1 Running Head: HIERARCHICAL REMOVAL MODELS · 1 3Present Address: Idaho Cooperative Fish and Wildlife Research Unit, Department of Fish and Wildlife Sciences, University of Idaho,
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1 3Present Address: Idaho Cooperative Fish and Wildlife Research Unit, Department of Fish and Wildlife Sciences, University of Idaho, 875 Perimeter Drive, MS 1141, Moscow, ID 83844, USA Author email: bstevens@uidaho.edu
Running Head: HIERARCHICAL REMOVAL MODELS 1
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A hierarchical framework for estimating abundance and population growth from 3
imperfectly observed removals 4
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BRYAN S. STEVENS1,3, JAMES R. BENCE1, DAVID R. LUUKKONEN1,2, AND WILLIAM F. PORTER1 6
1Department of Fisheries and Wildlife, Michigan State University, East Lansing, Michigan 7
48824, USA 8
2Michigan Department of Natural Resources, Rose Lake Research Center, 563 E. Stoll Road, 9
East Lansing, Michigan 48823, USA 10
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Abstract. Estimating abundance and growth of animal populations are central tasks in ecology 22
and natural resource management. Removal models for estimating abundance have a long history 23
in applied ecology, and recent developments provided hierarchical extensions that account for 24
spatially replicated sampling and heterogeneous capture probabilities. Measurement error is 25
common to removal data collected from many broad-scale monitoring programs, however, and a 26
general framework for population assessment using removal data in the presence of measurement 27
error is lacking. We developed a hierarchical framework for estimating abundance and 28
population trends from removal experiments that are replicated in space and time that 29
accommodates measurement error, as well as heterogeneity in capture probability and animal 30
density. We describe the model for variable-effort removal sampling and use it to estimate 31
region-specific abundance and population trends for wild turkeys (Meleagris gallopavo) in 32
Michigan, USA. We used a Bayesian approach for estimation and inference and fit models using 33
daily hunter harvest and effort estimates collected over 5 management regions for 14 annual 34
hunting seasons. Our analyses provide evidence for spatially-heterogeneous capture probabilities 35
among regions and turkey densities that were heterogeneous in both space and time, and show 36
that populations increased slightly over the study. Our framework provides a general approach 37
for population assessment using removal data that are collected over broad scales in resource-38
management contexts (e.g., animal harvesting), facilitating formal abundance estimation instead 39
of reliance on unverified indices for tracking populations of managed species. Thus, we provide 40
a useful tool for monitoring programs to assess populations over broad scales, and therefore 41
inform decision makers about population status at spatial scales similar to those for which 42
regulatory decisions are made. 43
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Key words: catch-effort, depletion sample, harvest management, hierarchical model, Meleagris 44
gallopavo, removal sample, wildlife monitoring, wild turkey 45
INTRODUCTION 46
Estimating abundance and population growth are central tasks in ecology, conservation, 47
and natural resource management. Ecology has been referred to as study of the distribution and 48
abundance of organisms (Kéry and Shaub 2012), and reliable information about population size 49
is central to understanding a broad array of theoretical and applied concepts, including density 50
dependence (Dennis 2002, Dennis et al. 2006, Lebreton 2009), harvest theory (Ricker 1954, 51
Schaefer 1954), extinction risks (Pimm et al. 1988), and biogeographic patterns and processes 52
(Brown 1984, Brown et al. 1996). Accurate abundance estimates are also vital in the context of 53
conservation and natural resource management, where they play a critical role in making and 54
evaluating the effectiveness of decisions (Williams et al. 2002, Nichols and Williams 2006). 55
Indeed, accurate information about population size is generally treated as a prerequisite for 56
decision-analytic procedures commonly advocated for state-dependent decision making in 57
conservation and resource management (Johnson et al. 1997, Kendall 2001, Martin et al. 2009). 58
Yet estimates of abundance, rather than just indices that are assumed proportional to abundance, 59
are often challenging for management agencies to generate at broad scales where decisions are 60
made because they are logistically difficult and costly to obtain. 61
Removal models are a well-established framework for estimating abundance of animal 62
populations (Moran 1951, Zippin 1956, Borchers et al. 2002). Removal models are developed 63
from sampling designs (hereafter referred to as removal experiments) under which individual 64
animals are captured and removed from a population over successive occasions (hereafter 65
referred to as trials), where the population is assumed to be closed to additions or other removals 66
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for the duration of the experiment (over all trials). A typical example records the number of 67
animals removed from a population over a series of days; however, removal trials can be defined 68
in other discrete units (e.g., multi-day periods, single electrofishing passes). Sampling intensity is 69
often assumed constant during each successive trial (fixed-effort sampling; Moran 1951, Zippin 70
1956, Dorazio et al. 2005), but models can be developed with variable sampling effort over trials 71
when data on that effort is available (DeLury 1946, Gould and Pollock 1997, St. Clair et al. 72
2013). Removal experiments thus measure change in the absolute number of removals or 73
removal rates over time, and as the remaining population is depleted the expected number of 74
removals for each trial gets smaller. Moreover, changes in both the observed and expected 75
number of removals across trials provides the information needed to estimate abundance at the 76
beginning of the experiment (DeLury 1946, Moran 1951, Zippin 1956). Given the nature of 77
removal sampling, collection of data under such designs is natural for harvested species. 78
Removal sampling is not limited to harvested animals, however, as a removal can be broadly 79
defined. As such, removal designs are commonly employed to estimate local abundance in 80
applied ecological studies (e.g., electrofishing surveys of stream fishes; Bohlin and Sundstrӧm 81
1977, Wyatt 2002). 82
Basic removal models have typically used a multinomial or conditional binomial 83
likelihood to model the data generating process, and in their original form assumed a constant 84
capture probability (Moran 1951, Zippin 1956, Borchers et al. 2002). In reality, capture 85
probabilities can be heterogeneous among individual animals, over trials within the same 86
experiment, or among experiments replicated in space and time (Lewis and Farrar 1960, Bohlin 87
and Sundstrӧm 1977, Gould and Pollock 1997, Borchers et al. 2002, Dorazio et al. 2005, 88
Mäntyniemi et al. 2005, St. Clair et al. 2013). Hierarchical statistical models have been used to 89
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accommodate many of these heterogeneities, and to explicitly model the distribution of 90
abundance and capture probability among sites and years for replicated experiments. For 91
example, Mäntyniemi et al. (2005) developed a model that accommodated heterogeneous capture 92
probabilities among animals by assuming the distribution of capture probabilities among 93
individuals followed a beta distribution. Others explicitly modeled the distribution of capture 94
probabilities and abundance among sites, years, or both for designs where entire experiments 95
were replicated (Dorazio et al. 2005, Royle and Dorazio 2006, Rivot et al. 2008). 96
Despite recent advances, there often remain practical limitations to the implementation of 97
removal models for estimating abundance over broad spatial scales. With some exceptions (e.g., 98
Rivot et al. 2008) there has been relatively little consideration of experiments that are replicated 99
in time, and data from individual years are often modeled separately (e.g., St. Clair et al. 2013) 100
despite the possibility of information sharing among temporal replicates. Existing models also 101
assume the number of animals removed and sampling efforts executed on each trial are perfectly 102
observed (Borchers et al. 2002). Perfect observation is not realistic for many broad-scale 103
monitoring programs collecting removal data, particularly for managed species where removals 104
occur through recreational or commercial harvest and monitoring produces estimates of harvest 105
and effort from samples. Measurement error in removal data results in overestimation of 106
abundance (Gould et al. 1997), and limits application of removal models for many existing data 107
sets. Thus, a pragmatic framework for the analysis of spatially- and temporally-replicated 108
removal experiments in the presence of measurement error is needed. To address this need, we 109
developed a hierarchical removal modeling framework for estimating abundance and population 110
growth when removal data are not perfectly observed. In addition, we sought to specifically 111
accommodate characteristics common to harvest monitoring programs implemented by resource-112
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management agencies: variable-effort sampling, heterogeneous abundance and capture 113
probabilities, and sampling error in the measurement of both the number of removals and 114
removal effort. Here we describe a hierarchical framework wherein models for fixed-effort or 115
perfectly observed data, as well as models for removal experiments replicated in either space or 116
time, can be considered as special cases. We demonstrate the methods to estimate region-specific 117
abundance and growth of wild turkey (Meleagris gallopavo) populations in southern Michigan, 118
USA, and use simulation to evaluate estimator performance under a variety of plausible 119
scenarios. 120
METHODS 121
Model description 122
We developed models for replicated removal experiments based on a model for variable-123
effort sampling replicated over S sites and T years that is extended to accommodate 124
heterogeneity in removal probability and animal density, and also measurement error in removal 125
data (Fig. 1; Table 1). Several individual components of the model we present were described 126
elsewhere (Borchers et al. 2002, Dorazio et al. 2005, Rivot et al. 2008, St. Clair et al. 2013). 127
However, our goal was to synthesize these components and extend the model to a flexible 128
approach for analysis that accommodated imperfectly observed data from spatially- and 129
temporally-replicated samples (i.e., under so-called metapopulation designs; Kéry and Royle 130
2010). 131
We started with a conditional binomial likelihood model for removals (Borchers et al. 132
2002): 133
𝐿𝐿�𝑁𝑁,𝜃𝜃� = ∏ ∏ ∏ �𝑁𝑁𝑗𝑗,𝑠𝑠,𝑡𝑡𝑦𝑦𝑗𝑗,𝑠𝑠,𝑡𝑡
� 𝑝𝑝𝑗𝑗,𝑠𝑠,𝑡𝑡𝑦𝑦𝑗𝑗,𝑠𝑠,𝑡𝑡�1 − 𝑝𝑝𝑗𝑗,𝑠𝑠,𝑡𝑡�
𝑁𝑁𝑗𝑗,𝑠𝑠,𝑡𝑡−𝑦𝑦𝑗𝑗,𝑠𝑠,𝑡𝑡𝐽𝐽𝑗𝑗=1
𝑆𝑆𝑠𝑠=1
𝑇𝑇𝑡𝑡=1 , 1) 134
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where 𝑁𝑁 is the vector of abundances at the start of each trial (j), 𝑝𝑝𝑗𝑗,𝑠𝑠,𝑡𝑡 is the probability an 135
individual animal present at the start of trial j in region s and year t is removed during that trial, 136
and 𝜃𝜃 represents a vector of parameters determining the removal probabilities. Thus 𝑁𝑁𝑗𝑗 137
represents the abundance of animals at the start of trial j and 𝑦𝑦𝑗𝑗 is the number of animals 138
removed on that trial, where 𝑁𝑁𝑗𝑗 = 𝑁𝑁𝑗𝑗−1 − 𝑦𝑦𝑗𝑗−1 for j > 1. Under variable-effort sampling, we can 139
model 𝑝𝑝𝑗𝑗,𝑠𝑠,𝑡𝑡 = 1 − (1 − 𝜑𝜑)𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 as a function of the magnitude of sampling effort (St. Clair et al. 140
2013), where 𝜑𝜑 is the per-unit-effort removal probability and 𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 is a forcing variable 141
representing the sampling intensity. As such, 1 −𝜑𝜑 represents the probability of not removing an 142
animal for one unit of effort, and (1 −𝜑𝜑)𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 is the probability of not removing an animal given 143
all 𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 units of effort. With this model for 𝑝𝑝𝑗𝑗,𝑠𝑠,𝑡𝑡, 𝜃𝜃 in equation 1 only has one element, the per-144
unit-effort removal probability (𝜑𝜑). 145
Equation 1 is easily extended when heterogeneity in removal probability or animal 146
density exists among animal groups (e.g., age or sex classes; see Example below): 147
