# Missing heritability: new statistical and algorithmic approaches

Mathematically, heritability is defined by considering a function $$F$$ mapping a set of (Boolean) variables, $$(x_1,.., x_n)$$ representing genotypes, and additional environmental or ’noise’ variables $$\epsilon$$, to a single (real or discrete) variable $$z$$, representing phenotype. We use the variance decomposition of $$F$$, separating the linear term, corresponding to additive (narrow-sense) heritability, and higher-order terms, representing genetic-interactions (epistasis), to explore several explanations for the ‘missing heritability’ mystery. We show that genetic interactions can significantly bias upwards current population-based heritability estimators, creating a false impression of ‘missing heritability’. We offer a solution to this problem by providing a novel consistent estimator based on unrelated individuals. We also use the Wright-Fisher process from population genetics theory to develop and apply a novel power correction method for inferring the relative contributions of rare and common variants to heritability. Finally, we propose a novel algorithm for estimating the different variance components (beyond additive) of heritability from GWAS data.