Using the Softplus Function to Construct Alternative Link Functions in Generalized Linear Models and Beyond

Response functions linking regression predictors to properties of the response distribution are fundamental components in many statistical models. However, the choice of these functions is typically based on the domain of the modeled quantities and is not further scrutinized. For example, the exponential response function is usually assumed for parameters restricted to be positive although it implies a multiplicative model which may not necessarily be desired. Consequently, applied researchers might easily face misleading results when relying on defaults without further investigation. As an alternative to the exponential response function, we propose the use of the softplus function to construct alternative link functions for parameters restricted to be positive. As a major advantage, we can construct differentiable link functions corresponding closely to the identity function for positive values of the regression predictor, which implies an quasi-additive model and thus allows for an additive interpretation of the estimated effects by practitioners. We demonstrate the applicability of the softplus response function using both simulations and real data. In four applications featuring count data regression and Bayesian distributional regression, we contrast our approach to the commonly used exponential response function.