Challenging the commonly used log-link in statistical models for count data with an application to infectious disease data

A response function is an essential part of any generalized linear model, but its choice
is rarely questioned. In particular, if the modeled expected value is restricted to be
greater than zero, the choice often falls on the exponential function. Even for a response
variable, for which the exponential function corresponds to the canonical link, there is
no indication that this is the true response function in general. Therefore, we propose to
take the softplus function as response function into consideration. The softplus function,
which is technically used in the context of neural networks, enables the modeling of the
conditional mean in an additive way and therefore ensures a linear interpretation of the
regression coefficients while respecting the positivity boundary of the conditional mean
at the same time. The central research question to be discussed in this study is: Does
the softplus activating function represent an adequate substitute of the commonly used
log-link with an application to infectious diseases? In the first step, a simulation study
gives insight into the robustness of the estimated coefficients under various circumstances.
Furthermore, the framework for the analysis of multivariate infection disease data yield
by Held et al. (2005) is self-implemented via the open source software R. By doing so,
the softplus function is introduced to the model class applied. The estimation results
from Held et al. (2005) are reproduced and compared to those concerning the softplus
link function with respect to the predictive quality. One-step-ahead-predictions build the
basis for mean-squared prediction errors and coverage frequencies of the upper prediction
limits. The results have been obtained using general optimisation routines via maximum
likelihood estimation.