We noted earlier that heterogeneity was an issue in some animal studies. The approach we took earlier was to try to model heterogeneity using finite mixture models. However, if individual animal attributes like age, mass, or other variable can predict catchability, then presumably these attributes could be measured on animals at capture and included in the model as covariates. In RMark this can be accomplished for closed models using a special class of models called the Huggins conditional models. In these, N does not appear in the likelihood but instead is a derived parameter. The likelihood is 'conditional' in the sense that it depends only on animals caught at least once, so that (unlike Lincoln-Petersen or other full likelihood approaches) the number of unmarked animals never appears in the model. This is a sensible approach for covariate analysis, since by definition we have no information on covariate values for animals that are never captured. In addition to allowing for individual covariates, the Huggins conditional models have been extended to include finite mixture models , with HugHet and HugFullHet the Huggins counterparts to HetClosed and FullHet unconditional models.
I have provided a data object and code that reads in capture histories and individual covariates computes Huggins estimates under a covariate heterogeneity model, as well as the Huggins models corresponding to M0, Mt, and Mb. Save the data object to your working directory; the R script will load the object into R memory (hint: these saved you a lot of screwing around reformatting data for proper input. You can look at the data object in R just by typing its name. ) . Comparison of the models via AIC indicates strong preference for the covariate model. I then use this to produce model-averaged estimates of capture probability and plot theses estimates (and unconditional CI) against the covariate values.
Next: Review exercises