Selection of a proper model as a basis for statistical inference from capture-recapture data is critical. This is especially so when using open models in the analysis of multiple, interrelated data sets (e.g., males and females, with 2-3 age classes, over 3-5 areas and 10-15 yr). The most general model considered for such data sets might contain 1000 survival and recapture parameters. This paper presents numerical results on three information-theoretic methods for model selection when the data are overdispersed (i.e., a lack of independence so that extra-binomial variation occurs). Akaike's information criterion (AIC), a second-order adjustment to AIC for bias (AIC(c)), and a dimension-consistent criterion (CAIC) were modified using an empirical estimate of the average overdispersion, based on quasi-likelihood theory. Quality of model selection was evaluated based on the Euclidian distance between standardized <(theta)over cap> and theta (parameter theta is vector valued); this quantity (a type of residual sum of squares, hence denoted as RSS) is a combination of squared bias and variance. Five results seem to be of general interest for these product-multinomial models. First, when there was overdispersion the most direct estimator of the variance inflation factor was positively biased and the relative bias increased with the amount of overdispersion. Second, AIC and AIC(c), unadjusted for overdispersion using quasi-likelihood theory, performed poorly in selecting a model with a small RSS value when the data were overdispersed (i.e., overfitted models were selected when compared to the model with the minimum RRS value). Third, the information-theoretic criteria, adjusted for overdispersion, performed well, selected parismonious models, and had a good balance between under- and overfitting the data. Fourth, generally, the dimension-consistent criterion selected models with fewer parameters than the other criteria, had smaller RSS values, but clearly was in error by underfitting when compared with the model with the minimum RSS value. Fifth, even if the true model structure (but not the actual parameter values in the model) is known, that true model, when fitted to the data (by parameter estimation) is a relatively poor basis for statistical inference when that true model includes several, let alone many, estimated parameters that are not significantly different from O.
