A first method to incorporate age-effects into superpopulation models is to define age-classes to characterize how long a carcass has been on the ground. Hereafter, we consider two such carcasses. The model is then based on a classes, or states, representing, for example, fresh and dry Markovian, stochastic transition between these two states (Hestbeck, Nichols & Malecki 1991). The fieldwork follows a standard multistate capture–recapture protocol with T sampling occasions (Hestbeck, Nichols & Malecki 1991). We adopt the maximum-likelihood approach and thus describe the model in terms of sufficient statistics (see below). Model parameters are presented in Table 1. An alternative and increasingly used approach is to describe capture–recapture models in terms of fundamental processes (e.g. Schofield & Barker 2008), which render their translation into Bayesian software (WinBUGS) programming code more straightforward (Bonner & Schwarz 2006; Royle 2009; Schofield & Barker 2009; Link & Barker 2010). Of interest is the superpopulation size NTot, defined as the number of distinct carcasses that are available during at least one of the sampling occasions:
Table 1. Multistate model: model parameters and data to be collected (statistics)
| Model parameters |
| ||The number of carcasses of age 1 already on the ground immediately before the first sampling occasion|
| ||The number of carcasses of age 2 already on the ground immediately before the first sampling occasion|
| ||The number of new carcasses that appear between sampling occasions i and i + 1 and that are still available for detection at time i + 1. i = 1,…,T−1. |
| ||Carcass persistence probability in age-class 1: the probability that a carcass of age 1 on the ground at time i will remain available for detection until time i + 1. i = 1,…,T−1. |
| ||The probability that a carcass of age 2 on the ground at time i will stay until time i + 1|
| ||The probability that a carcass of age 1 on the ground at time i and that persists until time i + 1 enters age-class 2 before time i + 1. i = 1,…,T−1. Reverse transition from 2 to 1 is impossible.|
| ||The probability that a carcass of age 1 on the ground at time i is detected then. i = 1,…,T|
| ||The probability that a carcass of age 2 on the ground at time i is detected then.|
| Statistics |
| ||the number of unmarked carcasses first detected when of age 1 at time i. i = 1,…,T|
| ||the number of unmarked carcasses first detected when of age 2 at time i|
| ||the number of marked carcasses detected when of age 1 at time i that were last detected as carcasses of age 1 at time j|
| ||the number of marked carcasses detected when of age 2 at time i that were last detected as carcasses of age 1 at time j|
| ||the number of marked carcasses detected when of age 2 at time i that were last detected as carcasses of age 2 at time j|
Likelihood of the numbers of unmarked carcasses
The probability for a carcass to enter in age-class 1 vs. in age-class 2 is akin to the initial proportion parameter of multievent models (Pradel 2009), but the only possibility to enter the system in age-class 2 is at the first sampling occasion. In other words, we consider the possibility that a carcass becomes dry before being detected, but we exclude the possibility that this event occurs within a single time interval. The probability to enter in age-class 1, denoted π, is thus defined by . Similarly, and following the standard formulation of superpopulation models, we introduce entry probabilities defined by . is, among carcasses that enter the superpopulation in age-class 1, the fraction that enters the system (individual dies) between i and i + 1 (i = 0,…,T−1). Following Schwarz & Arnason (1996), the are actual model parameters to be estimated, and the N's are estimated by backward transformation. Last, we define the intermediate quantities corresponding to the probability that a carcass enters the system in age-class s before time i, is still present and in age-class s at time i and is not seen before time i (i = 1,…,T), by the following recurrence relations:
Then, a multinomial distribution describes the u-statistics:
Likelihood of the recapture data
Following Brownie et al. (1993), we define the ‘survival’ transition matrix:
And the detection and non-detection matrices:
Then, Bij(rs), the probability of being detected in age-class s at occasion j given that previous detection was in age-class r at occasion i, can be found for any combination (i,j,r,s) in the (r,s)th cell of matrix Bij:
[if j=i + 1, the product is replaced by the identity matrix, and if j≤i, the matrix is filled with 0s]
The m-statistics follow multinomial distributions, for example for the ith line:
In the full time-varying model, which counts 6T-2 parameters in our multistate application, some parameters occur in the likelihood only as combinations and are thus not separately estimable, that is, the model is not identifiable (Schwarz et al. 1993). This is usually dealt with by putting some constraints on parameters describing the first and last time steps (Schwarz et al. 1993). In other words, a few assumptions about time variation are needed for the model to be identifiable. Alternatively, trial experiments can be designed to estimate the first and last detection probabilities directly with no identifiability problem. In the simulations presented below, we bypassed these issues by making the model time-constant. We also improved identifiability by replacing in the likelihood by its estimator .
