Graphical Modeling of the Joint Distribution of Alleles at Associated Loci

Pairwise linkage disequilibrium, haplotype blocks, and recombination hotspots provide only a partial description of the patterns of dependences and independences between the allelic states at proximal loci. On the gross scale, where recombination and spatial relationships dominate, the associations can be reasonably described in these terms. However, on the fine scale of current high-density maps, the mutation process is also important and creates associations between loci that are independent of the physical ordering and that can not be summarized with pairwise measures of association. Graphical modeling provides a standard statistical framework for characterizing precisely these sorts of complex stochastic data. Although graphical models are often used in situations in which assumptions lead naturally to specific models, it is less well known that estimation of graphical models is also a developed field. We show how decomposable graphical models can be fitted to dense genetic data. The objective function is the maximized log likelihood for the model penalized by a multiple of the model’s degrees of freedom. We also describe how this can be modified to incorporate prior information of locus position. Simulated annealing is used to find good solutions. Part of the appeal of this approach is that categorical phenotypes can be included in the same analysis and association with polymorphisms can be assessed jointly with the interlocus associations. We illustrate our method with genotypic data from 25 loci in the ELAC2 gene. The results contain third- and fourth-order locus interactions and show that, at this density of markers, linkage disequilibrium is not a simple function of physical distance. Graphical models provide more flexibility to express these features of the joint distribution of alleles than do monotonic functions connecting physical and genetic maps.