/vocab/InCNC/1.0/" target="_blank" rel="noreferrer noopener">In Copyright  NonCommercial Use Permitted</a>
Type
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Text
Format
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application/pdf
Language
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English
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Dissertation/Thesis
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North Carolina State University