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Denote the trace of a matrix as tr(��). Then ��= tr(W)/tr(W2) and df0= (tr(W))2/tr(W2), where tr(W) = tr(A��s) and tr(W2) = tr(A��sA��s). This method is referred to in this paper as the 2-cum approximation. The p-value is calculated under the null hypothesis H0: p=q. In this case, the true haplotype frequencies p and q are usually unknown, although the difference s=p?q is assumed to be zero. Therefore, both the 4-cum and 2-cum approximations can be used to find the p-value. We show only the results for the 4-cum approximation; the 2-cum approximation http://www.selleckchem.com/products/Adriamycin.html under the null hypothesis can be applied likewise. To find the corresponding ��1 and ��2 in the 4-cum approximation, we can use 0 to replace s and to replace ��s. Here is a consistent estimate of ��s with , and for i= 1,?��?, k. Note that the center parameter �� is always 0 under the null hypothesis. To prove this, it is sufficient to show s1��s2, which is equivalent to [tr((A��s)3)]2��[tr((A��s)2)][tr((A��s)4)], itself a direct conclusion from Yang et al. (2001). Then the p-value is estimated as (3) Power is usually calculated when p and q are known but not equal. In this case, the values of s=p?q and ��s=��p+��q= (P?ppT)/n+ (Q?qqT)/m are both known. Let d*�� be the critical value as defined in equation (4). The power to reject H0 at significance level �� is (5) We assume that the similarity matrix http://www.selleckchem.com/PD-1-PD-L1.html A is positive definite in formulas (1) to (5). However, in practice, A can be singular or have negative eigenvalues. If A is singular, that is, rank (A) = ra http://www.selleck.cn/products/MK-1775.html Therefore, even if A is singular, we can perform the above calculation to reduce its dimensionality and convert it into a non-singular matrix ��a. Then by replacing s with ��a, ��s with ��a, and A with ��a, all the above formulas can be applied as long as A does not have negative eigenvalues. We apply this method in the example of HapMap3 data, where the similarity matrices are often singular or nearly singular. If A is non-singular but has negative eigenvalues, equation (1) is still true although formulas (2) to (5) are not. In this case, we need to find the actual matrix W according to its definition. Next, we separate the eigenvalues of W into positive and negative groups. Assume that W has rp positive and rn negative eigenvalues, where rp+rn=r��. Without loss of generality, let and . Now define and . We get quadratic form , where A1 is positive definite. Therefore, its distribution can be approximated using formula (2). Similarly, define and . Then .