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Accurate knowledge of the null distribution of hypothesis tests is importantfor valid application of the tests. In previous papers and software, theasymptotic null distribution of likelihood ratio tests for detecting geneticlinkage in multivariate variance components models has been stated to be amixture of chi-square distributions with binomial mixing probabilities. Here weshow, by simulation and by theoretical arguments based on the geometry of theparameter space, that all aspects of the previously stated asymptotic nulldistribution are incorrect--both the binomial mixing probabilities and thechi-square components. Correcting the null distribution gives more conservativecritical values than previously stated, yielding P values that can easily beten times larger. The true mixing probabilities give the highest probability tothe case where all variance parameters are estimated positive, and the mixingcomponents show severe departures from chi-square distributions. Thus, theasymptotic null distribution has complex features that raise challenges for theassessment of significance of multivariate linkage findings. We propose amethod to generate an asymptotic null distribution that is much faster thanother empirical methods such as gene-dropping, enabling us to obtain P valueswith higher precision more efficiently.

Reconsidering the asymptotic null distribution of likelihood ratio tests for genetic linkage in multivariate variance components models