Two rules for the detection and quantification of epistasis and other interaction effects

Traditionally statistical interactions and epistasis are defined with respect to the ANOVA model of additive effects; that is, epistasis is defined as a deviation from the additive mode of combining main effects of gene substitutions. Furthermore, the definition is relative to a particular scale and epistasis can potentially be eliminated by a non-linear transformation of the underlying phenotype variables. The latter fact raises questions of the scientific validity of the concept if interaction, given its presumed arbitrariness. Here I am arguing that the arbitrariness in the definition and detection of epistasis, and any other interaction, can be eliminated if we observe measurement theoretical constraints on the treatment of quantitative data. I propose two principles for determining the appropriate reference model for the detection of epistasis. The first is the principle of effect propagation stating that the scale type of the effect measure determines the reference model for defining epistasis. For instance, if effects are measured as differences, then the reference model has to be additive. If the reference effects are fold differences, then the reference model has to be multiplicative. A mathematical justification for this rule is provided. The second principle is called irrelevant effects and derives from the principle of meaningfulness in measurement theory. In short, the rule says that the reference model is determined by the allowable scale transformations of the variables measured. The justification for this rule is that any mathematical model in which these variables figure have to be invariant to allowable scale transformations. These two rules can effectively eliminate the arbitrariness in the definition, detection, and quantification of epistasis or any other interaction effect.