An Alternative to the Beta-Binomial Distribution with Application in Developmental Toxicology

The beta-binomial distribution is resulted when the probability of success per trial in the binomial distribution varies in successive trials and the mixing distribution is from the beta family. For experiments with binary outcomes, often it may happen that observations exhibit some extra binomial variation and occur in clusters. In such experiments the beta-binomial distribution can generally provide an adequate fit to the data. Here, we introduce an alternative when the mixing distribution is assumed to be from the log-Lindley family. The properties of this new model are explored and it is shown that similar to the beta-binomial distribution, the log-Lindley binomial distribution can also be applied in modeling clustered binary outcomes. An example with real experimental data from a developmental toxicity experiment is utilized to provide further illustration.