When studying senescence in wildlife populations, we are often limited by the sparseness of the available information on the ages of the individuals under study. Additionally, heterogeneity between individuals can be substantial. Ignoring this heterogeneity can lead to biased estimates of the population parameters of interest and can mask senescence.
This article demonstrates the use of a recently developed capture–recapture model for extracting age-dependent estimates of survival probabilities for individuals of unknown age and extends the model by allowing for heterogeneity in survival and capture probabilities using finite mixtures.
Using simulation, we show that the estimates of age-dependent survival probabilities when age is unknown can be biased when heterogeneity in capture probabilities is not modelled, in contrast to the case of time-dependent survival probabilities when the estimates are robust to similar violations of model assumptions.
The methods are demonstrated using a long-term data set of female brushtail possums (Trichosurus vulpecula Kerr) for which age-specific models for survival probabilities indicating senescence are strongly favoured. We found no evidence of heterogeneity in survival but strong evidence of heterogeneity in capture probabilities.
These models have a wide range of applications for estimating age dependence in survival when the age is unknown as they can be applied to any capture–recapture data set, as long as it is collected over a period which is longer, and preferably considerably so, than the life span of the species studied.
A great limitation when it comes to studying senescence in wildlife populations results from the unavailability of information on the ages of the individuals under study. With many species of mammals and birds, it is not possible to identify the ages of the individuals unless these are caught as juveniles. As a result, one can assume a constant survival probability for all ages of adults and therefore only distinguish two age classes, namely juveniles and adults (Brownie et al. 1978). However, this does not enable the study of senescence and is, in many cases, unrealistic. Another approach would be to include only the individuals for which the ages are known with the disadvantage of discarding large parts of the data set or to use time since first capture as a proxy for age (Reed et al. 2008) which relies on the assumption that all individuals are of the same (unknown) age when first caught.
Two models have been proposed recently that aim to tackle the issue of estimating age-dependent survival probabilities for individuals of unknown age: Colchero & Clark (2012) developed a Bayesian hierarchical model, available in the R package BaSTA (Colchero et al. 2012), while McCrea et al. (2013) developed a mixture model for recovery data which requires that the ages of some of the individuals in the sample are known.
The same obstacles are presentedõ when modelling the transient presence of migratory birds at stop-over sites. In the stop-over setting, the probability of leaving the site is thought to depend, amongst other factors, on the time the individuals have spent at the site. However, this information is generally unavailable. Pledger et al. (2009) developed a model that builds on the unknown time of arrival of the individuals and allows the estimation of the probability of remaining until the next sampling occasion to be linked to the unknown time already spent at the site (see also Pradel (2009) for a different approach). They established that the age-dependent model, where age is the time since arrival in the stop-over setting, is identifiable even when all individuals are of unknown age (see Data S1 in the Supporting Information for the verification of this result using symbolic algebra methods in Maple) and applied their model to a stop-over data set of semipalmated sandpipers (Calidris pusilla).
The Pledger et al. (2009) model assumes that the time the individuals have spent at the site before the start of the study is negligible. This assumption is realistic for data sets collected at stop-over sites where it is common for the duration of the study to exceed the stop-over duration of the individuals. However, this is not the case when studying survival by sampling resident populations at permanent sites. Additionally, the model assumes homogeneous individuals that share common population parameters. However, it has been repeatedly shown that wildlife populations can be substantially heterogeneous, both in terms of survival (Cam et al. 2002) and capture probabilities (Gimenez & Choquet 2010; Pledger et al. 2010). In the first case, individuals with an overall higher probability of survival will prevail at older ages, which can result in the average probability of survival to increase for older ages, potentially masking the effect of senescence (Vaupel & Yashin 1985; Peron et al. 2010). In the latter case, ignoring any differences in capture probabilities between individuals can lead to ‘leakage’ between parameters as an over-simplistic model for capture probabilities can lead to the selection of a more complicated model for survival probabilities than necessary (Catchpole et al. 2004; Pledger et al. 2009) and/or to biased estimates of survival probabilities (Fletcher et al. 2012).
