Incorporating Auxiliary Data

Capture-recapture studies can benefit significantly from incorporating auxiliary data into the model structure. As noted in Section 2.2.1.1, the use of covariates enables a more parsimonious approach to be taken in preference to conducting entirely separate analy- ses for cohorts of animals that are assumed to exhibit homogeneity with regard to the parameters of interest. The parsimonious approach may also increases the precision of all parameter estimates as all of the data is used in the estimation process. Information on animal level phenotypic covariates or environmental variables may, when incorporated into the capture-recapture analysis, provide a much improved prediction of the capture and survival rates compared to the standard CJS model analysis. A second reason is that the investigator may wish to explore a variety of biological hypotheses about the relationships between the parameters being estimated and the covariates.

Most studies that incorporate auxiliary information (Pollock, 2002) use covariates that can commonly be classified as either individual animal level covariates or group or envi- ronmental covariates. The group or environmental covariates will often be assumed to be dynamic variables which vary between sampling occasions but remain constant over the animals in the group (e.g., environmental indexes such as the North Atlantic Oscillation). Individual animal level covariates are, by contrast, usually assumed to be static variables

which are constant over time (e.g., birth weight). In much of the existing literature on capture-recapture studies that incorporate auxiliary information the analyses are condi- tional on the observed values of the covariates; typically no distribution of the covariate is specified. If the auxiliary information consists of a static variable then the modelling is relatively simple. Since CJS models condition on marked animals no unmarked animals are included in the model. Therefore, for static variables that can be recorded at any point in time there will be no missing covariate values and the analysis can be relatively straightforward.

For continuous and stochastic covariates the analysis becomes more problematic. For variables of this type some essentially arbitrary discretisation can be imposed to create a series of states and the transitions between them can be modelled using the multistate approach. For example, if it was believed that an animal’s weight may influence its prob- abilities of survival and capture then this effect could be modelled by defining a sequence of discrete weight classes and specifying a model for the transition probabilities between them. These multistate models have the advantage that they can include time-specific covariates which vary on an individual level. Also, there is no distribution specified that constrains the covariate. The parameters (capture, survival and state transition rates) for animals in each individual state can vary independently from the parameters in any of the other states. However, this multistate approach of discretising continuous variables (e.g. Nichols et al. (1992)) can lead to a loss of information that may conceal any re- lationship, present in the data, between the variable and capture or survival rates. The large number of parameters required for the multiple states may mean it is not the most parsimonious model. Equally problems may arise with the fit of the chosen model. The choice of discretisation can be effectively arbitrary and the assumption of homogeneity within states may be violated if the continuous auxiliary variable is classified into too few

discrete states resulting in decreased precision for the parameter estimates. Conversely, if a large amount of grouping is imposed then the data may not contain enough information for all the parameters in the model, thus rendering some of them unidentifiable. As noted in Bonner and Schwarz (2006), another downside to this approach is that multistate mod- els implicitly assume that all animals in a single state behave identically but will behave differently if they belong to different states. Therefore, animals with observed covariate values either side of a state classification boundary will be assumed to behave differently when their covariate values may in fact be very similar.

An alternative approach (Lebretonet al., 1992) is to use a GLM-like modelling struc- ture which allows the capture and survival rates to depend on linear combinations of variables (or covariates) through a “link” function of specified form. Lebreton et al.

(1992) advocate using a scaled logistic link function as it constrains the estimated prob- abilities to lie in the plausible range of 0 to 1. The advantage to their approach over the multistate models is that it utilises the flexibility of GLMs allowing both discrete and continuous covariates to be incorporated in the linear predictor. An assumption implicit in this approach is that the relationships between the parameters of interest and the aux- iliary information can be described using these relatively simple link functions. Typically, the form of the relationship determined by the link function is assumed to be constant over time which may not always be realistic. Also, as noted in Bonner and Schwarz (2006) the common use of the logistic function imposes the constraint that the relationship between the parameter being estimated and an explanatory covariate is monotonic. They suggest exploring the use of piecewise functions such as polynomials or splines to obtain more flexible relationships. One of the main issues with the GLM-like modelling approach is that whilst it can be extremely effective for static covariates problems arise for dynamic time-dependent ones when an animal is not captured. For static covariates that can be

observed at any time the value remains the same, and therefore known, for the duration of the study regardless of whether the animal is captured again after the initial release or not. For dynamic covariates that evolve deterministically, such as age, the missing values of the covariate can be interpolated.

For dynamic covariates that evolve stochastically on an individual basis, any sampling occasion during which the individual is not captured will result in a missed value of the covariate. Equally, for static variables that can only be recorded at a specific time (e.g. birth weight) the issue of missing covariates makes analysis more problematic. A full model analysis requires some method of modelling the distribution of the missing values taken by the covariate over the duration of the study (Pollock, 2002; Bonner and Schwarz, 2006). Pollock (2002) suggests that, in addition to the discretising continuous variables approach of Nicholset al.(1992) in which the weights of meadow voles were classified into discrete weight states, an alternative would be to develop a full likelihood approach by integrating out unobserved covariates. As with the GLM-like approach (Lebreton et al., 1992) this method requires the distribution of the covariate to be specified with regard to how it evolves over time. Alternatively Pollock (2002) suggests conducting an anal- ysis conditional on the covariates in the context of a missing data problem. Under this scenario the EM algorithm (Dempsteret al., 1977) could be used to model the change in covariates over time.

Pollock (2002) also advocates Bayesian methods for the full likelihood approach and as a way of dealing with missing covariate values. The model likelihood can be evaluated using numerical integration to integrate out the missing values or Markov Chain Monte Carlo (MCMC) methods can be used to impute the missing values which can be treated as variables to be estimated in a Bayesian framework. A summary of Bayesian approaches

to the analysis of Capture-Recapture studies is given in Section 2.5. The approach of Bonner and Schwarz (2006) utilises a Bayesian methodology to provide a solution to the problem of modelling capture and survival rates using continuous stochastic covariates that vary with both time and individual.

The use of auxiliary information in capture-recapture models presents some issues regarding goodness-of-fit tests (White, 2002). If models are not fitting well then it is rec- ommended (Pollock, 2002) to adjust the AIC and the variance of the parameter estimates to correct for possible over-dispersion in the data. A correction parameter, c, is required but White (2002) notes that

no general, robust, procedures are currently available for estimating c.

White (2002) also warns against placing too much faith in the model-based relationships between auxiliary variables and estimated parameters unless the auxiliary variable is in- cluded as a component of a “manipulative investigation”. Without this the relationship can only be described as correlative.