  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.
