Reliable age estimation is important for management of many fisheries (Haddon 2011) and wildlife populations (Lyons et al. 2012). Models used to manage populations often rely on animals being assigned to age classes (Caswell 2001; Williams et al. 2002). In addition, age estimates are important for advocacy when attempting to conserve long-lived species such as chelonians. That is, evidence of long life spans may help convince people of chelonians' sensitivity to impacts of increased mortality of adults and older juveniles, leading to legal protection and attitude changes that will reduce such mortality.

Ages of animals can be known precisely if they can be marked at birth, but data are rarely collected over the lifetimes of chelonians (Medica et al. 2012). Growth rings on scutes have been used to age chelonians in numerous studies, but evaluations of the technique have often shown it to be a weak indicator of age, particular in mature animals (Wilson et al. 2003). Skeletochronology can potentially provide reliable ageing in chelonians (Curtin et al. 2008), but can only be performed on dead animals.

An alternative is to infer age from size based on fitted growth models. Such models can be fitted to measurements taken from known-age animals, but can also be fitted to successive measurements from recaptured animals of unknown age (Fabens 1965). Statistical problems associated with fitting models to recapture data (Sainsbury 1980) have been overcome by estimating individual variation in growth parameters, most recently using Bayesian hierarchical modeling (Zhang et al. 2009). Individual variation is particularly relevant to age estimation, because individuals of the same age can vary greatly in size. Hierarchical modeling allows this variation to be accounted for when estimating age based on size data.

Eaton and Link (2011) used a Bayesian hierarchical version of the von Bertalanffy (VB) growth model to infer age distributions for the central African dwarf crocodile (Osteolaemus tetraspis) as a function of length. Their modeling approach combined inferences from recapture data for unknown-age animals with data from known-age animals, and generated posterior age distributions that accounted for individual variation in growth parameters as well as uncertainty in parameter estimation. Here we use a similar approach to obtain age distributions as a function of size from a long-term data set for North American snapping turtles (Chelydra serpentina), but we extend the approach in 2 ways.

First, we use a modification of the VB model that is biphasic as well as incorporating individual variation in growth parameters. For animals with indeterminate growth, reductions in growth are often expected at sexual maturity due to the demands of reproduction, which means different functions are needed to describe pre- and postmaturity growth (Day and Taylor 1997). Age estimates from uniphasic growth models may therefore be biased, particularly in long-lived ectotherms with late maturity. That is, ages will be overestimated if based largely on data from mature animals, and underestimated if based mainly on data from juveniles. However, methods have been developed for fitting Bayesian hierarchical growth models that allow a change in growth rate at a critical age (Quince et al. 2008; Alós et al. 2010a, 2010b) or critical size (Armstrong and Brooks 2013).

Second, we combine our growth model with prior information on the expected age distribution. Although prior distributions must be specified in Bayesian inference, “uninformative priors” are typically used in the absence of information (Link and Barker 2010). It is questionable, however, whether any prior distribution for age structure could be truly uninformative. Eaton and Link (2011) used a uniform distribution as an uninformative prior for dwarf crocodile age, meaning crocodiles were assumed to have equal probability of being any age between zero and the maximum specified. This is an unrealistic assumption for most populations, and would be typically result in ages being overestimated to some extent. We nominated informative prior distributions for age based on independent survival estimates from our population, and assessed the sensitivity of age inference to the prior for turtles of different sizes.

METHODS

Species and Study Area

North American snapping turtles are omnivorous predators and scavengers that live in lakes, ponds, and slow-moving rivers (Steyermark et al. 2008). Our data were collected in the Wildlife Research Area (lat 45°35′N, long 78°30′W) of Algonquin Provincial Park, near the northern edge of the species' range. Female snapping turtles in this area predictably start egg-laying when their straight-line carapace length reaches 24 cm (Armstrong and Brooks 2013). Mature females lay 1 clutch annually, with these clutches buried in sandy soil or gravel near water (Congdon et al. 2008). Brooks et al. (1997) tested the reliability of growth ring counts in Algonquin Park snapping turtles by analyzing changes in counts between recaptures. They found the counts to be an inaccurate indicator of the number of years between captures for juveniles, and to be completely uninformative for mature turtles.