𝐿𝐿�𝑁𝑁,𝜃𝜃� = ∏ ∏ ∏ ∏ �𝑁𝑁𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡
� 𝑝𝑝𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡�1 − 𝑝𝑝𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡�
𝑁𝑁𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡−𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡𝐼𝐼𝑖𝑖=1
𝐽𝐽𝑗𝑗=1
𝑆𝑆𝑠𝑠=1
𝑇𝑇𝑡𝑡=1 2) 148
for I animal groups, where 𝑝𝑝𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡 = 1 − (1 − 𝜑𝜑𝑖𝑖)𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 . In this context the primary goal of 149
modeling is to estimate the initial abundances of each animal group for each site (s) and year (t), 150
𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡, and the per-unit-effort removal probability for each animal group 𝜑𝜑𝑖𝑖. Total abundance at 151
the start of each removal sequence is the sum of abundance across all groups (𝑁𝑁𝑠𝑠,𝑡𝑡 = ∑ 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡𝑖𝑖 ). 152
Lastly, un-modeled heterogeneity in removal probability is widely known to affect performance 153
of closed-population abundance estimators (Moran 1951, Bohlin and Sundstrӧm 1977, Gould 154
and Pollock 1997, Borchers et al. 2002, Mäntyniemi et al. 2005, St. Clair et al. 2013). To address 155
this, additional structure in 𝜑𝜑𝑖𝑖 can be modeled explicitly using the logistic function (i.e., for 156
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space, time, or trial-level covariates), as is well established in capture-recapture analyses 157
(Pollock et al. 1984, Huggins 1989, Alho 1990, Royle and Link 2002, Royle and Dorazio 2006). 158
In their basic form, equations 1 and 2 assume no specific structure to the number of 159
animals in the population at the start of removal sampling (i.e., 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡 are unconstrained). Rather 160
than estimating initial abundances for each removal experiment as an unconstrained (or nearly 161
unconstrained) parameter, we modeled the distribution of initial abundance as an inhomogeneous 162
Poisson process (Dorazio et al. 2005, Royle and Dorazio 2006, Rivot et al. 2008): 163
𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃(𝜇𝜇𝑖𝑖,𝑠𝑠,𝑡𝑡), 3) 164
𝜇𝜇𝑖𝑖,𝑠𝑠,𝑡𝑡 = 𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡 × 𝐴𝐴𝑠𝑠,𝑡𝑡, 4) 165
where 𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡 is the density of animals from group i at the start of sampling at site s, and 𝐴𝐴𝑠𝑠,𝑡𝑡 166
represents the area sampled for a given site and year (Rivot et al. 2008, St. Clair et al. 2013). The 167
hierarchical structure of equations 3 and 4 provides a constraint on the plausible values of 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡, 168
and also provides a flexible framework for modeling spatial-temporal heterogeneity in animal 169
density using log-linear regression (Royle and Dorazio 2008, St. Clair et al. 2013). 170
We modeled a stochastic observation process to accommodate imperfect observation of 171
variable-effort removal data. Equations 1-4 treat removals and effort as known without error, and 172
are thus applicable when perfect observations are feasible (e.g., stream electrofishing surveys, 173
harvest with mandatory reporting). Yet in many practical applications, especially for managed 174
populations where removals are coming from recreational or commercial harvest, observations of 175
𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡 and 𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡 are estimated from samples and therefore depart from their true values due to 176
measurement error. Thus, we augment our basic model with an observation model that treats true 177
values of removal and effort on each trial (𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡 and 𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡) as imperfectly observed latent 178
variables whose values determine the expectations of the observed data. Under this model, the 179
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true value of effort generates the true removal on each trial through a latent first-order Markov 180
process (i.e., equation 1 or 2), which in turn determines the expectation of the observed removal 181
for trial j. Specifically, we use a Poisson observation model to represent measurement error in 182
the observed values of removal (𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡∗ ) and effort (𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡
∗ ): 183
𝑦𝑦𝑖𝑖𝑗𝑗,𝑠𝑠,𝑡𝑡∗ ~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃�𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡�, 5) 184
𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡∗ ~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃�𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡�. 6) 185
Equations 5 and 6 can approximate many applications where over- and under-estimation of 186
removal and effort are both plausible, and the observation models could also be adapted for 187
specific applications if additional information on the measurement process were available. 188
Lastly, we employ a Bayesian paradigm for all model fitting and inference in order to 189
facilitate a straightforward and flexible framework for application. The equations and 190
distributional assumptions about the data and parameters ultimately determine the posterior 191
distribution of parameters of interest, upon which inferences will be made (Table 1). 192
Conceptually, the joint posterior distribution, given the observed data, is proportional to the joint 193
likelihood of the data and the latent states multiplied by the relevant prior distributions: 194
𝑓𝑓 �𝑁𝑁,𝜑𝜑, 𝛿𝛿|𝑦𝑦∗, 𝑒𝑒∗� ∝ 𝑓𝑓 �𝑦𝑦∗, 𝑒𝑒∗|𝑦𝑦, 𝑒𝑒� 𝑓𝑓 �𝑦𝑦|𝑁𝑁,𝜑𝜑�𝑓𝑓�𝑁𝑁|, 𝛿𝛿�𝜋𝜋�𝑒𝑒�𝜋𝜋 �𝜑𝜑�𝜋𝜋�𝛿𝛿�, 7) 195
where 𝜋𝜋�𝑒𝑒�, 𝜋𝜋 �𝜑𝜑�, and 𝜋𝜋�𝛿𝛿� represent the prior distributions for true sampling efforts, per-unit-196
effort capture probability parameters, and animal density parameters, respectively. More 197
specifically, equations 5 and 6 specify probability distributions for the observed data given the 198
latent removal and sampling effort (i.e., 𝑓𝑓 �𝑦𝑦∗, 𝑒𝑒∗|𝑦𝑦, 𝑒𝑒�). Equations 3 and 4 give probability 199
distributions for initial abundance given density (i.e., 𝑓𝑓�𝑁𝑁|,𝛿𝛿�) and equations 1 and 2 describe 200
how likely a sequence of true removals (the latent variable) are, conditioned on the initial 201
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abundances and per-effort capture probabilities (i.e., 𝑓𝑓 �𝑦𝑦|𝑁𝑁,𝜑𝜑�). In practice, however, the joint 202
distributions and priors of equation 7 become functions of additional parameters when structure 203
in 𝜑𝜑 or 𝛿𝛿 is modeled using hierarchical regression (e.g., Table 1). 204
Example using wild turkey harvest data 205
We fit hierarchical removal models to a wild turkey (hereafter turkey) dataset from 206
southern Michigan, USA (Appendix A). Turkeys are a popular game species in North America 207
and are managed to provide recreational hunting opportunities (Healy and Powell 2000, Harris 208
2010). The regulatory framework in Michigan consists of 2 discrete annual hunting seasons, with 209
male-only harvests during spring and either-sex harvests in the fall of each year (Kurzejeski and 210
Vangilder 1992, Healy and Powell 2000). Male-only spring harvest is considerably larger than 211
fall either-sex harvest in Michigan. Estimates of daily harvest and hunter effort across the spring 212
hunting season contain measurement error but enable development of variable-effort removal 213
models (Appendix A). We developed models using estimates of male-only harvest and hunter 214
effort from recreational spring hunting, and consequently estimated abundance and growth for 215
the male segment of the population. Male-focused monitoring is the current status quo of turkey 216
management (Lint et al. 1995, Kurzejeski and Vangilder 1992, Healy and Powell 2000), as it 217
remains challenging to collect reliable data from females. 218
We fit removal models using data replicated over 5 geographic sites for a period of 14 219
years (2002-2015; Fig. A1). The 5 sites (hereafter regions) are regions used to manage spring 220
harvest , where individual regions range in size from 6,739-16,148 km2 (Table A1). Hunting 221
occurs each spring from mid-April through the end of May, and hunting seasons varied annually 222
in length (range = 39-45 days; Table A2) but were consistent among regions each year. Each 223
hunter is legally allowed to harvest one male turkey each spring, but hunter efforts varied among 224
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regions (Table A2) and also by day within each hunting season (Fig. A2). Peaks in effort 225
occurred at the beginning of the season and on weekends thereafter but at a reduced magnitude 226
(Fig. A2, Table A3). Estimates of daily harvest and effort constitute a spatially- and temporally 227
replicated but imperfectly observed removal sample from each region and year, where individual 228
days and the number of hunters per day represented removal trials and sampling efforts, 229
respectively (Appendix A). 230
We fit multiple models to reflect hypothesized heterogeneity in removal probability and 231
turkey density (Table B1). The most complicated capture probability model we considered 232
contained age-group effects (i.e., juvenile and adult male turkeys) and within-experiment trends 233
over removal trials (i.e., 𝜑𝜑 changes linearly over a hunting season), a categorical spatial effect 234
representing differences among regions, and an annual random intercept to reflect yearly changes 235
in removal probability: 236
𝑙𝑙𝑃𝑃𝑙𝑙𝑃𝑃𝑙𝑙�𝜑𝜑𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡� = 𝛽𝛽0 + 𝛽𝛽1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛽𝛽2𝑗𝑗 + 𝛽𝛽𝑠𝑠 + 𝜂𝜂𝑡𝑡, 8) 237
where 𝜂𝜂𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,𝜎𝜎𝜂𝜂2) and 𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 is a binary variable indicating juvenile or adult turkeys. 238
Thus, in this application the regions were treated as removal trial sites. In addition, we 239
considered models with reduced versions of equation 8, without the annual effect (𝜂𝜂𝑡𝑡), and 240
without the site effect (both with and without 𝜂𝜂𝑡𝑡), resulting in 4 total capture probability models. 241
Both heterogeneity in removal probability among individual animals and behavioral responses to 242
initial removal create a pattern of declining removal probability over trials as more easily 243
captured animals are removed, resulting in negatively biased abundance estimates if not 244
accounted for (Moran 1951, Bohlin and Sunderstrom 1977, Gould and Pollock 1997, Borchers et 245
al. 2002, Mantyniemi et al. 2005, St. Clair et al. 2013). Our use of the within-experiment time 246
trend covariate (𝛽𝛽2𝑗𝑗) was intended to capture such a pattern if it existed, as has been successfully 247
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demonstrated elsewhere (Schnute 1983, Hayes et al. 2007, Kéry and Royle 2010), where we 248
expected a decline in removal probability over trials if individual heterogeneity or behavioral 249
responses to removal were present. We also considered 4 hypothesized models of turkey density, 250
where all density models included effects for age-group and linear time trends describing 251
changes in density over the duration of the study, as well as an interaction term that allowed for 252
age-specific trends in abundance over time. The most complicated density model also included a 253
categorical spatial effect and annual random intercepts to reflect hypothesized spatial-temporal 254
heterogeneity in average turkey density: 255
𝑙𝑙𝑃𝑃𝑙𝑙�𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡� = 𝛼𝛼0 + 𝛼𝛼1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛼𝛼2𝑙𝑙 + 𝛼𝛼3𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 ∗ 𝑙𝑙 + 𝛼𝛼𝑠𝑠 + 𝛾𝛾𝑡𝑡, 9) 256
where 𝛾𝛾𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,𝜎𝜎𝛾𝛾2). We also considered models with reduced versions of equation 9, 257
without the annual effect (𝛾𝛾𝑡𝑡), and without the site effect (both with and without 𝛾𝛾𝑡𝑡). We fit 16 258
total models (4 removal × 4 density; Table B1) and ranked their relative support using the 259
deviance information criteria (DIC; Spiegelhalter et al. 2002). 260
We used weakly informative Cauchy priors for all logit-scale regression coefficients 261
(Gelman et al. 2008), normal priors for all log-scale coefficients, and uniform priors for all 262
random effect variance parameters (Gelman et al. 2014; Table 1). Because per-unit effort capture 263
probabilities are close to zero for the levels of effort observed in this study, we used a Cauchy 264
prior with scale = 10 for the logit-scale intercept parameter, which produces a u-shaped 265
distribution on the real probability scale that is appropriate when mean probabilities are close to 266