Post hoc correction for early removals
NTot is the number of carcasses that are available for detection during at least one sampling occasion. However, scavengers can remove carcasses before the first sampling occasion following death; depending on the study design and persistence rate, these early removals can account for a large part of the total fatality number. Following Crosbie & Manly (1985) and Schwarz et al. (1993), it is here assumed that recruitment is uniformly distributed within each time interval. Then, the corrected estimate for the total number of fatalities occurring between the first and last searches is computed as:
where fU = 1 represents the density of the uniform distribution over one time interval. This correction is made post hoc using the maximum-likelihood estimates of model parameters, and the delta-method is used to compute sampling (co)variance. If recruitment is not uniformly distributed within time intervals (e.g. if most deaths occur on a particular night with exceptional weather or migration intensity), the corrected estimate can be seriously biased.
The multistate model was fitted to simulated data using the maximum-likelihood approach, in R (R Development Core Team, 2010) and with the optimization function nlm. We considered a few representative scenarios (Appendix S1, Supporting information part 1) but do not claim that this simulation study allows drawing conclusion for the rest of the parameter space.
Fatalities involving protected species are expected to be rare, and thus, NTot will usually be small. Two issues arise when sample size is small: imprecision of the estimates (small sample bias and instability of the numerical routine) and numerous zero detection events, that is, sampling occasions during which no new, unmarked carcass is detected. The first issue can be partially addressed by adding experimentally planted carcasses to the sample of naturally occurring ones. The second issue can lead to what is commonly referred to as the boundary parameter issue in capture–recapture models (Lebreton et al. 1992). To investigate whether this induced a bias on the estimation of NTot, we ran a second simulation analysis. We considered only one age-class, and a small sample of 12 carcasses, which entered in a staggered fashion over nine sampling occasions: . Carcass persistence was 0·8, and carcass detection was 0·5. This scenario was expected to loosely mimic a context of raptor management, with crew members searching an extensive area every 2 weeks, for example. We simulated 50 data sets under this scenario.
In this section, we assume a direct relationship between carcass age and model parameters, that is, deterministic instead of stochastic transitions between age-classes, with no need for onsite technicians to document the estimated age of the carcasses. Such regression models are a modification of the models developed by Kendall (2006) and Pledger et al. (2009). The modification consists in introducing a distribution for the age of carcasses present at the start of the study, whereas typical applications assume all individuals enter the system after or immediately before the start of the study (Matechou E., Pledger S., Efford M., Morgan B.J.T. and Thomson D.L., unpublished data). The likelihood is individually based. Model parameters are similar to the multistate model, except for the full age specificity (Table 2). In the following, the age of a carcass is the number of occasions it has been present on site (constant time interval between occasions).
Table 2. Regression model: notations for model parameters and data to be collected
| Model parameters |
| ||The total number of carcasses that are available to capture during at least one of the sampling occasions.|
| ||Entry probability: the proportion from the N carcasses that enters the system (individuals die) between sample occasions i and i + 1; i = 0,…,T−1. i = 0 represents carcasses already present at the first sampling occasion.|
| ||The probability that a carcass had entered the system x time periods before the start of the study In practice, we set a maximum age: x ≤ X.|
| ||Carcass persistence probability: the probability that a carcass of age a on the ground at time i persists until time i + 1. i = 1,…,T−1. In practice, we set a maximum age: |
| ||Carcass detection probability: the probability that a carcass of age a on the ground at time i is detected at that time. i = 1,…,T. |
| Data collected from naturally occurring carcasses |
| ||Detection history for kth individual carcass: ωk,i = 1 if the carcass is detected at sampling occasion i, and ωk,i = 0 if not detected.|
| ||The first sampling occasion at which carcass k is detected.|
| ||The last sampling occasion at which carcass k is detected.|
| C ||The total number of detected carcasses|
Probability of detection histories
For each detection history ωk, the probability of occurrence is the sum of the probabilities conditional on two disjoint events: entry before or during the study. To compute the former, we define negative time occasions, so that time occasion –x is situated x + 1 time intervals before the start of the study.
where X represents the maximum age of the carcasses that are present at first sampling occasion, arrival (k) is the first occasion after the entry of the kth carcass, and β′x denotes the probability that a carcass enters the study area between occasions –x and –x + 1, and survives up to occasion 1. The conditional probabilities are then expressed as functions of persistence and detection probabilities:
[In equations 10 and 11, the products are replaced by a one when the end index is lower than the start index].
To compute parameters β′x, we assumed that the entry rate was uniform before the first sampling occasion. We further assumed that age-specific persistence before the first sampling occasion could be approximated by the temporal average of age-specific persistence after the first occasion, which we denoted . Then, we could write:
with (0) = 1.
Probability of missing a carcass
Let P0 be the probability of missing a carcass given that it was available during at least one occasion. As in detected carcasses, P0 is the sum of the probabilities conditional on two disjoint events: entry before or during the study.
