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This article demonstrates that the choice of link function in binomial regression should be motivated by assumptions about the underlying latent abundance process generating binary ecological data. The authors derive a new generalized complementary log-log (gcloglog) link function appropriate for overdispersed count data following a negative binomial distribution, which encompasses both the standard cloglog link (for Poisson processes) and logit link (for geometric distributions) as special cases. Simulation studies show that misspecifying the link function leads to bias and poor coverage when the degree of overdispersion in the latent count process is incorrectly assumed, while the gcloglog link provides robust inference when replicate data are available.
Why it matters
This work provides ecologists with a principled framework for selecting link functions in binomial regression based on biological assumptions about abundance processes, potentially improving inference accuracy in presence-absence studies, occupancy modeling, and other applications involving binary ecological data. The new gcloglog link function offers a flexible tool for sensitivity analysis when overdispersion is uncertain.
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⚠️ Preprint – Noch nicht peer-reviewed
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Ecologists commonly apply binomial regression to study binary data. Often, the link function in binomial regression is not purposefully chosen, but instead selected on the grounds of interpretation or computation efficiency. However, if the binary response is the result of a latent abundance process, then the link function implies assumptions for this underlying process. The most well-known example of this is the complementary log-log (cloglog) link, which has direct connections with the probability of a positive count under a latent Poisson distribution. If the binary response can be assumed to represent a truncated latent count, the cloglog link can be a better choice than the logit link. On the other hand, counts in ecology are often overdispersed, meaning the assumption of an underlying Poisson abundance process is unrealistic. In this article, we provide a framework for consideration of the link choice in binomial regression, in terms of the latent abundance process that binary responses can represent. Aside from the Poisson case mentioned above, we show that the cloglog link is the right choice if the latent abundance process represents the biomass of individuals, or if the latent abundance process comes from an overdispersed count distribution with a linear mean-variance relationship. For the case where the latent count follows a negative binomial distribution with a quadratic mean-variance relationship, we derive a new generalized complementary log-log (gcloglog) link function, which includes the case of the cloglog link as well as that of the logit link, where the latter implies assuming moderate overdispersion in the form of a geometric distribution. Two simulation studies demonstrate that cloglog and logit link binomial regression can exhibit bias and undercoverage when the latent count is more overdispersed than assumed, while gcloglog link models overcome this issue but in practice require replication in the binomial response for reliable estimation. Two real ecological datasets further illustrate how to use the gcloglog link binomial regression in practice including as a means of sensitivity analysis. We encourage ecologists to give more thought to the choice of link function in binomial regression, based on considering the latent abundance process and its properties.
Source: On Motivating the Choice of Link Function in Binomial Regression Through A Latent Abundance Process