zero or one (Fig. B1; Gelman et al. 2008). Similarly, we used Cauchy priors with scale = 2.5 for 267
additional logit-scale regression coefficients, which produces a distribution on the real 268
probability scale with more support for intermediate values (Fig. B1; Gelman et al. 2008). We 269
used standard normal priors for log-scale fixed effects coefficients because when exponentiated 270
13
these produce realistic priors for the density of male turkeys (Fig. B2), which rarely exceeds 5 271
birds per km2 in excellent habitats (Vangilder 1992). We used uniform priors for all logit- 272
(𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,0.4)) and log-scaled (𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,1)) random-intercept variances to restrict their 273
distributions to plausible values, as variances approaching the upper bounds of these 274
distributions result in unrealistically large changes to removal probabilities and densities, 275
respectively (Appendix B). Lastly, because true daily efforts are unobserved latent states when 276
measurement error is included, prior distributions are also needed for daily efforts for all sites, 277
trials, and years (note that priors for the latent removals are specified indirectly through model 278
structure). We assumed a uniform prior for daily hunter effort that was sufficiently broad so-as to 279
encompass all plausible values (𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,8000)), given the daily efforts that were observed 280
(e.g., Fig. A2; Table A3). 281
We generated Markov chain Monte Carlo (MCMC) samples to make inferences from 282
posterior distributions of turkey abundance and population trends. We generated MCMC samples 283
using JAGS (version 4.0.1; Plummer 2003) called from within R (version 3.2.0; R Core Team 284
2015) using the R2jags package (Su and Yajima 2015). For each model we retained 200,000 285
MCMC samples from each of 3 chains (600,000 total posterior samples) after an initial burn-in 286
period of 2 million samples, and assessed convergence using multivariate Gelman-Rubin 287
statistics (Gelman and Rubin 1992) and traceplots of structural parameters (see Supplement 1 for 288
example JAGS model statement). In addition to monitoring structural model parameters and 289
initial abundances for each region and year, we also generated MCMC samples from posterior 290
distributions of finite rates of population change (i.e., 𝜆𝜆𝑖𝑖,𝑡𝑡 = 𝑁𝑁𝑖𝑖,𝑡𝑡𝑁𝑁𝑖𝑖,𝑡𝑡−1
) and annual harvest rates 291
(ℎ𝑖𝑖,𝑡𝑡 = 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝑒𝑒𝑠𝑠𝑡𝑡𝑁𝑁𝑖𝑖,𝑡𝑡
). 292
14
Lastly, to understand the implications of model structure for inferences, we conducted a 293
sensitivity analysis using alternative parameterizations of the top model identified as described 294
above. Specifically, we sought to determine if accommodating measurement error in the form of 295
an explicit observation model was necessary for inferences about population abundance and 296
trends from these data, and also the effect of assuming that regional (site-level) abundances 297
shared no structured stochastic dependencies (i.e., abundances not constrained by a hierarchical 298
structure). Thus, we re-fit the top model without measurement error, assuming daily harvest and 299
effort were perfectly observed. We also fit models where there was no explicit structure used to 300
constrain the spatial distribution of turkey abundance, but instead only vague a priori information 301
about population size (but still retaining the removal probability model identical to the original 302
model). The unstructured abundance model assumed discrete uniform priors for initial 303
abundance at the start of each removal experiment, and thus placed less constraint on estimates 304
of local abundance (Table B2). The model with unstructured abundance was fit both including 305
and excluding measurement error, and the sensitivity of abundance and population growth 306
estimates over the study duration was assessed among these alternative parameterizations. 307
Simulating model performance 308
We simulated data generation, model fitting, and parameter estimation to determine the 309
ability of our hierarchical models to accurately estimate population sizes and trends. We 310
simulated harvest data using observed region-specific hunter efforts as truth and the top model 311
from analyses described above as the data generating model, and used posterior mean estimates 312
of parameters from the top model as true values for generating harvest data. We simulated 313
imperfectly observed removal data for each of the 5 management regions under 2 scenarios 314
representing temporal replication: 1) one year of sampling, and therefore no temporal replication, 315
15
and 2) 10 consecutive years of removal sampling. Performance of removal models is known to 316
be sensitive to fraction of the population removed, where larger removals result in improved 317
accuracy (Zippin 1956, Gould and Pollock 1997, Dorazio et al. 2005, St. Clair et al. 2013). Thus, 318
we also simulated 6 scenarios where cumulative removal rates were altered through changes to 319
the number of removal trials and daily hunter efforts. We considered scenarios where the number 320
of removal trials (i.e., days of hunting) was approximately equal to (40 days), one half of (20 321
days), or one quarter of (10 days) the length of turkey hunting seasons in southern Michigan. We 322
replicated each of these scenarios over 2 scenarios of hunter effort: 1) high effort, where true 323
region-specific daily hunter efforts were those observed in Michigan, or 2) low effort, where true 324
region-specific daily efforts were half of those observed. 325
To understand performance of simplified and less constrained versions of the model 326
relative to the base model structure described above, we replicated model fitting and parameter 327
estimation for all 12 simulation scenarios described above using 4 different model 328
parameterizations: 1) using the true data generating model with hierarchical Poisson abundance 329
structure and measurement error in harvest and effort data (i.e., model correctly specified), 2) the 330
correct density model with a hierarchical Poisson abundance structure but excluding 331
measurement error (i.e., making an assumption that data were perfectly observed but the model 332
was otherwise correctly specified), 3) the unstructured abundance model including measurement 333
error, and 4) the unstructured abundance model excluding measurement error (Table B2). Thus, 334
our simulations assessed performance of hierarchical removal models under 48 combinations of 335
temporal replication of experiments, number of removal trials, magnitude of hunter efforts on 336
each trial, and estimation model structure (Table C1). For each fitted model we assessed 337
convergence (as described above), relative error (𝑒𝑒𝑠𝑠𝑡𝑡𝑖𝑖𝑒𝑒𝐻𝐻𝑡𝑡𝑒𝑒−𝑡𝑡𝐻𝐻𝑡𝑡𝑡𝑡ℎ𝑡𝑡𝐻𝐻𝑡𝑡𝑡𝑡ℎ
) for region-specific and total 338
16
abundance estimates, and region-specific and total finite rates of change over the entire study 339
(𝜆𝜆𝑖𝑖 = 𝑁𝑁𝑖𝑖,10𝑁𝑁𝑖𝑖,1
, for scenarios with temporal replication). We used posterior means as point estimates 340
of abundance and population growth, used the average relative error for each scenario to indicate 341
bias of these estimates, and also determined the fraction of 95% credible intervals that contained 342
the true values of abundance and finite rates of change for each scenario. Additional simulation 343
details are provided in Appendix C. 344
RESULTS 345
The top model estimated that the turkey abundance in southern Michigan was stable-to-346
increasing over the study, despite short-term annual fluctuations (Fig. 2). Total abundance 347
peaked at 52,911 males in 2008, which was followed by a short-term decline and then growth to 348
49,410 males in 2015 (Fig. 2). The best model included a time effect that shifted average turkey 349
density annually, spatial-heterogeneity in density among regions, and age-specific density trends 350
over the study (Table B1, Figs. B4 & B6). This model also estimated a larger removal 351
probability for adult males than juveniles and spatial heterogeneity in removal among 352
management regions, but little change in removal probabilities across trials within a hunting 353
season (Figs. B5 & B7-B8). Estimates of population growth from the top model were 354
synchronous among management regions but varied annually despite the spatial heterogeneity in 355
turkey density (Fig. 3). Estimated annual population growth for the entire study area ranged from 356
a low of 0.81 in 2010-2011 to a high of 1.16 in 2011-2012 (Fig. 3). Moreover, estimated annual 357
harvest rates ranged from a low of 0.27 for juvenile males in region ZA to a high of 0.79 for 358
adult males in region ZB (Fig. B9). Estimated harvest rates were generally larger for adult males 359
than juveniles, but the magnitude and temporal trends of harvest rates were region specific (Fig. 360
B9) and affected by patterns of hunter effort (Table A2). 361
17
Sensitivity analyses demonstrated that abundance estimates were generally robust when a 362
model was used to describe spatial-temporal structure in turkey density, whereas ignoring 363
measurement error changed the scale of abundance estimates when an unconstrained abundance 364
model was used (Fig. 4). Estimates of abundance and population growth were similar when the 365
hierarchical density model was used to constrain abundance, irrespective of the inclusion of 366
measurement error in the fitted model (Fig. 4). However, differences in estimates were more 367
pronounced when abundance lacked an explicit structure (i.e., when only constrained through 368
vague uniform priors), and the model lacking both measurement error and an abundance model 369
estimated smaller turkey abundance relative to the other three parameterizations (Fig. 4). Despite 370
differences in the absolute scale of abundance estimates, both models with unconstrained 371
abundance shared similar estimates of population growth over time, irrespective of inclusion of 372
measurement error (Fig. 3b), and these population growth estimates were more variable than the 373
models that included a Poisson abundance model. Region-specific estimates of abundance and 374
population growth usually showed similar sensitivity to model structure, but the magnitude of 375
differences in estimates among models varied by region (Figs. B10-B11). 376
Simulation results demonstrated that when specified correctly our model produced 377
estimates of abundance and population growth that are approximately unbiased at the sample 378
sizes and levels of hunter effort observed in this study (Table 2). Simulations assuming 10-years 379
of replicated removal experiments also suggested that the model with an abundance hierarchy 380
but ignoring measurement error produced similar abundance estimates as the true data generating 381
model that included measurement error, with a larger proportion of model fits resulting in 382
convergence (Table 2). However, the model including measurement error had better coverage of 383
credible intervals, which was 95% for all management regions. The corresponding models 384
18
without measurement error resulted in credible interval coverage less than 95% (range = 0.84-385
0.89), suggesting that ignoring measurement error likely inappropriately inflates perceived 386
precision. Reducing hunter effort by half resulted in worse performance for models including an 387
abundance hierarchy, with fewer model fits converging and larger magnitude of bias (Table 2). 388
The model including measurement error still had reasonable credible interval coverage under the 389
reduced effort scenario (0.90-0.95 for models that converged), however, the model ignoring 390
measurement error had degraded coverage (0.63-0.74; Table 2). Despite the bias of absolute 391
abundance estimates, estimates of population trends from the base hierarchical abundance 392
models under low effort scenarios were approximately unbiased (Table 2). Moreover, reducing 393
the number of removal trials degraded performance and reliability of the models with an 394
abundance structure, with decreased rates of convergence and typically increased bias of 395
abundance estimates for both scenarios of hunter effort (Tables C2-C3). 396
Simulations also demonstrated that when the unstructured abundance model was assumed 397
in the presence of measurement error, abundance estimates were biased for all scenarios (Table 398
2). Convergence was achieved for all models fit assuming an unstructured abundance model, 399
however, these models severely overestimated abundance, and abundance estimates more than 400
twice truth were common (Table 2). Unstructured models that included measurement error 401
