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Linear mixed models (LMMs) are widely used for heritability estimation ingenome-wide association studies (GWAS). In standard approaches to heritabilityestimation with LMMs, a genetic relationship matrix (GRM) must be specified. InGWAS, the GRM is frequently a correlation matrix estimated from the studypopulation's genotypes, which corresponds to a normalized Euclidean distancekernel. In this paper, we show that reliance on the Euclidean distance kernelcontributes to several unresolved modeling inconsistencies in heritabilityestimation for GWAS. These inconsistencies can cause biased heritabilityestimates in the presence of linkage disequilibrium (LD), depending on thedistribution of causal variants. We show that these biases can be resolved (atleast at the modeling level) if one adopts a Mahalanobis distance-based GRM forLMM analysis. Additionally, we propose a new definition of partitionedheritability -- the heritability attributable to a subset of genes or singlenucleotide polymorphisms (SNPs) -- using the Mahalanobis GRM, and show that itinherits many of the nice consistency properties identified in our originalanalysis. Partitioned heritability is a relatively new area for GWAS analysis,where inconsistency issues related to LD have previously been known to beespecially pernicious.

The Mahalanobis kernel for heritability estimation in genome-wide association studies: fixed-effects and random-effects methods