  Accurate knowledge of the null distribution of hypothesis tests is important
for valid application of the tests. In previous papers and software, the
asymptotic null distribution of likelihood ratio tests for detecting genetic
linkage in multivariate variance components models has been stated to be a
mixture of chi-square distributions with binomial mixing probabilities. Here we
show, by simulation and by theoretical arguments based on the geometry of the
parameter space, that all aspects of the previously stated asymptotic null
distribution are incorrect--both the binomial mixing probabilities and the
chi-square components. Correcting the null distribution gives more conservative
critical values than previously stated, yielding P values that can easily be
ten times larger. The true mixing probabilities give the highest probability to
the case where all variance parameters are estimated positive, and the mixing
components show severe departures from chi-square distributions. Thus, the
asymptotic null distribution has complex features that raise challenges for the
assessment of significance of multivariate linkage findings. We propose a
method to generate an asymptotic null distribution that is much faster than
other empirical methods such as gene-dropping, enabling us to obtain P values
with higher precision more efficiently.
