On Motivating the Choice of Link Function in Binomial Regression Through A Latent Abundance Process
- 1University of Bayreuth
Abstract
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.
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BibTeX
@misc{veen2026motivating,
title = {{On Motivating the Choice of Link Function in Binomial Regression Through A Latent Abundance Process}},
author = {B. van der Veen and F. K. C. Hui},
howpublished = {bioRxiv},
year = {2026},
month = oct,
doi = {10.64898/2026.10.16.759589},
url = {https://www.biorxiv.org/content/10.64898/2026.10.16.759589v1},
note = {CC BY},
}
RIS
TY - UNPB
TI - On Motivating the Choice of Link Function in Binomial Regression Through A Latent Abundance Process
AU - B. van der Veen
AU - F. K. C. Hui
PY - 2026
DA - 2026/10/16
PB - bioRxiv
DO - 10.64898/2026.10.16.759589
UR - https://www.biorxiv.org/content/10.64898/2026.10.16.759589v1
AB - 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.
ER -