Comparing methods for analyzing overdispersed binary data in aquatic toxicology.

Historically, death is the most commonly studied effect in aquatic toxicity tests. These tests typically employ a gradient of concentrations and exposure with more than one organism in a series of replicate chambers in each concentration. Whereas a binomial distribution commonly is employed for such effects, variability may exceed that predicted by binomial probability models. This additional variability could result from heterogeneity in the probabilities across the chambers in which the organisms are housed and subsequently exposed to concentrations of toxins. Incorrectly assuming a binomial distribution for the statistical analysis may lead to incorrect statistical inference. We consider the analysis of grouped binary data, here motivated by the study of survival. We use a computer simulation study to examine the impact of overdispersion or outliers on the analysis of binary data. We compare methods that assume binomial or generalizations that accommodate this potential overdispersion. These generalizations include adjusting the standard probit model for clustering/correlation or using alternative estimation methods, generalized estimating equations, or generalized linear mixed models (GLMM). When data were binomial or overdispersed binomial, none of the models exhibited any significant bias when estimating regression coefficients. When the data were truly binomial, the probit model controlled type I errors, as did the Donald and Donner method and the GLMM method. When data were overdispersed, the probit model no longer controlled type I error, and the standard errors were too small. In general, the Donald and Donner and the GLMM methods performed reasonably based on this study, although all procedures suffered some impact in the presence of potential outliers.