Quasi-regression for heritability

We show in an idealized model that the narrow sense (linear) heritability from d autosomal SNPs can be estimated without bias from n independently sampled individuals, by a method of moments strategy based on quasi-regression. The variance of this estimator is below C1/n+C2d /n for constants Cj <∞, uniformly in n > 2 and d > 1. In particular d n is allowed, and heritability can be consistently estimated in some limits where the effects of individual SNPs cannot be consistently estimated due to non-sparsity. Furthermore, when the C2d /n term dominates, then doubling the sample size reduces the variance by almost eight–fold. The idealization is that the model assumes complete linkage equilibrium, although it does allow for an arbitrary pattern of regression coefficients. In particular the coefficients need not be sparse or Gaussianly distributed, nor independent of the minor allele frequency. The method and the rate of convergence extend to additive heritability. For a full quadratic model encompassing pairwise interactions with d∗ = 2d + d(d− 1)/2 coefficients the error variance is O(d/n).