MOTIVATION: The multifactor-dimensionality reduction (MDR) method has been widely used in multi-locus interaction analysis. It reduces dimensionality by partitioning the multi-locus genotypes into a high-risk group and a low-risk group according to whether the genotype-specific risk ratio exceeds a fixed threshold or not. Alternatively, one can maximize the chi(2) value exhaustively over all possible ways of partitioning the multi-locus genotypes into two groups, and we aim to show that this is computationally feasible. METHODS: We advocate finding the optimal MDR (OMDR) that would have resulted from an exhaustive search over all possible ways of partitioning the multi-locus genotypes into two groups. It is shown that this optimal MDR can be obtained efficiently using an ordered combinatorial partitioning (OCP) method, which differs from the existing MDR method in the use of a data-driven rather than fixed threshold. The generalized extreme value distribution (GEVD) theory is applied to find the optimal order of gene combination and assess statistical significance of interactions. RESULTS: The computational complexity of OCP strategy is linear in the number of multi-locus genotypes in contrast with an exponential order for the naive exhaustive search strategy. Simulation studies show that OMDR can be more powerful than MDR with substantial power gain possible when the partitioning of OMDR is different from that of MDR. The analysis results of a breast cancer dataset show that the use of GEVD accelerates the determination of interaction order and reduces the time cost for P-value calculation by more than 10-fold. AVAILABILITY: C++ program is available at http://home.ustc.edu.cn/~zhanghan/ocp/ocp.html

MOTIVATION: It has been claimed in the literature that pooling DNA samples is efficient in estimating haplotype frequencies. There is, however, no theoretical justification based on calculation of statistical efficiency. In fact, the limited evidence given so far is based on simulation studies with small numbers of loci. With rapid advance in technology, it is of interest to see if pooling is still efficient when the number of loci increases. METHODS: Instead of resorting to simulation studies, we make use of asymptotic statistical theory to perform exact calculation of the efficiency of pooling relative to no pooling in the estimation of haplotype frequencies. As an intermediate step, we use the log-linear formulation of the haplotype probabilities and derive the asymptotic variance-covariance matrix of the maximum likelihood estimators of the canonical parameters of the log-linear model. RESULTS: Based on our calculations under linkage equilibrium, pooling can suffer huge loss in efficiency relative to no pooling when there are more than three independent loci and the alleles are not rare. Pooling works better for rare alleles. In particular, if all the minor allele frequencies are 0.05, pooling maintains an advantage over no pooling until the number of independent loci reaches 6. High linkage disequilibrium effectively reduces the number of independent loci by ruling out certain haplotypes from occurring. Similar calculations of efficiency for the case of no pooling justify the common belief that it is not worthwhile to use molecular methods to resolve the phase ambiguity of individual genotype data. AVAILABILITY: The R codes for the calculation are available at http://www.stat.nus.edu.sg/∼staxj/pooling CONTACT: [email protected].

MOTIVATION: Pooling large number of DNA samples is a common practice in association study, especially for initial screening. However, the use of expectation-maximization (EM)-type algorithms in estimating haplotype distributions for even moderate pool sizes is hampered by the computational complexity involved. A novel constrained EM algorithm called PoooL has been proposed recently to bypass the difficulty via the use of asymptotic normality of the pooled allele frequencies. The resulting estimates are, however, not maximum likelihood estimates and hence not optimal. Furthermore, the assumption of Hardy-Weinberg equilibrium (HWE) made may not be realistic in practice. METHODS: Rather than carrying out constrained maximization as in PoooL, we revert to the usual EM algorithm but make it computationally feasible by using normal approximations. The resulting algorithm is much simpler to implement than PoooL because there is no need to invoke sophisticated iterative scaling methods as in PoooL. We also develop an estimating equation analogue of the EM algorithm for the case of Hardy-Weinberg disequilibrium (HWD) by conditioning on the haplotypes of both chromosomes of the same individual. Incorporated into the method is a way of estimating the inbreeding coefficient by relating it to overdispersion. RESULTS: Simulation study assuming HWE shows that our simplified implementation of the EM algorithm leads to estimates with substantially smaller SDs than PoooL estimates. Further simulations show that ignoring HWD will induce biases in the estimates. Our extended method with estimation of inbreeding coefficient incorporated is able to reduce the bias leading to estimates with substantially smaller mean square errors. We also present results to suggest that our method can cope with a certain degree of locus-specific inbreeding as well as additional overdispersion not caused by inbreeding. AVAILABILITY: http://staff.ustc.edu.cn/ approximately ynyang/aem-aes