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||Missing data methods for linkage analysis of IBS and incomplete IBD from affected sib-pairs
||Buckman DW, Li ZH
||We derive linkage statistics for situations where the numberof marker alleles shared identical-by-descent (IBD) is incomplete,and the number of marker alleles shared identicalby-state (IBS) is known. The linkage statistics are based onthe assumption that the parental genotypes are missing atrandom (MAR). We first assume the marker IBD is unambiguousfor a sib-pair if the parental genotypes are available.Then we relax this assumption to assess the impact ofmarker ambiguity. The derivation of each statistic involves aTaylor series expansion of a log likelihood that is a functionof the recombination fraction and nuisance parameters andincorporates the missing data situation. The first derivativeof the log likelihood is zero under the null hypothesisof no recombination, so the Taylor series expansion of thelog likelihood reveals a linkage statistic that is proportional to the second derivative of the log likelihood evaluated at the null hypothesis value of the recombination fraction. We prove that the standardized linkage statistics have asymptotic normal distributions, and we provide required sample sizes and simulation results and consider the impact of parent availability and marker ambiguity.