Efficient Calculation of P‐value and Power for Quadratic Form Statistics in Multilocus Association Testing

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 take the general form D=X^TA X , where A is a general similarity matrix which may or may not be positive semi‐definite, and X follows the multivariate normal distribution with mean μ and variance matrix Σ, where Σ may or may not be singular. We show that D can be written as a linear combination of independent χ² random variables with a shift. Furthermore, its distribution can be approximated by a χ² or the difference of two χ² distributions. In the setting of association testing, our methods are especially useful in two situations. First, when the required significance level is much smaller than 0.05 such as in a genome scan, the estimation of p‐values using permutation procedures can be challenging. Second, when an EM algorithm is required to infer haplotype frequencies from un‐phased genotype data, the computation can be intensive for a permutation procedure. In either situation, an efficient and accurate estimation procedure would be useful. Our method can be applied to any quadratic form statistic and therefore should be of general interest.