resulted in more severe positive bias for abundance estimates than unstructured models that 402
simply ignored measurement error, and these biases were typically made worse by reducing 403
hunter effort or the number of removal trials (Tables 2, C2-C3). Despite their inability to 404
accurately estimate absolute abundance, unstructured models (both including and excluding 405
measurement error) were able to estimate population trends over time reasonably well when 406
hunter effort was high (i.e., at levels observed in this study; Table 2). Nonetheless, negative bias 407
19
in population growth estimates emerged for unstructured models when hunter effort or the 408
number of removal trials was reduced (Table C3). Lastly, performance was poor for all model 409
structures fit to data without temporal replication (i.e., only one year of data), where failed 410
convergence, biased abundance estimates, or both, were observed under all scenarios (Table C4). 411
DISCUSSION 412
We developed a hierarchical framework for estimating animal abundance and population 413
growth from replicated but imperfectly observed removal experiments. This framework 414
synthesizes existing removal methods and extends the models to accommodate measurement 415
error in sampling intensity and the number of removals per trial via an explicit observation 416
process. Estimates for our example with wild turkeys in Michigan, USA, suggested stable-to-417
increasing populations despite spatial-temporal fluctuations in turkey density. Sensitivity and 418
simulation analyses demonstrated these estimates were robust and approximately unbiased at the 419
levels of hunter effort observed. Simulations also demonstrated that accuracy of abundance 420
estimates was sensitive to inclusion of a model to explicitly account for heterogeneity in density 421
and abundance, implying weak ability to determine the absolute scale of abundance in the 422
presence of measurement error if heterogeneous density was not modeled directly but instead 423
abundances were unconstrained (i.e., the common approach where removal data from individual 424
years are analyzed separately; St. Clair et al. 2013). Broad-scale estimates of population growth 425
from our model were also robust and relatively insensitive to model parameterization. Moreover, 426
the modeling framework we describe is flexible, and additional data or prior information could 427
be used by tailoring the models to attributes of specific study systems and monitoring programs. 428
The hierarchical model we describe provided estimates that were robust and 429
approximately unbiased in the presence of measurement error under sample sizes and removal 430
20
rates observed in our example. The basic model used an inhomogeneous Poisson process to 431
explicitly model changes in density and induce stochastic dependence on the distribution of 432
abundances, and provided estimates of abundance and population growth that were 433
approximately unbiased. Surprisingly, this model also produced abundance estimates that were 434
robust to measurement error in the removal data, even when it was ignored by excluding the 435
observation model. The Poisson distribution and other count models (e.g., negative binomial) 436
have been used by numerous authors to provide structure and constraint to local abundance 437
estimates arising from studies under a metapopulation design (Dorazio et al. 2005, Royle and 438
Dorazio 2006, Royle and Dorazio 2008, Rivot et al. 2008, Kéry and Royle 2010, St. Clair et al. 439
2013). Our results imply such model structures can be used to constrain abundance estimates to 440
realistic values and ameliorate the bias that typically results from measurement error in the 441
recording of removal data. This appears to be a novel finding with respect to removal models, as 442
we are not aware of other studies evaluating the implications of measurement error when a 443
hierarchical model is used to represent deterministic (i.e., covariate effects) and stochastic 444
structure in regional abundance for spatially replicated removal sampling. However, excluding 445
the observation model did result in overestimation of precision in the presence of measurement 446
error. Thus, there remain tangible benefits of using the more complicated model that includes an 447
observation process, as it resulted in a more reliable assessment of uncertainty. 448
Absent a model inducing structure on the distribution of density and abundance, the 449
implications of measurement error for estimating abundance via removal methods were 450
substantial. Eliminating the model for abundance resulted in greater sensitivity of estimates to 451
observation model structure (i.e., including and excluding an observation process), and also 452
greater variability of estimates over time for turkey populations in Michigan (i.e., no shrinkage; 453
21
Royle and Link 2002). The effect of measurement error on estimator performance when 454
abundance was effectively unconstrained, except by a vague prior, was to inflate abundance 455
estimates substantially. Similarly, Gould et al. (1997) demonstrated positive bias in abundance 456
estimates when removal data from single experiments were subject to measurement error, and 457
found this directional bias was consistent over an order-of-magnitude change in the true 458
abundance. Collectively, these results imply that estimating the absolute scale of abundance via 459
removal models is more difficult when removals and effort are not perfectly observed, that 460
inappropriate scaling caused by measurement error results in large positive bias in abundance 461
estimates if ignored, and that without the structure induced by appropriately modeling spatial-462
temporal changes in density, the estimators are suspect. 463
Our analyses demonstrate that estimation problems induced by measurement error can be 464
ameliorated when information is shared among replicated experiments and the systematic 465
changes to animal density are explicitly modeled. Estimation challenges for closed population 466
abundance estimators with latent states are well described in the literature (e.g., mixture models 467
in capture recapture; Coull and Agresti 1999, Dorazio and Royle 2003). These challenges are 468
commonly used to justify modeling hierarchical structures on abundances under metapopulation 469
designs as we did in this study (Royle and Dorazio 2006, Royle et al. 2007, Royle and Dorazio 470
2008, Kéry and Royle 2010), which act along with the prior distributions to regulate parameter 471
estimates derived from posterior distributions under a Bayesian paradigm (Hooten and Hobbs 472
2015). This effectively allows researchers to mathematically constrain abundance estimates to 473
biologically plausible ranges. This is done through use of hierarchical structures and prior 474
distributions on animal abundance and density, respectively, as we did in this study to prevent 475
unrealistically large densities of turkeys. This approach is defensible for application of complex 476
22
hierarchical models, as one is effectively using what is already known to develop constraints on 477
parameter estimates in order that they remain biologically feasible (Hooten and Hobbs 2015). 478
Moreover, comparison of posterior and prior distributions from our example analyses with wild 479
turkeys (Appendix B) clearly shows the data were sufficient for estimating abundance under the 480
most complicated model that included both an abundance hierarchy and an explicit observation 481
model. Thus, while the overall model structure acted to constrain the posterior distributions of 482
abundance to plausible values, the data were also informative about spatial-temporal changes in 483
turkey abundance within the constraints of the model. 484
Despite the sensitivity of abundance estimator performance to inclusion of an explicit 485
model for density, estimates of population growth were robust to model structure. Our 486
simulations demonstrated that all of the model parameterizations considered were able to 487
reasonably estimate decadal-scale population trends in the presence of measurement error at the 488
observed sample sizes and removal rates. Even models with unstructured regional abundances 489
that overestimated the scale of abundance for a given year could reliably depict the finite rate of 490
change over a 10-year period (i.e., positive bias in abundance was consistent over time). That is, 491
the simulations showed that the models were capable of reliably estimating long-term lambda 492
even when the absolute abundance estimate was biased. Thus, assessing long-term population 493
trends may be easier than accurately scaling abundance when removal data are imperfectly 494
observed. Few studies have considered models for temporally-replicated removal experiments 495
(but see Rivot et al. 2008), and their usefulness for assessing population trends for managed 496
species appears to be underappreciated. Although trends may provide less information to 497
decision makers than absolute abundance estimates (Nichols and Williams 2006), we believe the 498
reliable trend estimates produced by our model are more informative and useful for decision 499
23
making than traditional indices (e.g., catch-per-unit effort) used to monitor species like the wild 500
turkey, as the models presented here can explicitly account for changes to both animal density 501
and the per-unit-effort removal probability. Moreover, estimates of population trends are often 502
used by practitioners to inform natural resource management decisions (e.g., when adjusting 503
harvest regulations), as estimates of absolute abundance can lack context to stakeholders and 504
decision makers without comparable historical estimates. 505
We demonstrated reasonable performance for hierarchical removal models that 506
accommodate measurement error under conditions similar to those observed for our example 507
analysis, but performance was sensitive to the fraction of the population removed and the 508
availability of replicated data. Reducing the fraction of the population removed, manifested 509
through reduction to either the magnitude of sampling effort or the total number of removal 510
trials, degraded performance considerably. Under these conditions, the models often failed to 511
converge and bias of abundance estimates increased. These issues are widely recognized in the 512
literature on capture-recapture and removal models, where performance is tightly linked to the 513
overall fraction of the population sampled (Zippin 1956, Gould and Pollock 1997, Coull and 514
Agresti 1999, Dorazio and Royle 2003, Dorazio et al. 2005, Mäntyniemi et al. 2005, St. Clair et 515
al. 2013). Our simulations indicate that problems associated with removing only a small fraction 516
of the population will likely manifest themselves through failed model convergence. In this light 517
and given the wide variety of contexts for which removal models may be applied, we suggest it 518
wise to use simulation to assess estimator performance for specific applications when removal 519
data are subject to measurement error. In addition, larger samples of simulated parameter 520
estimates would benefit future studies. We chose to simulate a large number of scenarios (48), 521
which came at the expense of large numbers of iterations of data generation and model fitting per 522
24
scenario (100) due to the computational limitations and demands of model fitting. While we 523
believe this was adequate for assessment of central tendency and bias of estimators, we 524
acknowledge that more simulation replicates would be useful to provide improved precision and 525
coverage of credible intervals. 526
While simulations indicated reasonable performance for removal experiments replicated 527
in space and time similar to our example dataset, our models were data hungry and did not 528
perform reliably in the face of measurement error when spatially-replicated removal experiments 529
were not also replicated in time. In general, this suggests substantial replication is likely a 530
prerequisite for applying these models, at least for data similar to those observed in our example, 531
which could limit application of the model. In many applied settings, the spatial replicates are 532
fixed and represent the entire area of interest (e.g., a set of management regions). Additional 533
replication of such removal experiments can only come in time, and our simulations suggest this 534
replication is necessary. It is not entirely clear, however, what amount of temporal replication 535
will be necessary for reliable application of the model under different levels of spatial 536
replication. It seems likely that replicated observations of the removal processes and information 537
