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this family is divided into two independent parent-child pairs {(H i , H j ), (H i , H k )} and {(H i , H j ), (H i , H i )} as described above, the number of occurences of H i , H j , and H k is 3, 2, 1. Therefore, the approximation used in Ding et al. [1] for multiple children families will bias the frequency estimates of some haplotypes. Despite this approximation was empirically shown to be still more efficient than GENEHUNTER [table 5 in 1] in the multiple children case. We suggest here a theoretically more justified method to handle the multiple children case. Recently we proposed another EMbased approach to handle full-sib families with two or more children and missing parents, termed FSHAP [2] , which can handle families with one parent and multiple offspring as well. Our simulation study (table 1) demonstrates that the performance of the PCHAP is not significantly improved by increasing the number of children, while the performance of FSHAP leads to results that (a) improve significantly with an increasing number of children and (b) significantly outperform the results obtained with PCHAP in almost all situations. FSHAP can handle one parent families with multiple children only and PCHAP can handle one parent families with a single child, but both of them are similar in the likelihood function and implemented via the EM algorithm. Thus we combine PCHAP and FSHAP in one framework to handle families with one available parent and 1 to n children. For the new method, named PMCHAP, the likelihood function of the population haplotype frequencies is the one used in PCHAP and described in [1] for single-child families or the one used in FSHAP and described in [2] for multiple-children families. In the implementation of the EM algorithm, the probabilities of parent-child haplotype pairs and full-sib haplotype sets are calculated according to the expectation step described in PCHAP and FSHAP, respectively [1, 2] . In the M-step a weighted average Recently we proposed a method for haplotype inference in nuclear families with only one parent available [1] . Our approach, which we call PCHAP (parent-child haplotyping) here for convenience, is exact for families consisting of one parent and one child only, in which parent and child share one haplotype. While the original PCHAP approach was designed to handle a single child only, we suggested as an approximation to handle two children by splitting the one parent and two children in two parent child haplotype pairs treated as independent. Obviously the decomposition of families with multiple children makes no correct and effective use of the family information. This is illustrated by the following example: consider a parent with diplotype (H i , H j ) (i 0 j), the first child with diplotype (H i , H k ) (k 0 i, j), and the second child with diplotype (H i , H i ). Correctly, H i is the joint haplotype of all three individuals and the correct number of occurences of H i , H j , and H k is 2, 1, 1. If Published online: December 15, 2008

human heredity

Haplotype Reconstruction and Estimation of Haplotype Frequencies from Nuclear Families with One Parent Available and Varying Numbers of Children Using the Exact Likelihood