# Determining Inheritance Distributions via Stochastic Penetrances

## Ola Hössjer

### Centre for Mathematical Sciences Mathematical Statistics Lund Institute of Technology, Lund University, 2001

##### ISSN 1403-9338
Abstract:
The conditional distribution of the inheritance vector given phenotypes is investigated for an inheritable disease at the disease locus under assumed perfect marker information. We use the fact that the inheritance probabilities depend on the unknown stochastic founder alleles and the penetrance factors. The inheritance distribution can be described as the moment generating function of an array of uncorrelated unit variance random variables $\xi$ (depending only on the disease allele probability), evaluated at an array of 'allele sharing statistics' $B$ (which also depends on the penetrance factors and disease allele probabilities). By allowing the penetrance to depend on an unknown parameter, we show that higher order moments and factors of $\xi$ and $B$ respectively are dominating when the genetic effect is strong, corresponding to simultaneous IBD sharing of many individuals. In contrast, lower order moments of $\xi$ and factors of $B$ are crucial for weak genetic effects, corresponding to pairwise IBD-sharing. We treat quantitative and dichotomous (binary) phenotypes in a unified framework and give explicit expressions for the local likelihood score function of Whittermore (1996). For inbred pedigrees, the local score function is dominated by individuals that are HBD at the disease locus, whereas for outbred pedigrees, the score function involves pairwise IBD sharing. Relations to existing score functions of nonparametric linkage ($S_{\scr{pairs}}$, $S_{\scr{all}}$ and $S_{\scr{rob dom}}$) and QTL (score functions based on multivariate normality) are highlighted. In the latter case, we show that the multivariate normal assumption is not needed for defining the local likelihood score function. Further, we give explicit expressions for the inheritance distribution in a survival analysis example.
Key words:
Founder alleles, IBD-sharing, inheritance distribution, linkage analysis, local score functions, stochastic penetrances.