sharing among experiments is what is needed, and thus the necessary amount of temporal 538
replication may depend on the number of spatial replicates, and vice versa. In other contexts the 539
number of spatial replicates could be much higher and require less temporal replication for 540
reliable inferences. Thus, while reliable performance of models required considerable replication 541
of removal data, we expect the necessary amount of replication to depend on the context of the 542
study. This further supports our suggestion that simulation-based assessment of estimator 543
performance for specific applications is wise when removal data are subject to measurement 544
error. 545
25
Although models for dealing with removal data influenced by measurement error have 546
some limitations, the hierarchical framework described here provides a high degree of flexibility 547
that will enable further tailoring of models to specific species and sampling protocols. In our 548
example we focused on relatively simple model structures to describe spatial-temporal 549
heterogeneity in removal and animal density (e.g., fixed effects and iid normal random 550
intercepts), yet the framework is flexible enough to accommodate a variety of model structures. 551
For example, this could include logit- or log-scale multivariate normal effects that induce a 552
correlation structure on realized values of density and removal probability (Royle and Link 2002, 553
Royle and Dorazio 2006, Royle and Dorazio et al. 2008), or conditional auto-regressive models 554
to induce spatial correlation among regions (Lichstein et al. 2002, Webster et al. 2008). In 555
addition, alternative observation models could be tailored to the characteristics of specific 556
monitoring programs. For instance, one might be able to use sampling theory (Cochran 1977) to 557
derive observation-error variances for estimated removal data for specific survey designs, instead 558
of assuming the variance-mean relationship of a Poisson distribution. Lastly, given the variety of 559
contexts for which removal models are applied, extensions for sampling specific taxonomic 560
groups could also be possible, where examples include analysis of removal data from avian point 561
counts and stream electrofishing with measurement error due to species misidentification. Thus, 562
the framework described here provides a flexible starting point for fine-tuning population 563
assessment for a variety of species that use imperfectly observed data arising from removal 564
sampling protocols. 565
Lastly, our analyses indicated male turkey populations in Michigan were stable-to-566
increasing, with heterogeneous density among management regions that also varied annually, 567
and heterogeneous per-unit-effort removal probabilities by animal group and among regions. We 568
26
estimated annual fluctuations in abundance consistent with known dynamics of turkey 569
populations, where previous authors suggested population fluctuations of up to 40% annually are 570
possible (Vangilder and Kurzejeski 1992, Healy and Powell 2000). These large annual 571
fluctuations were synchronous across southern Michigan despite a smaller magnitude of spatial 572
heterogeneity in density, which is again consistent with earlier work suggesting broad-scale 573
synchrony in turkey abundance is common (e.g., Fleming and Porter 2007). Higher removal 574
probabilities for adult males than for juvenile males were also anticipated, as multiple studies 575
reported that adult males are more vulnerable to harvest than juveniles (Vangilder et al. 1995, 576
Healy and Powell 2000, Chamberlain et al. 2012). Previous research also suggested harvest rates 577
for males might vary among management regions (Diefenbach et al. 2012), which would be 578
expected even in the absence of spatially heterogeneous capture probabilities given variation of 579
hunter effort (Clawsen et al. 2015, this study). 580
We suspected a priori that removal probability would decline over the hunting season due 581
to capture heterogeneity among individual animals, behavioral responses to harvest, or some 582
combination of these and other factors, yet we found little evidence of this pattern. It is possible, 583
however, that other parameterizations could better portray such systematic shifts to capture 584
heterogeneity over a hunting season. For example, a categorical effect over trials could capture 585
behavioral shifts of animals during the hunting season (i.e., early season vs late season), or linear 586
trends over the season may be modeled as a function of cumulative effort (and therefore 587
cumulative exploitation) up to a given trial, instead of declining at a constant rate over trials. 588
Nonetheless, our estimates appear robust and indicate turkey populations in Michigan are stable, 589
which directly contradicts, at least for our study region, a widespread belief common in the 590
eastern U.S. that turkey populations are in decline (Kobilinsky 2018). Such beliefs are often 591
27
based on monitoring of turkey populations using raw harvest or catch-per-unit effort indices that 592
cannot account for spatial-temporal changes in both density and removal probability. Thus, we 593
provide assessment tools that enable a more refined understanding of population trends for 594
harvested species like wild turkeys, which in this case facilitated a shifting perception of the 595
status of populations across the study region. Moreover, in the context of removal via 596
commercial or recreational harvest, our models provide the ability to estimate abundance and 597
exploitation rates arising from a given set of management regulations, both of which are useful 598
for simulation-based assessment of sustainable harvest strategies (Quinn and Collie 2005, 599
Deroba and Bence 2008, Punt et al. 2016). As such, our models provide value beyond the 600
assessment of status, and should further enable rigorous assessment of management decisions for 601
target species. 602
ACKNOWLEDGMENTS 603
Funding for this work was provided by the Michigan Department of Natural Resources (MDNR), 604
the Boone and Crockett Quantitative Wildlife Center, and Quantitative Fisheries Center at 605
Michigan State University. Partial funding for this project was provided through the Federal Aid 606
in Wildlife Restoration Act grant number W-155-R in cooperation with the U.S. Fish and 607
Wildlife Service, Wildlife and Sport Fish Restoration Program. We thank the Michigan Wild 608
Turkey Stakeholder Group, and especially M. Jones and A. Stewart, for conversations about 609
population assessment and turkey management that informed this project. This is publication 610
2020-XX of the Quantitative Fisheries Center at Michigan State University. 611
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SUPPLEMENTAL MATERIALS 760
Supplement 1 761
Example JAGS model statement for fitting hierarchical removal model with measurement error. 762
763
764
765
766
767
768
35
Table 1. Summary of model components for hierarchical removal model fit to wild turkey 769
harvest data from Michigan, USA. Subscripts refer to animal groups (i), trials (j) sites (s), and 770
years (t). 771
Model structure and distributions
Observation models
𝑦𝑦𝑖𝑖𝑗𝑗,𝑠𝑠,𝑡𝑡∗ ~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃�𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡�
𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡∗ ~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃�𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡�
Removal and abundance process models
𝑦𝑦𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡~𝐵𝐵𝑃𝑃𝑃𝑃𝑃𝑃𝑁𝑁𝑃𝑃𝑁𝑁𝑙𝑙�𝑁𝑁𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡,𝑝𝑝𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡�
𝑝𝑝𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡 = 1 − (1 − 𝜑𝜑𝑖𝑖)𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡
𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃�𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡𝐴𝐴𝑠𝑠,𝑡𝑡�
Capture probability and density models
𝑙𝑙𝑃𝑃𝑙𝑙𝑃𝑃𝑙𝑙�𝜑𝜑𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡� = 𝛽𝛽0 + 𝛽𝛽1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛽𝛽2𝑗𝑗 + 𝛽𝛽𝑠𝑠 + 𝜂𝜂𝑡𝑡
𝑙𝑙𝑃𝑃𝑙𝑙�𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡� = 𝛼𝛼0 + 𝛼𝛼1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛼𝛼2𝑙𝑙 + 𝛼𝛼3𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 ∗ 𝑙𝑙 + 𝛼𝛼𝑠𝑠 + 𝛾𝛾𝑡𝑡
Priors and hyper-priors
𝑒𝑒𝑗𝑗,𝑠𝑠,𝑡𝑡~𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,8000)
𝜂𝜂𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,𝜎𝜎𝜂𝜂2)
𝜎𝜎𝜂𝜂2~𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,0.4)
𝛾𝛾𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,𝜎𝜎𝛾𝛾2)
36
𝜎𝜎𝛾𝛾2~𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(0,1)
𝛽𝛽0~𝐶𝐶𝑁𝑁𝐶𝐶𝐶𝐶ℎ𝑦𝑦(10)
𝛽𝛽𝑘𝑘~𝐶𝐶𝑁𝑁𝐶𝐶𝐶𝐶ℎ𝑦𝑦(2.5)
𝛼𝛼0~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,1)
𝛼𝛼𝑘𝑘~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙(0,1)
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
37
Table 2. Results of simulation study used to assess performance of posterior mean estimates of abundance (𝑁𝑁�) and finite rate of 791
change (�̂�𝜆 = 𝑁𝑁�𝑡𝑡=10𝑁𝑁�𝑡𝑡=1
) for removal studies replicated over a ten-year period. The fraction of simulated model fits that resulted in 792
convergence (Convergence), as well as average relative error of abundance estimates from year 10 (Bias) and the fraction of 95% 793
credible intervals that contained true abundance from that year (for model fits that converged) are presented by turkey management 794
region (Region), and for all regions combined (Total). Results are presented for simulation scenarios defined by the magnitude of 795
hunter effort (High, Low), the hierarchical structure of the fitted abundance model (Poisson, Unstructured), and the presence of 796
measurement error in fitted model (Measurement error, No measurement error), for the simulation scenario where the number of trials 797
in each removal experiment was equal to 40 days of hunting. 798
Region
ZA ZB ZC ZE ZF Total
Metric & simulation scenarioa Convergence Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage
𝑁𝑁�
High effort
Measurement error
Poisson 0.79 0.03 0.95 0.05 0.95 0.04 0.95 0.04 0.95 0.04 0.95 0.04 0.95
Unstructured 1.00 2.03 0.00 4.89 0.00 4.59 0.00 2.03 0.00 2.08 0.00 2.73 0.00
No measurement error
Poisson 0.89 0.02 0.88 0.03 0.84 0.02 0.89 0.02 0.87 0.03 0.87 0.02 0.89
Unstructured 1.00 1.85 0.00 3.97 0.00 3.77 0.00 1.84 0.00 1.89 0.00 2.36 0.00
Low effort
Measurement error
38
Poisson 0.38 -0.09 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.90
Unstructured 1.00 2.25 0.00 5.21 0.00 5.03 0.00 2.40 0.00 2.18 0.00 3.01 0.00
No measurement error
Poisson 0.68 -0.12 0.63 -0.10 0.74 -0.11 0.66 -0.10 0.69 -0.11 0.68 -0.11 0.66
Unstructured 1.00 2.16 0.00 5.10 0.00 4.97 0.00 2.33 0.00 2.09 0.00 2.93 0.00
�̂�𝜆
High effort
Measurement error
Poisson 0.79 -0.01 0.94 0.01 0.99 -0.02 0.97 0.00 1.00 0.02 0.91 0.00 1.00
Unstructured 1.00 0.02 0.97 0.01 1.00 -0.02 0.99 -0.01 1.00 0.01 1.00 0.00 0.99
No measurement error
Poisson 0.89 -0.01 0.80 0.01 0.93 -0.01 0.94 0.00 0.91 0.01 0.78 0.00 0.85
Unstructured 1.00 0.01 0.92 0.00 0.95 -0.03 0.87 -0.02 0.83 -0.01 0.97 -0.01 0.88
Low effort
Measurement error
Poisson 0.38 -0.03 0.89 0.00 1.00 -0.04 0.74 -0.01 1.00 0.01 1.00 -0.01 0.97
Unstructured 1.00 -0.02 0.98 -0.03 0.96 -0.12 0.55 0.00 1.00 -0.06 0.81 -0.05 0.59
No measurement error
Poisson 0.68 -0.02 0.76 0.00 0.97 -0.04 0.34 -0.01 0.96 0.01 0.97 -0.01 0.78
Unstructured 1.00 -0.02 0.86 -0.03 0.87 -0.12 0.29 0.00 0.97 -0.06 0.53 -0.05 0.26 a Mathematical details of each model structure and simulation scenario are described in Tables B1-B3 and Appendix C. 799
800
801
802
39
Figure 1. Conceptual diagram of hierarchical removal model fit to wild turkey harvest data from 803
Michigan, USA. Boxes indicate fixed, observed quantities, whereas circles indicate random 804
(unobserved) quantities or processes. Subscripts denote animal group (i), removal trial (j), sites 805
(s), and year (t), and π denotes a prior distribution for a random quantity (priors for regression 806
coefficients excluded for visual clarity). Symbol definitions are provided in text, and further 807
mathematical details of model structure is provided in Table 1. 808
809
810
811
812
40
Figure 2. Region-specific and total population trajectories for male turkey populations (2002-813
2015) in southern Michigan, USA. Posterior mean estimates are provided by solid lines, whereas 814
dashed lines indicate percentile-based 95% credible intervals (equal tail probabilities). 815
816
817
818
819
820
821
822
41
Figure 3. Annual estimates of the finite rate of change (Lambda) for male turkey populations in 823
individual regions and the entire southern Michigan, USA, study area for 2002-2015. a) 824
