Determining Inheritance Distributions via Stochastic Penetrances
Ola Hössjer
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2001
ISSN 14039338

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 IBDsharing. 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, IBDsharing, inheritance distribution, linkage analysis,
local score functions, stochastic penetrances.