Data Set

The data set consisted of 1996 straight-line carapace length measurements taken from 317 individually marked turtles from 1972 to 2005. Nesting females were usually found by patrolling known nest areas daily from late May to early July, and were caught by hand once their clutches were buried. This patrolling generally resulted in annual recaptures of mature females. Mature males and juveniles were captured less regularly via trapping or opportunistic encounters. Mature turtles can be sexed based on the length of the precloacal tail section (from posterior end of plastron to cloaca) in relation to the length of the posterior lobe of the plastron (Ernst 2008), but juveniles cannot be sexed externally. Most (288) of the turtles in the data set were of known sex but unknown age, but there were smaller samples of individuals of known sex and age (5), unknown sex but known age (19), and unknown sex and age (5). Turtles of unknown age were only included if they were measured at least twice. See Armstrong and Brooks (2013) for further details.

Growth Model

Armstrong and Brooks (2013) fitted a baseline hierarchical VB model to the above data set, then used the deviance information criterion to compare its predictive value to that of alternative models. The baseline model was similar to the VB model fitted by Eaton and Link (2011) in that it incorporated random individual variation as well as sex differences in parameters, and integrated inferences from first measurements of known-aged animals with those from recapture data. Minor differences include the use of a VB function for first measurements as well as recaptures (Eaton and Link [2011] used a linear function for the former, as the VB function was expected to approximate linearity up to the first measurements), the substitution of k_i/a_i for k_i in the VB function (see Armstrong and Brooks [2013] for rationale), and the modeling of individual variation and random error as normal rather than gamma processes. Comparison to alternative models gave strong evidence that individual variation in parameters should be retained, and that sex-specific biphasic growth should be incorporated (Armstrong and Brooks 2013).

Under the best model, which received unambiguous support, the average growth of males and females is similar until they reach 24 cm, after which females change trajectory toward a smaller asymptotic length. For males the model takes the form

for first measurements, and

for recaptures, where L_ij is the expected length of individual i at measurement j, a_i is the individual's asymptotic length, k_i determines its initial growth rate, t_j is its age at measurement j, t₀ is the theoretical age at which its length would be zero, y is the year of measurement, and ε_ij is random error. Individual variation in parameters a_i and k_i was taken to be normally and log-normally distributed, respectively, with means μ_a and μ_k and variances σ²_a and σ²_k, and ε_ij was taken to be normally distributed with mean 0 and variance σ²_e.

For females > 24 cm the model takes the form

for first measurements, and

for recaptures, where β_a is the change in the asymptotic size parameter, and t′ and y′ are age and time at which 24 cm was reached. The values of t′_i and y′_i are estimated as part of the modeling, with their expected values given by

and

Consequently, in an individual Markov chain Monte Carlo (MCMC) iteration, a recaptured female's length at measurement j could be predicted by Equation 2, Equation 4, or a combination of the 2 (see WinBUGS code in Appendix 1). The model can be used to generate probability distributions for carapace length as a function of age and sex (Armstrong and Brooks 2013:fig. 2).

Age Estimation

From Bayes' theorem (Link and Barker 2010), the probability of an individual being a particular age (z) based on its length is

where Pr(L_ij | t_z) is the likelihood (the probability of the individual being that length at age z based on the model), and Pr(t_z) is the prior probability of being that age. The discrete form of the theorem is appropriate for our scenario because turtles are measured at about the same time of year, so their ages fall into whole numbers of years. Because the model generates individual distributions for parameters a_i and k_i, it is possible to generate individual posterior age distributions that exploit the information gained from repeated measurements. Alternatively, posterior distributions can be generated for hypothetical individuals of any length by sampling a_i and k_i from distributions incorporating the range of individual variation. Eaton and Link (2011) only attempted the latter, as their data set contained only 5 observations of animals captured more than twice. In contrast, most of the 317 turtles in our data set were captured more than twice, and 155 were captured at least 5 times. We therefore used the model above to estimate the age at initial capture for all unknown-age turtles in our data set as well as ages of hypothetical males and females of various lengths.

Ages are estimated by inserting unknown ages in Equations 1 and 3, and modeling these unknown ages as missing values (Link and Barker 2010). However, the “cut” function in WinBUGS (Spiegelhalter et al. 2007) is applied to all parameters in the relevant lines of code (see supplementary material online at http://dx.doi.org/10.2744/CCB-1055.1.s.1) to prevent the parameter estimation from being influenced. Bayes' theorem is implemented by assigning a prior distribution to each unknown age. It is reasonable to assume that survival probability is constant with respect to size and age among mature snapping turtles (Brooks et al. 1988), as turtles appear to have negligible senescence (Girondot and Garcia 1998; Guerin 2004). Consequently, the age distribution among mature turtles is expected to approximate the negative binomial distribution

where S is the annual survival probability. This probability was estimated to be 0.966 for large (> 24-cm) snapping turtles in Algonquin Park (Galbraith and Brooks 1987), whereas annual survival of smaller turtles was estimated to be 0.754 (Brooks et al. 1988).