In this paper, we demonstrate how the Pledger et al. (2009) model can be applied to data sets collected at permanent sites where individuals are born and die, and we show a possible way to model the birth times of the individuals that may have been born prior to the start of the study. Using this model allows the estimation of age-dependent survival probabilities, enabling the study of senescence, without any information on the ages of the individuals being required. Furthermore, we extend the model to allow for heterogeneous probabilities of survival and capture using finite mixtures, following the work of Pledger et al. (2010). Therefore, the two major innovations in the present paper are as follows: (i) combining heterogeneity with age structure when age is unknown in the same model and (ii) modelling entry terms before the first sample that is required for populations already well established before the start of the study.
The models and methods are presented in 'Materials and methods' section. The 'Simulations' section demonstrates the importance of extending the possible birth times of the individuals beyond the study time limits, especially when the study length is not considerably greater than the lifespan of the species, as well as the importance of accounting for heterogeneous capture probabilities when the focus is on obtaining unbiased estimates of age-dependent survival probabilities. The application of the methods to a long-term data set of brushtail possums is given in the 'Application to real data: possums' section, and a discussion of the results is found in 'Discussion' section.
Materials and methods
The model presented in this section applies to the class of capture–recapture (CR) data sets which are collected by repeatedly capturing individuals from the population of interest and uniquely marking them before releasing them back to the population. Each of the D individuals caught in at least one of the K sampling occasions has its own CR history. Since detection is usually imperfect, an individual available for capture might be missed on one, more than one or even all capture occasions, so that D can be considerably lower than the population size. Additionally, the time of first possible capture of an individual, b, does not necessarily equal the time of first capture, f, and the same holds for the times of last possible capture, d, and last capture, l, that is, b ≤ f ≤ l ≤ d. If individual i, i = 1,…,D, is caught on occasion j, j = 1,…,K, then the entry in CR history i is equal to 1 and 0 otherwise. The number of individuals with a common capture history h is denoted by , so that and all unique CR histories form the rows of CR matrix X. If individuals with capture history h are caught at sample j, then and 0 otherwise.
Each individual comes independently from one of G groups with probability distribution . Group membership is a latent variable and therefore unknown.
Generalising the Schwarz & Arnason (1996) model, the parameters of our model are as follows:
N, the ‘super-population’ size which is equal to the number of individuals that became available for capture at least once during the study,
, which denotes the proportion of N that are new additions to the population at time j, ,
, which is the probability that an individual that belongs to group g and is of age a at time j will survive to time j + 1 and therefore to age a + 1 and finally,
, which is the probability that an individual that belongs to group g and is of age a at time j will be caught at time j.
The unique capture histories observed, together with the capture history with all entries equal to 0 shared by N−D individuals, form the cells of a multinomial distribution with probabilities equal to the probability of observing each of the capture histories.
The probability of having capture history h and belonging to group g and being born just before time b and dying just after time d is equal to:
Since the times of birth and death, as well as the group to which the individuals belong, are unknown, to derive the unconditional total probability of capture history h, one needs to sum over all possible values of b, d and g:
where, and are, respectively, the known times of first and last capture of individuals with capture history h.
As can be seen above, b takes values that are the sampling occasions 1 to . Therefore, it is assumed that the time the individuals have spent at the site before the start of the study is negligible or, in the context of a permanent study site, that all of the individuals caught at the first sample are newborns, all individuals caught at the second sample are at most 2 years old, etc. As mentioned above, this is a realistic assumption when modelling stop-over data since the stop-over is usually included within the study span. However, when a sample is taken at a permanent study site, it is expected that the individuals caught will be a mixture of different ages. Treating these older individuals as newborns can considerably bias the estimates of age-dependent survival probabilities, as shown in 'Simulations' section, unless the study span is considerably longer than the life span of the species modelled.
To derive unbiased estimates of survival, the possible birth times of the individuals need to be extended backwards by occasions. The value of depends on the length of the study as well as on the life span of the species of interest, as is discussed in 'Simulations' section. The ‘super-population’, N, corresponds to those individuals that became available for capture at least once during the study. Hence, if an individual was born prior to the start of the study, in order to be part of N, it needs to have survived until the first sampling occasion.