The conditional probabilities are expressed as functions of persistence and detection probabilities:
[In equations 14 and 15, the products are replaced by a one when the end index is lower than the start index].
In this model, the age specificity in persistence and detection probabilities is used to make inference about the ages of the carcasses at first detection. This model is thus similar to a mixture model with as many heterogeneity classes as there are possible ages at first detection (Pradel 2009; Pledger, Pollock & Norris 2010), and is therefore identifiable. However, in practice, low sample size and lack of strong age specificity lead to ‘plateaus’ in the likelihood, that is, large areas of the parameter space for which the likelihood hardly varies. Such ill-conditioning can make it difficult for the optimization procedure to find the maximum likelihood when that maximum is not different enough from surrounding values. Furthermore, if the assumption of uniform entry rate before first occasion is not met, then additional parameters are needed to describe the β′x [equation 12], in which case the model becomes truly non-identifiable. These identifiability issues can be addressed by using known-age data, that is, carcasses of which the age at first detection is perfectly known. Trial experiments (planting fresh carcasses at known times and locations) provide such a data set. For each trial carcass, the recorded data are then: the first and last occasion the carcass was present, and its detection history (detections by a person different from the one that planted the carcass). Then, the probability of observing history is as follows:
where and are the first and last sampling occasions a fake carcass is present on site.
If C naturally occurring carcasses have been found and fake ones have been planted, and under the usual assumption of independence of fate, the joint likelihood is as follows:
where θ is a vector containing all model parameters, including NTot.
In our simulations (Appendix S1, Supporting information part 2), we used the three-parameter logit-quadratic function (equation 18): first persistence probability may increase with age (representing the activity of scavengers, heterogeneity and drying) and then it may decrease with age (decay).
where, A, B, Ci are parameters to estimate. We note that in actual estimation situations, this parametric form can yield flawed results by constraining symmetry on the logit scale (i.e. when the persistence probability increases at young ages but does not decrease afterwards, this model often nonetheless estimates a spurious decline at old, poorly documented ages). We did not consider age specificity in detection probability in our simulations.
Current study designs and New Jersey application case
Trial experiments are often organized to deal with the detection probability. Externally obtained carcasses are planted in the search area; the same technicians that would look for ‘naturally occurring’ carcasses report whether they found the trial carcasses. Trial carcasses are also used to document persistence rate (Smallwood et al. 2010; Huso 2011; Barrientos et al. 2012). The typical study design that combines carcass searches and trial experiments, however, differs from the situations covered in the previous two sections: (1) naturally occurring carcasses are not marked and left where found but are instead permanently removed upon detection, (2) trial experiments and carcass searches are not always simultaneous, and (3) carcasses are not assigned to an age-class. We describe below a modified version of the superpopulation model that combines both types of data, at the cost of a model simplification. This model is available in R-package carcassCMR (Appendix S3, Supporting information).
Removing the age specificity (superscripts) from the multistate model, but otherwise maintaining the same notation, we write for i < T:
Then, a multinomial distribution describes the u-statistics:
Detection trials consist of planting carcasses (or any item of which the detection probability is similar to that of a carcass) prior to a scheduled search. Noting Di the number of planted items at time i, and di the number of those items detected during the carcass search, detection probability pi is then estimated following:
New Jersey application case
The study was conducted by the New Jersey Audubon Society at the Jersey Atlantic Wind, LLC/Atlantic City Utilities Authority wind power facility, near Atlantic City, NJ (39°22′N, 74°27′W). There were five turbines. Below each turbine, we defined a search plot as a square with sides of 130 m. Only 24–85% of each search plot was searchable; however, the rest consisting of inaccessible rooftops, clarifying and aerating pools, and tidally influenced marsh areas. The superpopulation thereby corresponded to carcasses falling within the searchable area. The latter did not vary through time. For this study, we focused on the months of August and September 2007 (post-breeding migration period). This included a total of T = 23 sampling occasions separated by intervals ranging from 2 to 5 days. We selected only the data from bats: Hoary bat Lasiurus cinereus and Eastern red bat Lasiurus borealis, which we pooled together.
Trial experiments were conducted between July 2007 and August 2010 by placing a total of 57 small bird or bat carcasses (<35 g) in the searchable area within 24 h of a scheduled search (with a maximum of one trial per month and two trial carcasses per turbine and trial). Forty seven of these carcasses persisted until the next scheduled search. The detection trial consisted of recording whether carcasses were detected during that scheduled search. The persistence trial consisted of visiting carcass locations every day for 7 days, to monitor their continued presence or their removal.
