Application of approximate variances of variance components and their ratios in genetic tests

The application and underlying assumptions of formulae used to estimate the variance of variance components and ratios of variance components are fully described for (1) variance components estimated using Henderson's Method 3 (HM3) and Restricted Maximum Likelihood (REML) and (2) ratios of variance components commonly used in genetic tests - biased and unbiased heritabilities. A first-order Taylor series approximation is often used to estimate the variance of a ratio of two random variables (e.g., heritability), however the formula is complicated, thus making calculations prone to errors. Dickerson's approximation is considerably simpler, though relatively rarely used. In case studies using data from 148 slash pine full-sib progeny tests, Dickerson's method was found to be slightly more conservative than the Taylor series approximation when estimating the variance of heritability estimates, regardless of test size, age, or the trait (volume, which is a continuous trait, and rust resistance, which is a bernoulli trait). Both the Taylor series and Dickerson approximations compared favorably with an empirical estimate of the variance of heritability estimates, however there is some evidence of small-sample bias associated with the use of the asymptotic variance-covariances from REML variance component estimation.