The Measurement Theory of Fitness: a Definition and its Implications for Epistasis

wants to thank the ESI for the financial and administrative support during his tenure at the institute. 2 ABSTRACT Measurement theory is a branch of applied mathematics that clarifies how quantitative concepts can be defined and what their mathematical properties are. In this contribution ideas from measurement theory are applied to the fitness concept. It is shown that fitness can be quantified by using and extending results from utility theory. Wrightian fitness can be derived as a so-called strict utility model. Furthermore the selection equation predicting the change in gene and genotype frequency can also be derived directly from the strict utility model. The only requirement for the applicability of the classical selection theory is the absence of genotype by genotype interactions. Fitness is measured on a ratio scale, i.e. is unique up to a multiplication by a constant. The latter property clarifies the notion of gene interaction, also known as epistasis: if a change in the genetic background only changes the fitness values of the alleles at a locus by a constant factor, the genetic background does not affect the selection dynamic at the focal locus. The reason is that multiplication by a constant factor is an admissible scale transformation and thus has no consequences for the biological meaning of these measures. Hence multiplicative fitness is non-epistatic for Wrightian fitness. It is concluded that measurement theory is a useful approach for clarifying the meaning of quantitative concepts also in biology.