regionally, b) all regions combined. Dots indicate posterior mean point estimates, whereas 825
whiskers represent percentile-based 95% credible intervals (equal tail probabilities). 826
a) 827
828
829
830
831
832
42
b) 833
834
835
836
837
838
839
840
841
43
Figure 4. Differences in estimates of abundance (N) and of finite rate of population change (λ) 842
for turkeys in southern Michigan arising from corresponding models fit accounting for and 843
ignoring measurement error in daily hunter effort and harvest data, and using different 844
hierarchical structures for the spatial distribution of abundance. This figure demonstrates 845
sensitivity of estimates to changes in model structure. 846
a) 847
848
849
850
851
44
b) 852
853
854
855
856
857
858
859
860
861
45
APPENDIX A. Supplemental description of wild turkey harvest management and data. 862
863
Estimates of region-specific hunter effort and harvest were produced annually using 864
randomized mail surveys and survey sampling methodology. Each year thousands of surveys are 865
mailed to individual hunters who record their spatially-explicit daily effort and harvest 866
information, as well as the age category (i.e., juvenile or adult) of any harvested male turkeys. 867
These surveys are used to generate regional estimates of total harvest and effort over the entire 868
hunting season, which are reported annually through management reports (e.g., Frawley and 869
Boone 2015). Individual hunter survey records also provided information on daily effort and 870
harvest, and thus the fraction of total effort and harvest (by age category) that occurred each day 871
of the hunting season within each management region. Samples of daily harvest and effort from 872
individual hunters were combined with regional total harvest and effort estimates to produce 873
estimates of daily total hunter effort and age-specific harvest for each region and for each spring 874
hunting season. We did not consider non-response bias to be problematic for survey-based 875
estimates of harvest and effort in this study because annual response rates were generally high 876
(range = 56%-83%), resulting in > 100,000 individual hunter records used to estimate daily 877
harvest and effort (range = 3,546-12,125 records per year; Table A4). 878
We used estimated harvests and efforts obtained from mail surveys and survey sampling 879
methods to estimate spatially-explicit abundances for male turkeys in management regions of 880
southern Michigan, USA (Fig. A1). We did not include the two-county management region (ZD) 881
located in the far southeast corner of Michigan in our analyses because this region consists of the 882
Detroit metropolitan area, where hunting efforts and the number of turkeys harvested annually 883
are sparse. Every year mail surveys are sent to Michigan turkey hunters shortly following the 884
46
close of hunting to assess hunter harvest and efforts (e.g., Frawley and Boone 2015). When 885
responding to surveys hunters were asked to provide the days hunted, the day of harvest if 886
successful (bag limit of only 1 male turkey in place for entire study duration), and information 887
used to indicate age of any harvested turkey. To indicate age of harvested birds, each successful 888
hunter records length of the harvested turkey’s beard, where the beard is a morphological 889
characteristic describing a bundle of hairs protruding from the chest (Kelly 1975). Specifically, 890
beards < 6 inches are used to indicate a juvenile (i.e., 1-year old) turkey, whereas beards ≥ 6 891
inches indicate an adult turkey (i.e., ≥ 2-year old; Kelly 1975). Survey responses were used to 892
generate estimates of total harvest and effort at a county scale (e.g., Frawley and Boone 2015), 893
which we scaled up to regional estimates by summing over all counties contained within each 894
region (Table A2). 895
We used information contained in individual hunter records (Table A4) to convert 896
regional estimates of harvest and efforts over the entire hunting season to estimates of daily 897
efforts and age-specific harvests within each season (Table A3). Individual mail survey 898
responses provided information about the temporal distribution of harvest and effort within each 899
hunting season, as well as the age composition of harvests. We used all individual hunter records 900
collected from 2002-2015, except those that did not include the information needed for the 901
analysis (e.g., no date of harvest or beard length measurement) or contained an obvious data 902
entry error (e.g., indicated hunting dates when season was not open; >100,000 records used; 903
Table A4). 904
Survey records provided annual estimates of proportional harvest by age category 905
(juvenile or adult) for each region, which was multiplied by total harvest estimates to produce 906
estimates of total age-specific harvest for each region and year. To estimate daily harvest and 907
47
effort across each hunting season for each management region, we determined the fraction of 908
regional total harvest or efforts represented by the individual hunter record data, and used these 909
sampling fractions to scale up raw summaries of daily harvest and effort described by hunter 910
records to regional estimates of daily harvest and effort. Specifically, we summed daily harvest 911
and efforts recorded by individual hunter records for each region and year, and determined the 912
fraction of total harvest or effort these raw summaries represented by dividing by the estimated 913
regional totals for the entire hunting season described above. For example, if the estimated total 914
effort over the entire hunting season for ZA in 2002 was 72,378 hunter days, and the sum of 915
daily efforts from raw hunter records for that region in the same year was 14,179 hunter days, 916
then we assumed efforts recorded from raw survey data represented approximately 0.196 of the 917
regional total efforts for that year (14,179/72,378), and consequently multiplied raw daily effort 918
totals from individual hunter records by approximately 5.1 (72,378/14,179) to produce estimates 919
of daily total effort at the scale of the management region. This approach was also used to scale 920
up raw daily harvest summaries from individual hunter records to regional estimates of daily 921
total harvest by age-group for each year. 922
LITERATURE CITED 923
Frawley, B. J., and C. E. Boone. 2015. 2014 Michigan spring turkey hunter survey. Wildlife 924
Division Report No. 3607, Michigan Department of Natural Resources, Lansing, 925
Michigan, USA. 926
Kelly, G. 1975. Indices for aging eastern wild turkeys. Proceedings of the National Wild Turkey 927
Symposium 3:205-209. 928
929
930
48
Table A1. Spatial area (km2) of management regions considered in this study. 931
Management region Area
ZA 16,148
ZB 6,739
ZC 8,671
ZE 14,670
ZF 13,445
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
49
Table A2. Estimated total harvest of male wild turkeys and total hunter efforts (hunter days) for wild turkey management regions of 947
southern Michigan, USA. Season length (days) varied among years but was consistent across space within years. 948
ZA ZB ZC ZE ZF
Year Season length Harvest Effort Harvest Effort Harvest Effort Harvest Effort Harvest Effort
2002 40 6,559 72,378 1,758 21,934 1,383 16,915 3,472 44,508 4,643 60,177
2003 41 7,275 75,132 1,930 25,701 1,822 21,967 4,056 48,342 5,266 62,096
2004 43 7,690 78,257 2,271 25,995 1,999 22,947 4,984 52,621 6,288 66,424
2005 44 7,720 79,467 2,101 25,830 2,323 27,234 4,980 55,754 5,903 64,251
2006 45 7,987 95,024 2,429 32,001 2,710 32,819 5,875 66,068 6,286 73,979
2007 39 7,731 76,774 2,641 24,247 2,938 31,180 5,942 55,042 6,600 67,828
2008 41 8,073 71,457 2,678 24,642 3,777 35,481 6,461 56,285 7,341 63,261
2009 42 8,644 76,717 2,706 28,616 3,411 34,439 6,828 65,994 6,903 72,575
2010 43 7,908 71,771 2,522 27,611 3,593 36,094 6,406 56,413 6,557 64,785
2011 44 6,746 69,882 1,769 21,183 2,531 35,040 5,220 54,830 5,157 64,521
2012 39 6,521 59,217 2,166 19,939 2,945 30,602 5,044 47,348 5,712 52,375
2013 40 6,253 56,371 2,340 21,506 3,037 31,032 5,537 46,388 5,409 58,563
2014 41 5,526 52,528 2,392 19,234 2,729 25,178 5,461 47,775 4,784 50,672
2015 42 5,840 46,159 2,423 19,276 2,649 21,777 5,353 43,542 5,016 42,753
949
950
951
50
Table A3. Example removal samples for wild turkeys in southern Michigan, USA. Estimated daily harvest of juvenile and adult male 952
turkeys and hunter effort (no. hunters) for 3 management regions for the first 10 days of the 2015 season. 953
Day of hunting season
Region / Metric 1 2 3 4 5 6 7 8 9 10
ZA
Juvenile 107 27 40 0 40 94 27 27 13 13
Adult 817 228 241 121 281 348 375 54 134 80
Effort 4,067 3,318 1,151 1,606 2,596 3,465 1,913 9,50 7,49 1,097
ZC
Juvenile 25 25 0 0 76 25 13 13 0 13
Adult 431 114 114 101 139 190 165 38 127 51
Effort 2,023 1,827 540 1,091 1,373 1,631 932 736 613 441
ZF
Juvenile 92 39 52 52 52 52 39 13 39 13
Adult 642 275 170 118 210 236 262 92 92 65
Effort 3,422 3,042 845 1,845 2,394 2,985 1,591 943 1,042 718
954
955
51
Table A4. Total number of wild turkey mail survey records (i.e., individual survey responses) 956
and total number of age samples of harvested male turkeys retrieved from those records that 957
were used in calculation of daily hunter harvest and effort estimates used to fit hierarchical 958
removal models. 959
Year Hunter records Age samples
2002 9,353 3,572
2003 10,357 4,776
2004 10,868 5,252
2005 12,125 5,784
2006 11,131 5,145
2007 9,518 4,581
2008 6,266 3,381
2009 6,315 3,110
2010 6,539 3,337
2011 5,381 2,483
2012 5,410 2,596
2013 4,920 2,448
2014 3,719 1,642
2015 3,546 1,652
960
961
962
963
964
965
966
967
52
Figure A1. Wild turkey harvest management regions in southern Michigan, USA, where we 968
estimated spatially-explicit abundance and population trends (2002-2015) for male wild turkeys 969
using hierarchical removal models. Regions (ZA, ZB, ZC, ZE, and ZF) are outlined in bold to 970
denote their location within the lower peninsula of Michigan (thin black outline). 971
972
973
974
975
976
977
978
53
Figure A2. Example (region ZA over the 2015 spring hunting season) of changes to daily hunter 979
effort (no. hunters) over a hunting season for a management region in southern Michigan, USA. 980
981
982
983
54
APPENDIX B. Supplemental tables and figures. 984
985
Table B1. Hierarchical removal models fit to daily turkey harvest and effort data from hunting 986
regions in southern Michigan, USA, from 2002-2015. Models included combinations of fixed 987
and random effects used to model heterogeneity in turkey density (δ) and per-unit effort capture 988
probabilities (φ) through space and time, and among animal groups (indicator of juvenile or adult 989
male turkeys) and removal trials (trend over days within the hunting season). The base fixed 990
effects model for δ included an age-group effect, a linear time trend, and an interaction between 991
age group and time trend, whereas the base fixed effects model for φ included an age-group 992
effect and a linear time trend over removal trials within a hunting season (Table 1). The effects 993
included in the base models are designated by (.), and were included in all models that were fit. 994
Fixed regional effects (Space) and annual random intercepts (Time) were added to these base 995
models to accommodate spatial-temporal heterogeneity in density and capture probability (Table 996
1). Models that converged were ranked using the deviance information criterion (DIC), whereas 997
models that failed to converge were not considered further (-). 998
999
Model description DIC ΔDIC
δ(.) φ(.) 103016.7 5666.1
δ(Space) φ(.) 99460.2 2109.6
δ(Time) φ(.) 101734.6 4384
δ(Space + Time) φ(.) - -
δ(.) φ(Time) 101663.6 4313
δ(Space) φ(Time) 98412.8 1062.2
δ(Time) φ(Time) - -
δ(Space + Time) φ(Time) - -
55
δ(.) φ(Space) 99117.7 1767.1
δ(Space) φ(Space) 98564.4 1213.8
δ(Time) φ(Space) 98035.4 684.8
δ(Space + Time) φ(Space) 97350.6 0
δ(.) φ(Space + Time) - -
δ(Space) φ(Space + Time) 97473.2 122.6
δ(Time) φ(Space + Time) - -
δ(Space + Time) φ(Space + Time) - -
1000
1001
56
Table B2. Model structures used to determine sensitivity of abundance and population growth 1002
estimates for male turkeys in southern Michigan, USA, from 2002-2015. Sensitivity analyses 1003
compared estimates among models that differed in measurement error (present vs. absent) and 1004
abundance model structures (base hierarchical model vs. unstructured abundance model). 1005
Modela Abundance model structureb
Basec 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡~𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃(𝜇𝜇𝑖𝑖,𝑠𝑠,𝑡𝑡)
𝜇𝜇𝑖𝑖,𝑠𝑠,𝑡𝑡 = 𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡 × 𝐴𝐴𝑠𝑠
