Page 113 - Applied Probability

6
Applications of Identity
Coefficients

6.1 Introduction

The current chapter discusses some applications of kinship and condensed
identity coefficients. We commence with the simplest problem of genetic
risk prediction involving just two relatives. This setting is artificial because
practical genetic counseling usually takes into account information on a
whole pedigree rather than information on just a single relative. We will
revisit the question of genetic counseling when we explore algorithms for
computing pedigree likelihoods.
Our applications of identity coefficients to the correlations between rel-
atives, to risk ratios for qualitative diseases, and to robust linkage analysis
are more relevant. Calculation of correlations between relatives forms the
foundation of classical biometrical analyses of quantitative traits such as
height, weight, and cholesterol level [3]. Due to the advent of molecular ge-
netics and positional cloning strategies and to the controversies surrounding
race and IQ, biometrical genetics has fallen out of fashion. Nonetheless, it
is still a useful tool for exploratory analysis of quantitative traits. If one is
mindful of its untestable assumptions and treats its results with caution,
then biometrical genetics can offer remarkable insights into the nature and
strength of genetic influences on quantitative traits.
Calculation of genetic risk ratios brings genetics into the mainstream
of epidemiological thinking on qualitative diseases. Although the models
employed to interpret risk ratios are simplistic, it is helpful to have simple
models for benchmarks. If these models are ruled out for a disease, then
geneticists should adopt robust methods for mapping genes predisposing
people to the disease. This chapter ends by explaining one such robust
technique for linkage analysis. Section 9.13 takes up this topic again and
offers better statistics.

6.2 Genotype Prediction

One application of condensed identity coefficients involves predicting the
genotype of person j based on the observed genotype of person i.At an
autosomal locus in Hardy-Weinberg equilibrium, suppose allele a k has pop-
ulation frequency p k . To obtain the genotypic distribution of j at this locus