A comparison of identity-by-descent and identity-by-state matrices that are used for genetic evaluation and estimation of variance components

The genetic covariance matrix conditional on pedigree is proportional to the pedigree-based additive relationship matrix (PARM), which is twice the matrix of identity-by-descent (IBD) probabilities. In genomic prediction, IBD probabilities in the PARM, which are expected genetic similarities between relatives that are derived from the pedigree, are substituted by realized similarities that are derived from genotypes to obtain a genomic additive relationship matrix (GARM). Different definitions of similarity lead to different GARMs, and two commonly used GARMS are the matrix G, which is based on an allele substitution effect model, and the matrix T, which is based on an allele effect model. We show that although the two matrices T and G are not proportional, they give identical predictions of differences between breeding values. When genomic information is used for variance component estimation, the GARM G_x is computed from genotype covariates that have been standardized to have unit variance. That approach is equivalent to fitting a random regression model using the same standardized covariates. We show that under Hardy-Weinberg and linkage equilibria (LE) that the genetic variance is kσγ2, where σγ2 is the variance of a randomly sampled element from the vector of k substitution effects. However, if linkage disequilibrium (LD) has been generated through selection, covariances between genotypes at different loci will be negative, and therefore, the additive genetic variance will be lower than kσγ2. When the GARM G_x is assumed to be proportional to the genetic covariance matrix, the parameter being estimated is kσγ2. We have demonstrated by simulation that kσγ2 overestimates the additive genetic variance when LD is generated by selection. We argue that unlike the PARM, GARMs are not proportional to a genetic covariance matrix conditional on the observed causal genotypes. The objective here is to recognize the difference between these covariance matrices and its implications.