Analytical results for directional and quadratic selection gradients for log-linear models of fitness functions

Log-linear models are widely used for assessing determinants of fitness in empirical studies, for example, in determining how reproductive output depends on trait values or environmental conditions. Similarly, theoretical works of fitness and natural selection employ log-linear models, often with a negative quadratic term, generating Gaussian fitness functions. However, in the specific application of regression-based analysis of natural selection, such models are rarely employed. Rather, OLS regression is the predominant means of assessing the form of natural selection. OLS regressions allow specific evolutionary quantitative parameters, selection gradients, to be estimated, and benefit from the fact that the associated statistical models are easily applied. We examine whether selection gradients can be directly expressed in terms of the coefficients of models using exponential fitness functions with linear or quadratic arguments. Such models can be easily fitted with generalised linear models (GLMs). The expressions we obtain coincide with those for Gaussian functions, but relax the major constraint that the (log) fitness function is concave (downwardly curved). Additionally these results lead to univariate and multivariate analyses of both linear and quadratic selection that potentially incorporate pragmatic and interpretable models of fitness functions, where the parameters can be related analytically to selection gradients, and that can be operationalised using widely-available statistical tools.