Denote by the probability that a new entrant is in group g, assumed to be constant over time, and by the proportion of N which has just arrived at occasion b and is in group g. The proportion of N in group g is equal to . However, unless some form of extrapolation is used, we do not have any information to estimate for . For example, the β probabilities can be modelled in terms of known covariates, such as weather, and the results extended to times prior to the start of the study. Another possibility is to assume that the number of individuals born between occasions up to the start of the study is constant and equal to the average number, as estimated by the model, during the study period. If that is a reasonable assumption to make for the population of interest, then the proportion of N that were born at time and are in group g is equal to , where is the average estimated for b ∈ [1,K] and, for example, corresponds to the proportion of N that belong to group g that were newborns at time 0, which is one time interval before the start of the study. The time-intervals are equal and correspond to natural time units, for example, years. Time-specific survival probabilities corresponding to times are not estimable, unless they are constrained, for example, using parametric curves. Then, becomes . Finally, the proportion of N that were new arrivals at occasion b when b ≤ 0 is equal to
As was noted above, for an individual born prior to the start of the study to be part of the ‘super-population’, it has to survive to the first sample, which is why, even though the possible birth times of the individuals are back-tracked to times before time 1, the possible death times of the individuals are after the start of the study, that is, d ≥ 1 as shown in the likelihood of eqn 3, which combines the work of Pledger et al. (2009) and Pledger et al. (2010) and estimates age-dependent parameters when no information on the ages of the individuals is available, while at the same time allowing for heterogeneous survival and capture probabilities.
We note that survival probability is not estimated when d = K as there is no information on the survival of the individuals after the end of the study. Therefore, the term that appears in the likelihood is evaluated only for values of d < K, as shown by the indicator variables. Summing over the possible death times of individuals is similar to using the χ term in the Jolly–Seber type models (Pollock et al. 1990; Lebreton et al. 1992; Schwarz & Arnason 1996), where is the probability of not being caught again after sample j; only in this case, this probability will not only depend on the value of j but also on the value of a, that is, the age of the individual at time j and of g. Survival probabilities that correspond to times before the start of the study cannot be modelled as time-varying unless parametric curves are used, for example, linking survival to weather or other time-varying covariates.
Our suggested model should be preferred over the Pledger et al. (2009) model when heterogeneity in survival and/or capture probabilities is suspected and over the Pledger et al. (2010) model when the interest is in the estimation of age-specific survival. Modelling the capture histories as shown in (eqn 1) instead of eqn 3 is sufficient if the number of adults present at the start of the study is expected to be low compared with the number of newborns, for example, by excluding older individuals caught in the first few samples, or if the duration of the study is considerably longer than the life span of the species. Simulations presented in the 'Simulations' section show that the model is robust to fluctuations of the entry probabilities before the start of the study as long as these are not systematic, for example, showing an increasing or decreasing pattern over time.
The model assumptions are those of the Pledger et al. (2010) model together with the additional assumption that the proportions of individuals belonging to each group are the same for all cohorts.
The likelihood function is written in C and is maximised using function optim in R with the built-in quasi-Newton optimisation algorithm. R code, under the name ‘HETAGE Workshop’, is available at the second author's personal homepage, accessible from http://homepages.ecs.vuw.ac.nz/~shirley/. An R package, ‘hetage’, is in progress and will soon be available on the CRAN website.
This section assesses the bias that can potentially be introduced in the estimates of survival probabilities if we ignore heterogeneity in capture probability and/or we assume that all individuals are born within the study time limits.
Ignoring heterogeneity in capture probability
Treating a heterogeneous population as homogeneous with regard to capture probability can lead to considerably biased estimates of population size (Carothers 1973; Pollock 1982; Pollock et al. 1990; Link 2003). In this section, we demonstrate, using simulation, that ignoring heterogeneity in capture probability can also lead to biased estimates of survival probabilities when survival depends on age and age is unknown. We generated 100 data sets using N = 1000, β = (0·10, 0·09, 0·14, 0·14, 0·08, 0·11, 0·07, 0·06, 0·08, 0·13), indicating that all individuals are born within the study limits, and , where is the probability of surviving from age a to a + 1 which is considered constant over time and γ = 6, κ = 5 are the scale and shape parameters of a Weibull distribution (Pledger et al. 2009). Two groups were defined in terms of capture with probabilities and , respectively. Various values of were used, specifically and in addition, we set . We modelled the probabilities of the different capture histories using (eqn 1), since all individuals are born within the study limits in this case, and we used a Weibull survivor function to model survival probabilities for both one and two groups of individuals in terms of capture probabilities.