We therefore generated separate age estimates using negative binomial priors based on these 2 survival probabilities, and compared these to assess the sensitivity to the choice of prior. Although it would be desirable to construct an overall age distribution based on both survival probabilities, this would require using our growth model to infer age, meaning the prior would not be based on independent data. We compared the estimates generated using the 2 negative binomial priors with those generated using the uniform prior

which was expected to give unrealistically high age estimates for at least the larger animals.

We added the lines of code generating age estimates to the WinBUGS code used to fit the growth model (Appendix 1), and ran the whole model simultaneously to allow covariance in parameter estimation to be accounted for in age estimation. We used uninformative priors for all parameters and hyperparameters in the growth model (Armstrong and Brooks 2013), and generated posterior distributions from 500,000 MCMC iterations after a burn-in of 10,000 iterations.

RESULTS

Asymptotic carapace length was estimated to have a median value of 38.2 cm in males, and to fall between 35.6 and 40.8 cm in 95% of individuals. For females the estimated median was 30.9, with 95% of individuals falling between 28.3 and 33.5 cm. The largest individuals recorded were 40.7 cm and 35.7 cm for males and females, respectively.

If a negative binomial prior with 0.966 annual survival is used (NB[1,0.034]), as is most realistic for large turtles, the estimated age of a hypothetical 38-cm male is 73 yrs (95% prediction interval [PI]  =  27–173) and the estimated age of a hypothetical 31-cm female is 65 yrs (95% PI  =  23–162) (Fig. 1). These rise to 299 yrs (95% PI  =  76–490) and 288 yrs (95% PI  =  63–489) when the uniform prior is used (U[0,500]), and fall to 19 (95% PI  =  8–37) and 17 yrs (95% PI  =  7–34) when a negative binomial prior based on the juvenile survival rate is used (NB[1,0.246]). A hypothetical 24-cm turtle (the size when females begin egg-laying) of either sex has an estimated age of 25 yrs (95% PI  =  8–70) when NB(1,0.034) is used as the prior. This rises to 36 yrs (95% PI  =  9–126) when U(0,500) is used, and falls to 11 yrs (95% PI  =  6–24) when NB(1,0.246) is used. A hypothetical 10-cm turtle has an estimated age of 6 yrs (95% PI  =  2–14) under the most realistic prior of NB(1,0.246). This rises to 10 yrs (95% PI  =  3–28) using NB(1,0.034) and 11 yrs (95% PI  =  3–37) using U(0,500).

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Posterior age distributions for real turtles (Fig. 2) were quite different from those for hypothetical turtles, with the degree of difference depending on the prior used and the size of the turtle. The credible intervals for ages of large turtles corresponded fairly closely to the prediction intervals for hypothetical turtles when the uniform prior was used, but were high in relation to the prediction intervals for hypothetical turtles when negative binomial priors were used. For example, when NB(1,0.034) was used as the prior, the estimates for the 6 males near 38 cm (37–39) averaged 172 yrs, compared with the median 73 yrs for a hypothetical 38-cm male, and the estimates for the 32 females near 31 cm (30–32) averaged 127 yrs, compared with the median 65 yrs for a hypothetical 31-cm female. The credible intervals for ages of small turtles fit the predicted age distributions for hypothetical turtles when NB(1,0.256) was used as the prior, but were lower and much tighter than the distributions for hypothetical turtles when the other priors were used.

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The choice of prior had less influence on age estimates for real turtles than those for hypothetical turtles (Fig. 3), as is to be expected given that the former exploit additional data on individual-specific growth parameters. The influence of the prior on individual age estimates was greatest for large turtles, and had negligible effect for turtles first measured when < 24 cm (Fig. 3). Among large turtles, the influence of the prior was highest for turtles estimated to be atypically old for their size (Fig. 2). The influence of the prior was unrelated to the number of captures, but depended on the interval between the first and last capture (i.e., it had greater influence if this interval was short).

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Using age estimates for individual turtles, it is possible to infer individual growth histories in relation to age (Fig. 4). However, such histories are hypotheses that are highly sensitive to estimation uncertainty, particular among the larger turtles.

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DISCUSSION

Our results illustrate that it is possible to combine a hierarchical biphasic growth model with informative priors to obtain posterior age distributions as a function of size, either for hypothetical turtles measured once or for recaptured turtles with 2 or more measurements. However, the results also illustrate the inherent uncertainty involved in attempting to infer age from size, mainly due to individual variation in growth. Uncertainty in parameter estimation and model selection will also be important in many cases. However, the large sample sizes for our case study meant that the standard errors for mean parameter values were quite small in relation to the estimated individual variation, and the choice of model was also unambiguous (Armstrong and Brooks 2013).

