arXiv2009
We address the asymptotic and approximate distributions of a large class of test statistics with quadratic forms used in association studies. The statistics of interest do not necessarily follow a chi-square distribution and take the general form $D=X^T A X$, where $X$ follows the multivariate normal distribution, and $A$ is a general similarity matrix which may or may not be positive semi-definite. We show that $D$ can be written as a linear combination of independent chi-square random variables, whose distribution can be approximated by a chi-square or the difference of two chi-square distributions. In the setting of association testing, our methods are especially useful in two situations. First, for a genome screen, the required significance level is much smaller than 0.05 due to multiple comparisons, and estimation of p-values using permutation procedures is particularly challenging. An efficient and accurate estimation procedure would therefore be useful. Second, in a candidate gene study based on haplotypes when phase is unknown a computationally expensive method-the EM algorithm-is usually required to infer haplotype frequencies. Because the EM algorithm is needed for each permutation, this results in a substantial computational burden, which can be eliminated with our mathematical solution. We assess the practical utility of our method using extensive simulation studies based on two example statistics and apply it to find the sample size needed for a typical candidate gene association study when phase information is not available. Our method can be applied to any quadratic form statistic and therefore should be of general interest.