When replicate count data are overdispersed, it is common practice to incorporate this extra-Poisson variability by including latent parameters at the observation level. For example, the negative binomial and Poisson-lognormal (PLN) models are obtained by using gamma and lognormal latent parameters, respectively. Several recent publications have employed the deviance information criterion (DIC) to choose between these two models, with the deviance defined using the Poisson likelihood that is obtained from conditioning on these latent parameters. The results herein show that this use of DIC is inappropriate. Instead, DIC was seen to perform well if calculated using likelihood that was marginalized at the group level by integrating out the observation-level latent parameters. This group-level marginalization is explicit in the case of the negative binomial, but requires numerical integration for the PLN model. Similarly, DIC performed well to judge whether zero inflation was required when calculated using the group-marginalized form of the zero-inflated likelihood. In the context of comparing multilevel hierarchical models, the top-level DIC was obtained using likelihood that was further marginalized by additional integration over the group-level latent parameters, and the marginal densities of the models were calculated for the purpose of providing Bayes' factors. The computational viability and interpretability of these different measures is considered.
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