Our results support the belief of many turtle biologists (e.g., Carr and Goodman 1970; Congdon et al. 2001) and other herpetologists (Halliday and Verrell 1988) that size is weakly related to age. Congdon et al. (2001) state that the relationship between size and age may be strong in juveniles, weaker in adults, and become weakest or absent in the oldest individuals. This trend will occur in any species where animals approach an asymptotic size that varies among individuals, as the effect of the individual variation will progressively overwhelm the effect of age as animals get larger. Our results suggest there is great uncertainty in age estimates even for small turtles if they are only captured once. For example, although the median age prediction for a 10-cm juvenile is more than twice that of a 5-cm juvenile, there is huge overlap in the prediction intervals, and these intervals even overlap with those for mature (> 24-cm) individuals. However, there was generally much tighter estimation for juveniles that were recaptured. The initial ages of these animals could be estimated to within 1–3 yrs as long as they had grown at least 3 cm between the first and last capture.

Our results also show the importance of choosing appropriate priors and assessing sensitivity to the choice of prior. Eaton and Link (2011) compared uniform priors with different maxima, and concluded that the lower limits of prediction intervals were robust to the choice of prior. Although we also found lower limits to be less affected than medians or upper limits, even the lower limits differed between the uniform and negative binomial priors, and between the 2 different negative binomial priors reflecting adult vs. juvenile survival rates. We did find that posterior distributions generated using the 2 negative binomial priors were identical for recaptured turtles first measured when < 24 cm. We therefore suggest that the prior based on the adult survival rate [NB(1,034)] is appropriate for all recaptured turtles. The appropriate prior for age estimates of hypothetical turtles is less straightforward because the posterior distribution is sensitive to the prior at all sizes. However, reasonable inferences can probably be made by focusing on NB(1,034) for large turtles and NB(1,0.256) for small turtles, and the combination of the 2 for intermediate sizes. The ideal prior would be the expected overall age distribution based on age-specific survival and fecundity estimates (Schwarz and Runge 2009), but because ages of most turtles are unknown in our study population, it is impossible to obtain these estimates independently of the growth model.

Although size-based age estimation is particularly difficult for large turtles, this is where the information can have most impact, both because it is impossible to age old turtles from growth rings due to wear and because age estimates for old animals are valuable for advocacy. The North American snapping turtle is a good example of a species needing such advocacy, as their populations are subject to road mortality, harvesting, and persecution (Brooks et al. 1988; Congdon et al. 1994; Committee on the Status of Endangered Wildlife in Canada 2008). In Ontario, despite being classified as a species at risk in the province and nationally, snapping turtles are still subject to a harvest that is almost certainly unsustainable. Information on the long life spans and slow growth of these animals is critical for highlighting their long-term vulnerability to impacts, and the posterior distributions reported in this article have already been used for this purpose (Armstrong 2009).

The prediction intervals for hypothetical turtles with NB(1,0.034) as prior probably give a realistic guide to the ages of large turtles. That is, the largest male and female snapping turtles in our population were probably 30–190 yrs old at first capture, with a best estimate of about 80 yrs. Ten of the recaptured turtles had median age estimates > 190 yrs, but the credible intervals all ranged < 190 except for 1 extremely large (35.7-cm) female that showed no detectable growth over 14 yrs. An underlying assumption of these estimates is that the individual variation in growth detected during the study can be extrapolated back throughout the turtles' lives. This assumption may be questionable when the age estimates are many times greater than the duration of the study. Even 190 yrs may seem an implausibly long life, but it is important to keep in mind that turtles are particularly long-lived animals (Shine and Iverson 1995), and that the study population was at the northern edge of its range where the animals are only active for about 5 mo of the year.

Future attempts to estimate age from size must continue to account for individual variation in growth parameters and, for chelonians and other long-lived ectotherms, should also account for biphasic growth and use appropriate priors for expected age distributions. As noted by Eaton and Link (2011), Bayesian hierarchical modeling frameworks facilitate flexible model structures allowing for individual variation, but also accommodate multi-model inference and assimilation of independent data sets into a single framework. Therefore, an obvious step for advancing size-based aging is to assimilate the data with those from other techniques such as growth ring counts. Given that there are many different methods for aging animals and most of them are problematic, the ideal approach will be to incorporate all relevant data for a species into a unified framework that fully accounts for the error associated with each technique.