Abstract
A defining feature of capture-recapture is missing data due to imperfect detection of individuals. The standard approach used to deal with the missing data is to integrate (or sum) over all the possible unknown values. The missing data is completely removed and the resulting likelihood is in terms of the observed data. The problem with this approach is that often biologically unnatural parameters are chosen to make the integration (summation) tractable. A related consequence is that latent variables of interest, such as the population size and the number of births are only available as derived quantities. As they are not explicitly in the model they are not available to be used in the model as covariates to describe population dynamics. Therefore, models including density dependence are unable to be examined using standard methods.
Instead of explicitly integrating out missing data, we choose to include it using data augmentation. Instead of being removed, the missing data is now present in the likelihood as if it were actually observed. This means that we are able to specify models in terms of the data we would like to
have observed, instead of the data we actually did observe. Having the complete data allows us to separate the processes of demographic interest from the sampling process. The separation means that we can focus on specifying the model for the demographic processes without worrying about the sampling model. Therefore, we no longer need to choose parameters in order to simplify the removal of missing data, but we are free to naturally write the model in terms of parameters that are of demographic interest. A consequence of this is that we are able write complex models in terms of a series of simpler conditional likelihood components. We show an example of this where we fit a CJS model that has an individual-specific time-varying covariate as well as live re-sightings and dead recoveries.
Data augmentation is naturally hierarchical, with parameters that are specified as random effects treated as any other latent variable and included into the likelihood. These hierarchical random effects models make it possible to explore stochastic relationships both (i) between parameters in the model, and (ii) between parameters and any covariates that are available.
Including all of the missing data means that latent variables of interest, including the population size and the number of births, can now be included and used in the model. We present an example where we use the population size (i) to allow us to parameterize birth in terms of the per-capita birth rates, and (ii) as a covariate for both the per-capita birth rate and the survival probabilities in a density dependent relationship.