Figure 1 shows the true Weibull survivor function as well as the median curve derived from the 100 simulations under the homogeneous and heterogeneous models. The figure illustrates that ignoring heterogeneity can lead to biased estimates of survival probabilities, especially when the difference between the capture probabilities of the two groups is considerable. This bias naturally decreases as the difference between the capture probabilities of the two groups decreases.
We also simulated data under exactly the same scenario but with survival probabilities depending on time instead of age. Decreasing survival probabilities with time are likely to be observed for stop-over data where all individuals leave the study site by the end of the stop-over period. As expected, the bias in estimating the survivor function when heterogeneity in capture is ignored in this case is negligible, as seen in Fig. 2.
Finally, we generated data by setting survival probabilities as constant across time and age and all other parameters as specified above but fitted models with ϕ modelled using a Weibull survivor function for both G = 1 and G = 2. The resulting estimates are shown in Fig. 3 where it is seen that the unmodelled heterogeneity in p leaks into the estimation of ϕ, especially when the difference between the two groups is substantial. In fact, the model with age-dependent ϕ when G = 1 has a lower AIC value than the model with constant ϕ in 99 and 61 out of the 100 simulations when and , respectively, while the corresponding numbers when G = 2 are equal to 9 and 19.
Assuming all individuals are born within the study limits
To show the effect on the estimates of survival probabilities from assuming that all individuals are born within the study time limits, we generated 100 data sets for 20 occasions but only the last 10 occasions were sampled. If these occasions are thought of as years, then we start sampling a population 10 years after it has been introduced in an area. As a result, individuals caught at each time are a mixture of ages. There is a common capture probability P = 0·7 and varying values for the scale parameter of the Weibull, γ = 4,5,6 and 10 while the shape parameter is equal to 5; the greater γ is, the longer the life span.
Similar entry probabilities
The true β parameters that are used to generate the presence histories over the 20 sampling times are equal to: (0·04, 0·06, 0·03, 0·07, 0·04, 0·06, 0·03, 0·01, 0·05, 0·06, 0·03, 0·05, 0·08, 0·03, 0·08, 0·07, 0·08, 0·02, 0·09, 0·02); therefore, they are similar to random draws from a uniform(0·01, 0·09) distribution.
Figure 4 shows the true Weibull survivor function together with the median estimate derived from the 100 simulations when birth times are not extended to times prior to the start of the study and when birth times are extended up to 10 occasions prior to the start of the study, that is, .
The figure illustrates that, especially when the span of the study is not longer than the life span of the species under study, the estimates of age-specific survival probabilities can be considerably biased unless the likelihood of eqn 3 is used. The bias in ϕ is more obvious as γ increases. In the case of γ = 10, even after extending the possible birth times of the individuals by 10 occasions before the start of the study, the resulting median estimate for the survivor function is slightly lower than the true curve, which demonstrates that it is important that the duration of the study is longer than the life span of the species. In this case, the value of was set equal to 10 because it was known that for this simulated data set, the possible birth times of the individuals could only be between −9 and 10. For real data sets, should be chosen using knowledge of the life span of the species. In practice, there is no value for which is too large but the only drawback is then the time required for the likelihood optimisation algorithm to converge. We have observed that after a certain value for , which depends on the duration of the study and the species life span, the estimates obtained do not differ except in the third or fourth decimal place.
Different entry probabilities
The true β parameters that are used to generate the presence histories over the 20 sampling times are equal to: (0·01, 0·02, 0·12, 0·04, 0·02, 0·03, 0·02, 0·15, 0·04, 0·05, 0·02, 0·03, 0·16, 0·02, 0·02, 0·04, 0·04, 0·11, 0·05, 0·01); therefore, in this case, there are four occasions with β probability greater than 0·1, while the remaining probabilities are in the [0·01, 0·05] range.
Figure 5 illustrates that even though additions to the population are not constant prior to or after the start of the study, the estimates derived from the model of eqn 3 are highly satisfactory and certainly preferable to those derived when capture histories are modelled using (eqn 1) instead.
