Modeling binomial amphibian roadkill data in distance sampling while accounting for zero-inflation,serial correlation and varying cluster sizes simultaneously

Roadkill is of ecological importance so that there is increasing academic research to understand the causes and patterns of roadkills and their impact on ecosystems. This work is motivated by the study on roadkills of endangered Bufo calamita (B. calamita) (The natterjack toad) out of amphibian roadkills. The status of B. calamita is regarded as unfavorable due to large population declines. In the mentioned study, B. calamita and total amphibian roadkills were recorded via distance sampling on a National Road of Southern Portugal between March 1995 and March 1997. The traditional binomial modeling of these data are challenged by three issues. First, the zeros in B. calamita counts far exceeded its nominal level. Second, there is likely serial correlation among observations along the road. Finally, there is varying number of total amphibian roadkills at each sampling location; therefore, there is likely randomness in the number of total amphibian roadkills. All these features may contribute to overdispersion in the binomial observations. These three issues are routinely addressed one at a time separately, the first through zero-inflated binomial models, the second, for example, by means of random effects models for serially correlated binomial data and the third by models for binomial data with random cluster sizes. Therefore the data cannot be adequately modeled by any of these separate models. In this paper, we propose a new model to tackle these three issues simultaneously in the binomial analysis of B. calamita roadkills out of amphibian roadkills. Our approach is generally applicable to other binomial data with these three features.