𝑙𝑙𝑃𝑃𝑙𝑙�𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡� = 𝛼𝛼0 + 𝛼𝛼1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛼𝛼2𝑙𝑙 + 𝛼𝛼3𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 ∗ 𝑙𝑙 + 𝛼𝛼𝑠𝑠 + 𝛾𝛾𝑡𝑡
𝛾𝛾𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙�0,𝜎𝜎𝛾𝛾𝑡𝑡�
Unstructured 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡~𝑈𝑈𝑃𝑃𝑃𝑃𝑓𝑓𝑃𝑃𝑁𝑁𝑁𝑁(𝐻𝐻𝑁𝑁𝑁𝑁𝐻𝐻𝑒𝑒𝑃𝑃𝑙𝑙𝑖𝑖,𝑠𝑠,𝑡𝑡, 30000) a Models were fit assuming measurement error in observed harvest and effort estimates 1006
(equations 5-6), and also assuming no measurement error (i.e., harvest and effort 1007
estimates perfectly observed). Capture probability model for all sensitivity analyses: 1008
𝑙𝑙𝑃𝑃𝑙𝑙𝑃𝑃𝑙𝑙�𝜑𝜑𝑖𝑖,𝑗𝑗,𝑠𝑠,𝑡𝑡� = 𝛽𝛽0 + 𝛽𝛽1𝐴𝐴𝑙𝑙𝑒𝑒𝑖𝑖 + 𝛽𝛽2𝑗𝑗 + 𝛽𝛽𝑠𝑠 + 𝜂𝜂𝑡𝑡. All fixed-effect covariates and random 1009
effects are defined in Table 1. 1010
b Abundance model quantities are: 𝑁𝑁𝑖𝑖,1,𝑠𝑠,𝑡𝑡 = abundance at the start of the removal 1011
experiment in year t for animal group i at region s; 𝜇𝜇𝑖𝑖,𝑠𝑠,𝑡𝑡 = average abundance for animal 1012
group i at region s in year t, 𝐴𝐴𝑠𝑠 = area of site s; 𝛿𝛿𝑖𝑖,𝑠𝑠,𝑡𝑡 = density for animal group i at region 1013
s in year t; and 𝐻𝐻𝑁𝑁𝑁𝑁𝐻𝐻𝑒𝑒𝑃𝑃𝑙𝑙𝑖𝑖,𝑠𝑠,𝑡𝑡 = estimated total harvest of animal group i at region s in year 1014
t. 1015
c Base model was model identified as top model using DIC (Table B1). 1016
1017
1018
57
Figure B1. Example of prior distributions on the real probability scale for logit-scale Cauchy 1019
priors with scale equal to: a) 10, and b) 2.5. 1020
a) 1021
1022
b) 1023
1024
58
Figure B2. Example of prior distributions for regression coefficients for log density based on 1025
standard normal prior distributions. The prior is displayed on the real scale of density using 1026
exponentiated slope coefficients. 1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
59
Figure B3. Examples of spatial-temporal shifts on the real probability scale for random 1039
realizations of logit-scale intercepts whose standard deviation was set at the maximum value of 1040
the uniform prior on the logit-scale (i.e., 𝜎𝜎 = 0.4). Note that the log-scale random effects with 1041
maximum variance (𝜎𝜎 = 1) would produce shifts in real-scale densities similar to figure B2. 1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
60
Figure B4. Posterior distributions of density model (Table 1) regression coefficients from the top 1053
hierarchical removal model (Table B1) for the following covariates: a) Age, b) Trend, c) 1054
Age*Trend interaction (i.e., time trend slope difference for juveniles, relative to adults), and 1055
regional effects of d) ZB, e) ZC, f) ZE, and g) ZF relative to h) ZA (treated as intercept). 1056
a) 1057
1058
1059
1060
1061
1062
1063
61
b) 1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
62
c) 1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
63
d) 1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
64
e) 1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
65
f) 1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
66
g) 1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
67
h) 1130
1131
1132
1133
1134
1135
1136
1137
1138
68
Figure B5. Posterior distributions of capture model (Table 1) regression coefficients from the top 1139
hierarchical removal model (Table B1) for the following covariates: a) Age, b) Trial, and 1140
regional effects of c) ZB, d) ZC, e) ZE, and f) ZF relative to g) ZA (treated as intercept). 1141
a) 1142
1143
1144
1145
1146
1147
1148
1149
69
b) 1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
70
c) 1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
71
d) 1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
72
e) 1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
73
f) 1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
74
g) 1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
75
Figure B6. Posterior distributions of density model (Table 1) regression coefficients (black) 1215
compared to their prior distributions (red; to demonstrate information gained from the data) for 1216
the top hierarchical removal model (Table B1), for the following covariates: a) Age, b) Trend, c) 1217
Age*Trend interaction (i.e., time trend slope difference for juveniles, relative to adults), regional 1218
effects of d) ZB, e) ZC, f) ZE, and g) ZF relative to h) ZA (treated as intercept), and i) the 1219
variance of the annual random effect terms. 1220
a) 1221
1222
1223
1224
1225
76
b) 1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
77
c) 1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
78
d) 1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
79
e) 1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
80
f) 1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
81
g) 1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
82
h) 1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
83
i) 1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
84
Figure B7. Posterior distributions of capture model (Table 1) regression coefficients (black) 1313
compared to their prior distributions (red; to demonstrate information gained from the data) for 1314
the top hierarchical removal model (Table B1) for the following covariates: a) Age, b) Trial, and 1315
regional effects of c) ZB, d) ZC, e) ZE, and f) ZF relative to g) ZA (treated as intercept). 1316
a) 1317
1318
1319
1320
1321
1322
1323
85
b) 1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
86
c) 1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
87
d) 1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
88
e) 1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
89
f) 1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
90
g) 1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
91
Figure B8. Posterior distributions of capture model (Table 1) regression coefficients from the top 1390
hierarchical removal model (Table B1), on the real probability scale (i.e., applying the inverse-1391
logit function to the continuous logit-scaled values) for the following covariates: a) Age, b) Trial, 1392
and regional effects of c) ZB, d) ZC, e) ZE, and f) ZF relative to g) ZA (treated as intercept). 1393
These posterior distributions can be compared to prior distributions shown in Figure B1 (a: 1394
intercept; b: regression coefficients) to demonstrate information gained from the data for each 1395
capture model parameter on the real probability scale. 1396
a) 1397
1398
1399
1400
92
b) 1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
93
c) 1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
94
d) 1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
95
e) 1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
96
f) 1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
97
g) 1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
98
Figure B9. Estimated age-specific annual harvest rates for male turkeys in management regions 1466
of southern Michigan, USA: a) ZA, b) ZB, c) ZC, d) ZE, and e) ZF. 1467
a) 1468
1469
1470
1471
1472
1473
1474
1475
1476
99
b) 1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
100
c) 1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
101
d) 1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
102
e) 1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
103
Figure B10. Sensitivity of abundance estimate at regional scale: a) ZA, b) ZB, c) ZC, d) ZE, e) 1521
ZF. Plotted are differences between point estimates (posterior means) produced by each model 1522
and the base model, scaled by the base-model estimates. 1523
a) 1524
1525
1526
1527
1528
1529
1530
1531
104
b) 1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
105
c) 1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
106
d) 1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
107
e) 1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
108
Figure B11. Sensitivity of estimates of finite rate of change (λ) for populations at the regional 1576
scale: a) ZA, b) ZB, c) ZC, d) ZE, e) ZF. Plotted are differences between point estimates 1577
(posterior means) produced by each model and the base model, scaled by the base-model 1578
estimates. 1579
a) 1580
1581
1582
1583
1584
1585
1586
109
b) 1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
110
c) 1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
111
d) 1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
112
e) 1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
113
Figure B12. Comparison of estimated posterior distributions of total male turkey abundance 1631
(across all management regions in southern Michigan, from 2002-2015) to derived prior 1632
distributions for total abundance (simulated as a function of priors and hyper-priors for other 1633
model parameters) to demonstrate information contained within the observed data. Comparisons 1634
are shown for the top model used to make inference about turkey abundance (Table 1), for 1635
abundance in year 1 (a: prior; b: posterior), year 7 (c: prior; d: posterior), and year 14 (e: prior; f: 1636
posterior). Comparisons are also shown for the unstructured abundance model used to assess 1637
implications of model structure for inferences about turkey abundance (Table B2), for abundance 1638
in year 1 (g: prior; h: posterior), year 7 (i: prior; j: posterior), and year 14 (k: prior; l: posterior). 1639
a) 1640
1641
114
b) 1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
115
c) 1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
116
d) 1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
117
e) 1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
118
f) 1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
119
g) 1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
120
h) 1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
121
i) 1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
122
j) 1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
123
k) 1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
124
l) 1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
125
APPENDIX C. Description of analyses and results of simulation study. 1763
1764
We simulated data and then model fitting and parameter estimation for each simulation 1765
scenario, and assessed precision and bias of resulting abundance estimates. For simulation 1766
scenarios with no temporal replication of removal experiments (i.e., one year of data), we used 1767
the observed region-specific hunter efforts from 2015, as well as parameter estimates and the 1768
capture probability structure from the top model to generate removal data. We used posterior 1769
mean estimates of stage-specific abundance and capture probability parameters from the top 1770
model as truth, and used the conditional binomial removal model (equation 3) to generate true 1771
removal data sets for each scenario of hunter effort. Specifically, the high hunter effort scenario 1772
assumed observed region-specific hunter effort data from 2015 were truth, whereas the low 1773
hunter effort scenario assumed observed 2015 efforts halved (rounded to the nearest integer 1774
value) were truth, and for each of these scenarios of true effort we generated 100 random data 1775
sets of true removals for j = 40 removal trials within each management region. We then 1776
simulated measurement error in harvest and effort data by generating 100 datasets of observed 1777
effort and removal, one for each true removal trial sequence, using the Poisson observation 1778
model described in text (equations 7-8). These simulated observed harvest and effort data sets 1779
were used for all parameter estimation for scenarios that lacked temporal replication, where the 1780
first 40, 20, or 10 trials were used to fit models and assess parameter estimates, depending on the 1781
scenario (Table C1). 1782
We simulated ten years of population change and removal data that were used to assess 1783
estimator performance under conditions with temporally replicated removal experiments. We 1784
generated removal data using 10 years’ worth of observed hunter efforts (2006-2015), as well as 1785
126
parameter estimates and model structures from the top model (see tables B1-B2). We started 1786
with posterior mean region- and stage-specific abundance estimates as true abundances in the 1787
first year of study, and simulated population change for nine additional years into the future 1788
using the hierarchical abundance structure from the top model (Tables B1-B2). To generate log-1789
scale random intercepts affecting density annually (see model structures, Tables B1-B2), we 1790
generated nine realizations of 𝛾𝛾𝑡𝑡 (𝛾𝛾𝑡𝑡~𝑁𝑁𝑃𝑃𝑁𝑁𝑁𝑁𝑁𝑁𝑙𝑙�0,𝜎𝜎𝛾𝛾𝑡𝑡�) using the posterior mean estimate of 𝜎𝜎𝛾𝛾𝑡𝑡 1791
from the top model as the true process standard deviation. This resulted in region- and stage-1792
specific abundances that were generated over a ten-year period that were subsequently used in 1793
conjunction with effort data observed from 2006-2015 to generate variable effort removal 1794
samples for each region and year. 1795