However, if the differences between the β probabilities before and after the start of the study are more systematic, as in this example where all of the β probabilities before the start of the study are greater than those after the start, β = (0·08, 0·05, 0·06, 0·09, 0·08, 0·07, 0·09, 0·09, 0·08, 0·06, 0·03, 0·01, 0·02, 0·04, 0·01, 0·02, 0·03, 0·02, 0·04, 0·03), then unless the study is considerably longer than the life span of the species, the results are equally biased as those derived when capture histories are modelled using (eqn 1) (Fig. 6).
Application to real data: possums
We consider data collected for 23 years on female Australian brushtail possums (Trichosurus vulpecula Kerr), near Wellington, New Zealand, (Efford 1998).
Weibull survivor functions are used to model survival probabilities, as was shown in the 'Simulations' section considering the possibility that the parametric curve applies to the survival of ages greater than 1, 2,…, 6. Using parametric curves to model survival of older ages requires a considerably smaller number of parameters to be estimated by the model, especially in the case of long-lived species such as possums, compared to estimating one survival probability for each age. Furthermore, because of the small number of individuals that survive to older ages, the corresponding estimates of survival probabilities are often unidentifiable or imprecise, unless a parametric curve is used to extrapolate to these older ages. To account for potential heterogeneity in survival probabilities, we allowed for different groups to have a different scale parameter for the Weibull curve and different survival probabilities corresponding to the young ages not modelled using the Weibull function. was set equal to the maximum observed age in the sample, that is, the maximum observed difference between the times of first and last capture, which is 16 years.
Previous work on the specific data set has found strong evidence of a need to allow for more than one group of possums in terms of capture probability (Pledger et al. 2010), and we have therefore considered the cases of G = 1…4 for this paper while also introducing time dependence in p. Time and group membership are assumed to have an additive effect on capture probabilities.
The models with AIC values within 6 of the best model are presented in Table 1. Although boundary regularity conditions do not hold in the case of finite mixture models such as this one, the use of AIC as a model selection tool has gained support in the literature (Pledger et al. 2010; Cubaynes et al. 2012). There is clear evidence of heterogeneity in capture probability for the possum data set which agrees with the findings of Pledger & Efford (1998), Pledger et al. (2003) and Pledger et al. (2010) as no models with G = 1 appear in Table 1.
Table 1. List of models with the lowest AIC values together with number of parameters estimated (ν), AIC and Δ AIC values. Survival probability of the G groups is modelled using a Weibull curve or using Weibull curve and separate parameters for ages 1:m denoted by . Capture probability of the G groups is modelled as constant or time-dependent, c and t, respectively
t,G = 3
t,G = 3
t,G = 3
t,G = 3
t,G = 4
t,G = 4
t,G = 4
t,G = 3
t,G = 4
The model with the lowest AIC value suggests that the animals are homogeneous in terms of their lifetime survival patterns with separate parameters needed to model survival of ages 1–3. Additionally, it suggests heterogeneity, with G = 3, and time dependence in capture probabilities. The three competing models with similar AIC values differ only by the age after which the Weibull function is used to model the age-specific survival probabilities. All four models provide practically identical estimates for ages 1–10 and very similar estimates for older ages.
The estimated survival probabilities up to age 16 from the selected model are presented in Fig. 7. The confidence intervals are derived using 100 nonparametric bootstrap samples, obtained by sampling with replacement the individual capture histories of the marked possums. The shape parameter of the Weibull is estimated equal to 3·12 with a 95% bootstrap percentile confidence interval equal to (2·40, 4·00). The scale parameter is estimated equal to 9·11 with a corresponding 95% bootstrap percentile confidence interval equal to (7·95, 9·90). It can be seen that juvenile survival is considerably lower than adult survival up to about age 8. There is some juvenile migration in this population, so the apparently low juvenile survival is due in part to emigration (Efford 1998). As expected, there is more variability in the estimates for older ages. The estimates of age-dependent survival probabilities from our selected model follow closely those derived using possums of known age from the same population reported by Efford (2000).
