Determining Inheritance Distributions via Stochastic Penetrances

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Abstract
The aim of linkage analysis is to map the position of a gene contributing to an inheritable disease. The statistical model contains the disease allele frequency and penetrance parameters. Here I investigate the inheritance distribution, that is, the conditional distribution of the inheritance vector given phenotypes at the disease locus. Based on this, the likelihood and likelihood score function of Whittemore can be defined. As a result, a general semiparametric methodology of choosing score functions in linkage analysis is proposed. The proposed approach is valid for arbitrary pedigrees, and I treat quantitative, dichotomous (binary), and other phenotypes in a unified framework. The resulting score functions can be easily incorporated into existing software for multipoint linkage analysis. I use the fact that the inheritance distribution depend on unknown founder alleles. These are treated as hidden data and give rise to “stochastic penetrance factors.” Certain uncorrelated unit variance random variables that are functions of the founder alleles are introduced. I show that the moment-generating function and moments of these play crucial roles in choosing likelihoods and likelihood score functions. Lower-/higher-order moments are more important when the genetic effect is weak/strong, and this corresponds to simultaneous identical by descent (IBD) sharing of few/many individuals. For inbred pedigrees and nonadditive models, the likelihood score function is dominated by individuals homozygous by descent at the disease locus. For outbred pedigrees, the local score function involves pairwise IBD sharing. Relations to existing score functions of nonparametric linkage (Spairs, Sall, Srobdom) and quantitative trait loci (QTL) are highlighted.

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