A model for co-occurrent assortative mating and vertical cultural transmission and its impact on measures of genetic associations

Assortative mating for a given phenotype is the phenomenon by which mates select each other based on their phenotypic similarity. Other phenomena can create positive correlation between the parents and the offsprings environment: vertical cultural transmission, or dynastic effects. When these phenomena occur together, they induce a gene-environment correlation at the population scale. It will impact genetic measures of associations such as SNP effect size and SNP-heritability. In this paper, we provide a complete mathematical modelling of both assortative mating and vertical cultural transmission in the classical framework of the polygenic additive model. We establish for the first time the theoretical evolution and equilibrium values of all quantities of interest involved. We then derive its consequences on typical genetic epidemiology analyses; including both population and family-based study designs. We show the consequences on heritability estimation, Genome-Wide Association Studies (GWAS), and the variance explained by polygenic scores. We validate our calculations through simple forward-time simulations.