The estimated β parameters are presented in Fig. 8. The estimates corresponding to times after the start of the study do not show any systematic differences and are randomly fluctuating in the 0·01, 0·08 interval. The estimates corresponding to times prior to the start of the study have an asymptote at zero because only animals surviving until the start of the study are included in N. Even though the maximum observed age was equal to 16, only cohorts born about 8–10 years before the start of the study actually contribute to the ‘super-population’. In fact, if is set to 10, then the results are indistinguishable from those derived when , suggesting that the model is robust with respect to the exact choice of provided the extension back in time covers the lifetime of most animals.
A goodness-of-fit test, where the observed frequencies of the unique capture histories are compared against the observed, is not possible in this case since 121 of the 187 unique capture histories are only observed once. As an informal assessment of the fit of the model, we simulated 100 data sets under the selected model, and for each data set, we counted the number of histories with difference between the times of last and first capture of the individuals equal to 0, 1, …, 10 or >10. The box plots in Fig. 9 present the results of these Monte Carlo simulations and demonstrate a satisfactory fit of the model since the observed frequencies are always within the range of the fitted frequencies and in most cases, within the interquartile range.
The methods presented in this article provide estimates of age-specific survival probabilities for individuals of unknown age while allowing for heterogeneity in survival and capture probabilities. Additionally, the unwarranted assumption that all individuals are born within the study time limits is avoided by back-tracking the possible birth times of the individuals to times prior to the start of the study. The advantages of these models are that they allow the study of senescence for wildlife populations when direct information on age is unavailable and at the same time correct for possible bias introduced by differences in the survival and capture probabilities between the individuals.
The duration of the study is crucial in deciding which of our models to employ. Previous models assuming the animal was born (first available for capture) at the sample when it was first caught are well known to have substantial bias in age-related parameters. If the study is long in relation to the typical life span of the animals, most of the capture histories will belong to animals born during the study (i.e. just before the first sample, or later). In this case, summing over possible birth times from sample 1 onwards (eqn 1) will remove most of the bias in age-related parameters. However, if the study is short in relation to the life span, and if no useful covariates are available for guessing the age, it is possible to remove more bias by also summing over possible birth times before the start of the study eqn 3. For this to be possible, we have assumed that entry and survival parameters are stable over time. This stability assumption may be given some support if information criteria select a model with entry and survival parameters constant over time (i.e. to constant after a high , and to constant). Even using average estimates and assuming constant entry and departure before the first sample is better than assuming all animals present at the first sample have just been born since estimation of age-dependent survival probabilities is robust to violation of the assumption of constant entry and departure, unless the differences are systematic, for example, the number of births is always increasing or decreasing.
When fitted to a long-term data set of brushtail possums, the model strongly supported age-dependent survival and heterogeneous capture probabilities which agrees with previous studies of this population. As explained in Pledger et al. (2010), this heterogeneity is spatially induced because the traps are always set in the same locations, making it more likely for possums with a home range including several traps to be caught.
Parametric curves for modelling survival are well established in the literature. In this case, a Weibull curve for ages greater than 3 was found to be sufficient to model survival in comparison with a fully age-dependent model which requires one parameter to be estimated for each age. Comparison of the two approaches was also recommended by Nussey et al. (2008) who suggested comparing different models of age dependence. Future analyses could also consider different parametric curves such as Gompertz or even nonparametric smoothers (Gimenez et al. 2006).
Our simulation results suggest that unmodelled heterogeneity in capture probabilities can result in biased estimates of age-specific survival probabilities, especially if the differences between the groups are substantial, highlighting the importance of the model of this paper.
Although this aspect has not been pursued in this article, these likelihood-based models can be combined with known-age data using joint likelihoods in order to maximise the use of available information.
We have essentially assumed that the population is closed to migration. Emigration is included in the mortality probabilities, but immigration causes a lowering of the capture probability estimates of younger animals, since they may have been outside the study area while young. If there is substantial juvenile dispersal, the data and model could be restricted to adult animals, which have more settled home ranges with ‘age’ being time since reaching maturity.
The first author would like to thank the following institutions/centres for their financial support as well as for providing a stimulating environment for conducting research: the Max Planck Institute for Demographic Research in Rostock, Germany, the National Centre for Statistical Ecology, the University of Kent in Canterbury, UK, and Victoria University of Wellington, New Zealand. We are grateful to Lloyd Pledger for helping us with R code and to Roger Pradel for his encouraging and perceptive comments.