After generating 10 years of true population change, we generated true removal data sets 1796
for each year of sampling using estimates of capture probability parameters from the top model 1797
(Tables B1-B2) and the conditional binomial removal model (equation 3) as truth. For the high 1798
hunter effort scenario we assumed region-specific effort data observed from 2006-2015 were 1799
truth, and again halved these values for the low hunter effort scenarios. However, the turkey 1800
hunting season in Michigan was only 39 days for two years of our study (2007 and 2012), and 1801
thus we simulated true efforts for day 40 in those years as Poisson random variables with 1802
expectation equal to the observed effort in day 39 for the corresponding region. We again 1803
generated 100 random data sets of true removals for j = 40 removal trials for each scenario of 1804
hunter effort, and simulated observed effort and removal data using the Poisson observation 1805
model. Simulated observations of harvest and effort were used for all parameter estimation 1806
similar to one-year simulations, where the first 40, 20, or 10 trials were used to fit models and 1807
assess parameters, depending on the scenario. For each simulation scenario, we fit each of the 1808
127
four model structures (Table C1) 100 times (one for each simulated dataset) following the model 1809
fitting methods described above. 1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
128
Table C1. Simulation scenarios used to assess performance of abundance and population growth estimators generated from 1831
hierarchical removal models fit to variable-effort removal data with measurement error. Two scenarios of temporal replication 1832
included simulation of removal experiments that were not replicated over time (1-year) and removal experiments that were replicated 1833
for each year in a 10-year period (10-year). Model fitting and parameter estimation were replicated for studies with 40, 20, and 10 1834
removal trials each year, for 2 levels of assumed sampling effort (High and Low; see Methods and Appendix C for additional 1835
description). For each combination of temporal replication, number of removal trials, and magnitude of hunter effort, four different 1836
model structures were fit to the data to estimate abundance (and population growth for 10-year studies) and assess performance with 1837
simplified (i.e., no measurement error) or less constrained (unstructured abundance model) versions of the model, relative to the top 1838
hierarchical removal model with measurement error described in text (referred to as base model here). 1839
Trials
Temporal replication and model structurea 40 days 20 days 10 days
1-year
Base model High Low High Low High Low
Base model - no measurement error High Low High Low High Low
Unstructured model High Low High Low High Low
Unstructured model - no measurement error High Low High Low High Low
10-year
Base model High Low High Low High Low
129
Base model - no measurement error High Low High Low High Low
Unstructured model High Low High Low High Low
Unstructured model - no measurement error High Low High Low High Low
a Model structures: Base model = top model described in Tables B1-B2; Base model – no measurement error = top 1840
model described in tables B1-B2 but assuming no measurement error in removal sampling data; Unstructured model = 1841
model without hierarchical Poisson structure on abundance (Table B2); Unstructured model – no measurement error = 1842
model without hierarchical Poisson structure on abundance (Table B2) but assuming no measurement error in removal 1843
sampling data.1844
130
Table C2. Results of simulation study used to assess performance of abundance estimates for removal studies replicated over a ten-1845
year period. The fraction of simulated model fits that resulted in convergence (Convergence), as well as average relative error of 1846
abundance estimates from year 10 (Bias) and the fraction of 95% credible intervals that contained true abundance from that year (for 1847
model fits that converged) are presented by turkey management region (Region), and for all regions combined (Total). Results are 1848
presented for simulation scenarios defined by the magnitude of hunter effort (High, Low), the number of trials in each removal 1849
experiment (40, 20, 10 days of hunting), the hierarchical structure of the fitted abundance model (Poisson, Unstructured), and the 1850
presence of measurement error in fitted model (Measurement error, No measurement error). 1851
Region
ZA ZB ZC ZE ZF Total
Scenarioa Convergence Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage
High effort
Measurement error
Poisson
40 trials 0.79 0.03 0.95 0.05 0.95 0.04 0.95 0.04 0.95 0.04 0.95 0.04 0.95
20 trials 0.58 0.06 0.95 0.08 0.95 0.06 0.95 0.06 0.95 0.07 0.95 0.06 0.95
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 2.03 0.00 4.89 0.00 4.59 0.00 2.03 0.00 2.08 0.00 2.73 0.00
20 trials 1.00 2.32 0.00 5.46 0.00 5.02 0.00 2.28 0.00 2.35 0.00 3.06 0.00
10 trials 1.00 2.38 0.00 5.05 0.00 4.75 0.00 2.34 0.00 2.33 0.00 3.00 0.00
131
No measurement error
Poisson
40 trials 0.89 0.02 0.88 0.03 0.84 0.02 0.89 0.02 0.87 0.03 0.87 0.02 0.89
20 trials 0.68 0.05 0.87 0.07 0.82 0.05 0.85 0.05 0.87 0.06 0.85 0.05 0.87
10 trials 0.08 0.05 0.88 0.06 0.88 0.04 0.88 0.05 0.88 0.05 0.88 0.05 0.88
Unstructured
40 trials 1.00 1.85 0.00 3.97 0.00 3.77 0.00 1.84 0.00 1.89 0.00 2.36 0.00
20 trials 1.00 2.17 0.00 5.10 0.00 4.67 0.00 2.09 0.00 2.19 0.00 2.83 0.00
10 trials 1.00 2.31 0.00 4.89 0.00 4.62 0.00 2.28 0.00 2.27 0.00 2.92 0.00
Low effort
Measurement error
Poisson
40 trials 0.38 -0.09 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.95 -0.08 0.95
20 trials 0.00 - - - - - - - - - - - -
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 2.25 0.00 5.21 0.00 5.03 0.00 2.40 0.00 2.18 0.00 3.01 0.00
20 trials 1.00 2.26 0.00 4.81 0.00 5.20 0.00 2.41 0.00 2.22 0.00 3.00 0.00
10 trials 1.00 2.34 0.00 4.73 0.00 5.52 0.00 2.50 0.00 2.29 0.00 3.12 0.00
No measurement error
Poisson
40 trials 0.68 -0.12 0.63 -0.10 0.74 -0.11 0.66 -0.10 0.69 -0.11 0.68 -0.11 0.66
20 trials 0.07 0.00 1.00 0.01 1.00 0.00 1.00 0.01 1.00 0.01 1.00 0.00 1.00
132
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 2.16 0.00 5.10 0.00 4.97 0.00 2.33 0.00 2.09 0.00 2.93 0.00
20 trials 1.00 2.21 0.00 4.70 0.00 5.10 0.00 2.36 0.00 2.16 0.00 2.94 0.00
10 trials 1.00 2.31 0.00 4.62 0.00 5.41 0.00 2.43 0.00 2.25 0.00 3.05 0.00 a Mathematical details of each model structure and simulation scenario are described in Tables B1-B3 and Appendix C. 1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
133
Table C3. Results of simulation study used to assess properties of estimates of the finite rate of change for removal studies replicated 1865
over a ten-year period. The fraction of simulated model fits that resulted in convergence (Convergence), as well as average relative 1866
error (Bias) of population growth estimates over the entire study (𝜆𝜆𝑖𝑖 = 𝑁𝑁𝑖𝑖,𝑡𝑡=10𝑁𝑁𝑖𝑖,𝑡𝑡=1
for region i) and the fraction of 95% credible intervals 1867
that contained the true population growth (for model fits that converged) are presented by turkey management region (Region), and for 1868
all regions combined (Total). Results are presented for simulation scenarios defined by the magnitude of hunter effort (High, Low), 1869
the number of trials in each removal experiment (40, 20, 10 days of hunting), the hierarchical structure of the fitted abundance model 1870
(Poisson, Unstructured), and the presence of measurement error in fitted model (Measurement error, No measurement error). 1871
Region
ZA ZB ZC ZE ZF Total
Scenarioa Convergence Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage
High effort
Measurement error
Poisson
40 trials 0.79 -0.01 0.94 0.01 0.99 -0.02 0.97 0.00 1.00 0.02 0.91 0.00 1.00
20 trials 0.58 -0.02 0.97 0.01 1.00 -0.03 0.95 -0.01 0.98 0.01 0.97 -0.01 0.97
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 0.02 0.97 0.01 1.00 -0.02 0.99 -0.01 1.00 0.01 1.00 0.00 0.99
20 trials 1.00 0.01 0.99 0.00 1.00 -0.03 0.00 -0.03 0.98 0.00 0.99 -0.01 1.00
10 trials 1.00 0.00 0.96 -0.08 0.99 -0.08 0.90 -0.03 0.97 -0.04 0.99 -0.05 0.63
134
No measurement error
Poisson
40 trials 0.89 -0.01 0.80 0.01 0.93 -0.01 0.94 0.00 0.91 0.01 0.78 0.00 0.85
20 trials 0.68 -0.02 0.78 0.01 0.97 -0.02 0.88 -0.01 0.88 0.01 0.91 -0.01 0.87
10 trials 0.08 -0.02 0.75 -0.01 0.88 -0.04 0.50 -0.03 0.75 0.00 0.75 -0.02 0.75
Unstructured
40 trials 1.00 0.01 0.92 0.00 0.95 -0.03 0.87 -0.02 0.83 -0.01 0.97 -0.01 0.88
20 trials 1.00 0.00 0.89 0.00 0.96 -0.03 0.90 -0.03 0.79 -0.01 0.92 -0.01 0.82
10 trials 1.00 0.00 0.90 -0.09 0.69 -0.08 0.65 -0.03 0.83 -0.04 0.76 -0.05 0.25
Low effort
Measurement error
Poisson
40 trials 0.38 -0.03 0.89 0.00 1.00 -0.04 0.71 -0.01 1.00 0.01 1.00 -0.01 0.97
20 trials 0.00 - - - - - - - - - - - -
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 -0.02 0.98 -0.03 0.96 -0.12 0.55 0.00 1.00 -0.06 0.81 -0.05 0.59
20 trials 1.00 -0.06 0.78 -0.11 0.79 -0.08 0.86 0.00 1.00 -0.07 0.76 -0.06 0.35
10 trials 1.00 -0.05 0.91 -0.14 0.78 -0.03 1.00 0.01 1.00 -0.08 0.74 -0.06 0.68
No measurement error
Poisson
40 trials 0.68 -0.02 0.76 0.00 0.97 -0.04 0.34 -0.01 0.96 0.01 0.97 -0.01 0.78
20 trials 0.07 -0.01 1.00 0.01 1.00 -0.03 0.71 0.01 1.00 0.02 0.71 0.00 1.00
135
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 -0.02 0.86 -0.03 0.87 -0.12 0.29 0.00 0.97 -0.06 0.53 -0.05 0.26
20 trials 1.00 -0.06 0.62 -0.11 0.62 -0.09 0.58 0.00 0.95 -0.07 0.53 -0.07 0.08
10 trials 1.00 -0.06 0.68 -0.15 0.53 -0.04 0.89 0.01 0.97 -0.08 0.55 -0.06 0.27 a Mathematical details of each model structure and simulation scenario are described in Tables B1-B3 and Appendix C. 1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
136
Table C4. Results of simulation study used to assess performance of abundance estimates for removal studies replicated over a one-1885
year period. The fraction of simulated model fits that resulted in convergence (Convergence), as well as average relative error of 1886
abundance estimates (Bias) and the fraction of 95% credible intervals that contained true abundance (for model fits that converged) 1887
are presented by turkey management region (Region), and for all regions combined (Total). Results are presented for simulation 1888
scenarios defined by the magnitude of hunter effort (High, Low), the number of trials in each removal experiment (40, 20, 10 days of 1889
hunting), the hierarchical structure of the fitted abundance model (Poisson, Unstructured), and the presence of measurement error in 1890
fitted model (Measurement error, No measurement error). 1891
Region
ZA ZB ZC ZE ZF Total
Scenarioa Convergence Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage Bias Coverage
High effort
Measurement error
Poisson
40 trials 0.00 - - - - - - - - - - - -
20 trials 0.00 - - - - - - - - - - - -
10 trials 0.00 - - - - - - - - - - - -
Unstructured
40 trials 1.00 2.15 0.00 4.22 0.00 3.96 0.00 2.18 0.00 2.17 0.00 2.66 0.00
20 trials 1.00 2.08 0.00 4.39 0.00 4.03 0.00 2.10 0.00 2.07 0.00 2.62 0.00
10 trials 1.00 1.91 0.00 4.60 0.00 4.16 0.00 1.94 0.00 1.90 0.00 2.54 0.00
137
No measurement error
Poisson
40 trials 0.14 0.22 0.79 0.20 0.79 0.21 0.79 0.22 0.79 0.21 0.79 0.22 0.79
20 trials 0.04 0.75 0.87 0.75 0.82 0.75 0.86 0.75 0.86 0.75 0.84 0.75 0.85
10 trials 0.11 -0.50 0.64 -0.52 0.55 -0.50 0.55 -0.51 0.55 -0.51 0.55 -0.51 0.55
Unstructured
40 trials 1.00 2.20 0.00 3.99 0.00 3.71 0.00 2.22 0.00 2.22 0.00 2.61 0.00
20 trials 1.00 2.14 0.00 4.28 0.00 3.93 0.00 2.16 0.00 2.14 0.00 2.64 0.00
10 trials 1.00 1.90 0.03 4.64 0.00 4.20 0.00 1.92 0.03 1.88 0.03 2.55 0.00
Low effort
Measurement error
Poisson
40 trials 0.00 - - - - - - - - - - - -
20 trials 0.00 - - - - - - - - - - - -
10 trials 0.01 0.46 1.00 0.41 1.00 0.46 1.00 0.45 1.00 0.44 1.00 0.45 1.00
Unstructured
40 trials 1.00 1.93 0.00 4.25 0.00 4.29 0.00 1.97 0.00 1.98 0.00 2.56 0.00
20 trials 1.00 1.93 0.00 4.22 0.00 4.17 0.00 1.89 0.00 1.83 0.00 2.47 0.00
10 trials 1.00 1.97 0.00 4.22 0.00 4.09 0.00 1.90 0.00 1.81 0.00 2.48 0.00
No measurement error
Poisson
40 trials 0.04 -0.02 1.00 -0.06 1.00 -0.03 1.00 -0.04 1.00 -0.04 1.00 -0.03 1.00
20 trials 0.02 -0.04 1.00 -0.08 1.00 -0.04 1.00 -0.05 1.00 -0.06 1.00 -0.05 1.00
138
10 trials 0.01 -0.70 0.00 -0.71 0.00 -0.71 0.00 -0.71 0.00 -0.71 0.00 -0.71 0.00
Unstructured
40 trials 1.00 1.92 0.01 4.32 0.00 4.41 0.00 1.96 0.01 1.95 0.01 2.57 0.00
20 trials 1.00 1.93 0.02 4.21 0.00 4.23 0.00 1.88 0.02 1.83 0.02 2.47 0.00
10 trials 1.00 2.01 0.00 4.18 0.00 4.11 0.00 1.93 0.00 1.85 0.00 2.50 0.00 a Mathematical details of each model structure and simulation scenario are described in Tables B1-B3 and Appendix C. 1892
1893
1894
1895
1896
1897
1898
1899
1900
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