Gain in efficiency from using generalized least squares in the Haseman-Elston test

Techniques that test for linkage between a marker and a trait locus based on the regression methods proposed by Haseman and Elston [1972] involve testing a null hypothesis of no linkage by examination of the regression coefficient. Modified Haseman-Elston methods accomplish this using ordinary least squares (OLS), weighted least squares (WLS), in which weights are reciprocals of estimated variances, and generalized estimating equations (GEE). Methods implementing the WLS and GEE currently use a diagonal covariance matrix, thus incorrectly treating the squared trait differences of two sib pairs within a family as uncorrelated. Correctly specifying the correlations between sib pairs in a family yields the best linear unbiased estimator of the regression coefficient [Scheffe, 1959]. This estimator will be referred to as the generalized least squares (GLS) estimator. We determined the null variance of the GLS estimator and the null variance of the WLS/OLS estimator. The correct null variance of the WLS/OLS estimate of the Haseman-Elston (H-E) regression coefficient may be either larger or smaller than the variance of the WLS/OLS estimate calculated assuming that the squared sib-pair differences are uncorrelated. For a fully informative marker locus, the gain in efficiency using GLS rather than WLS/OLS under the null hypothesis is approximately 11% in a large multifamily study with three siblings per family and 25% for families with four siblings each.