Analogues of the fundamental and secondary theorems of selection, assuming a log-normal distribution of expected fitness

It is increasingly common for studies of evolution in natural populations to infer the quantitative genetic basis of fitness (e.g., the additive genetic variance for relative fitness), and of relationships between traits and fitness (e.g., the additive genetic covariance of traits with relative fitness). There is a certain amount of tension between the theory that justifies estimating these quantities, and methodological considerations relevant to their empirical estimation. In particular, the additive genetic variances and covariances involving relative fitness are justified by the fundamental and secondary theorems of selection, which pertain to relative fitness on the scale that it is expressed. However, naturally-occurring fitness distributions lend themselves to analysis with generalized linear mixed models (GLMMs), which conduct analysis on a different scale, typically on the scale of the logarithm of expected values, from which fitness is expressed. This note presents relations between evolutionary change in traits, and the rate of adaptation in fitness, and log quantitative genetic parameters of fitness, potentially reducing the discord between theoretical and methodological considerations to the operationalization of the secondary and fundamental theorems of selection.