Resolving discrepancies between chimeric and multiplicative measures of higher-order epistasis

Epistasis - the interaction between alleles at different genetic loci - plays a fundamental role in biology. However, several recent approaches quantify epistasis using a chimeric formula that measures deviations from a multiplicative fitness model on an additive scale, thus mixing two scales. Here, we show that for pairwise interactions, the chimeric formula yields a different magnitude but the same sign of epistasis compared to the multiplicative formula that measures both fitness and deviations on a multiplicative scale. However, for higher-order interactions, we show that the chimeric formula can have both different magnitude and sign compared to the multiplicative formula. We resolve these inconsistencies by deriving mathematical relationships between the different epistasis formulae and different parametrizations of the multivariate Bernoulli distribution. We argue that the chimeric formula does not appropriately model interactions between the Bernoulli random variables. In simulations, we show that the chimeric formula is less accurate than the classical multiplicative/additive epistasis formulae and may falsely detect higher-order epistasis. Analyzing multi-gene knockouts in yeast, multi-way drug interactions in E. coli, and deep mutational scanning of several proteins, we find that approximately 10% to 60% of inferred higher-order interactions change sign using the multiplicative/additive formula compared to the chimeric formula.