Detection and daily persistence probabilities were assumed constant through time, similar between bats and birds of similar size and independent of carcass age. The first two assumptions were required in order to use trial data in lieu of capture–recapture data. We considered three models with different distributions of carcass entries through time: (1) full time dependence, (2) depends logit-linearly on the length of the interval between occasions i and i + 1 and (3) linear increase with time on the logit scale. The varying length of the intervals between sampling occasions was accommodated by introducing a daily survival probability and raising it to the appropriate power.
Sampling protocols with marking of carcasses
In the first two sections, we described models for which carcasses are marked upon detection and left where found to be subjected to the detection process again. Currently, those models are deemed inapplicable because of the stationary nature of carcasses (‘Current study design and New Jersey application case’). Yet, the loss of information about carcass location between successive sampling occasions can be achieved by dividing a large survey area into multiple subunits and assigning crew members to different units at each occasion, so that no crew member surveys the same sample unit twice in a row. Alternatively, one may use a double observer protocol, where the efficiency of the second observer is estimated using the carcasses found by the first observer, and reciprocally (e.g. Nichols et al. 2000).
Various methods can be used to mark naturally occurring carcasses, from sticker notes to passive integrated transponders (PIT tags), individual alphanumeric tags and geo-location. We also note that because the likelihood of the multistate model is based on summary statistics , it is not strictly necessary that carcasses are individually marked. ‘Batch marks’ may be used, that is, time-specific colour marks (spray paint, wool threads, etc.) can be applied to carcasses upon detection. One consideration for any form of marking is that the mark should not draw the attention of the crew members to the marked carcasses.
Four variables then describe the sampling effort: (1) the proportion of turbines that are monitored, denoted K; (2) the number of sampling occasions, denoted T, that is, the number of times field crews visit the search area; (3) the interval between successive sampling occasions, denoted I; and 4) optionally, the number of trial carcasses used to boost sample size and document persistence and detection probabilities, denoted F. For a fixed daily removal rate by scavengers, persistence probability decreases exponentially with I.
We were then interested in the optimal design of a capture–recapture carcass survey, that is, the optimal combination of values. For this purpose, we devised equation 23 to describe the utility of a carcass survey: utility increases with K because small K's induce a risk of sampling a non-representative set of turbines; utility increases with study duration I × T because short durations induce a risk of sampling a non-representative period (e.g. miss migration peak); utility increases with diminishing bias (in our case bias is linked to the cross multiplication needed to extrapolate from the sampled turbines to the whole farm); and utility diminishes with increasing standard error of the estimate for superpopulation size (Mackenzie & Royle 2005).
where was the expected bias (in absolute proportional value), was the expected coefficient of variation of the estimate of fatality number, and u1, u2 and u3 were parameters of which the values depend on the objectives of the stakeholders.
We then defined the components of the cost function:
where c0 was the fixed overhead component, c1 the price of visiting the wind farm once, c2 the price of searching under one additional turbine once technicians were on site, and c3 the price of acquiring and deploying one trial carcass. We assumed that there was no additional cost when increasing the interval between sampling occasions.
We focused on the optimal study design for the following set of three constraints: (1) Fixed cost: we wanted to find the optimal allocation of resources that maximized utility for a given budget. (2) Fixed study duration: we assumed that among the requirements and specifications was the fact that the study had to document fatality number over a predetermined period of time, for example one migratory season. (3) Fixed number of trial carcasses: we assumed that the number of trial carcasses was constrained by factors other than cost. Under these three constraints, there was a single set of K, T, I and F values that maximized utility. For illustration purposes, we computed this set of optimal values for c0 = 10, c1 = 10, c2 = 1, c3 = 0 (since F was fixed), u1 = 0 (since duration was fixed), u2 = 1 and u3 = 10. Study duration was fixed to I × T = 90 days and number of trial carcasses to F = 100 carcasses.
We needed an explicit function describing and as a function of K, T, I F. Our model was, however, deemed too complex for an analytic approach, in particular because the analytic computation of would be based on asymptotic variances, whereas in the situations of interest here, sample sizes were expected to be low (Carother 1973; Crosbie & Manly 1985). Therefore, we conducted a series of simulations in which we fixed K, T, I, F, and directly estimated and from these data sets, and then extrapolated to obtain a continuous function. Simulations were conducted under the following scenario: there were 100 turbines; one animal died each day; the turbine causing death was selected at random (uniform distribution of carcasses); the carcasses had a constant 5% daily probability of being removed by scavengers ( = 0·95, resulting in expected persistence time of 20 days); at each of T equally spaced sampling occasions, the carcasses had a 70% probability of being detected (P = 0·70). K was allowed to vary from 5 to 95%, T from 3 to 15 occasions, I from 1 to 90 days and F from 10 to 1000 carcasses. We simulated and analysed 20,736 data sets. We then investigated the dependence of and on K, T, I, F using linear models (generalized additive models indicated that nonlinearities were weak once explanatory variables K, T, I, F were log-transformed).