Random-effects models are a popular tool for analysing total narrow-sense heritability for simple quantitative phenotypes on the basis of large-scale SNP data. Recently, there have been disputes over the validity of conclusions that may be drawn from such analysis. We derive some of the fundamental statistical properties of heritability estimates arising from these models, showing that the bias will generally be small. We show that that the score function may be manipulated into a form that facilitates intelligible interpretations of the results. We use this score function to explore the behavior of the model when certain key assumptions of the model are not satisfied -- shared environment, measurement error, and genetic effects that are confined to a small subset of sites -- as well as to elucidate the meaning of negative heritability estimates that may arise. The variance and bias depend crucially on the variance of certain functionals of the singular values of the genotype matrix. A useful baseline is the singular value distribution associated with genotypes that are completely independent --- that is, with no linkage and no relatedness --- for a given number of individuals and sites. We calculate the corresponding variance and bias for this setting.
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