Page 118 - Applied Probability

6. Applications of Identity Coefficients
102
6.4 Risk Ratios and Genetic Model Discrimination
The correlation patterns among relatives provide a simple yet powerful
means of discriminating between genetic models for a trait. Following Risch
[10], let us explore these patterns for a genetic disease characterized by two
states, normal and affected. To any person in a population there corre-
sponds an indicator random variable X such that X = 0 if the person is
normal and X = 1 if the person is affected. In this notation the prevalence
of the disease is K =Pr(X =1)= E(X).
The disease may have both genetic and environmental determinants. For
the sake of simplicity, we assume that the disease indicators X i and X j
of two relatives i and j are independent given their genotypes. We further
suppose that the prevalence of the disease does not vary with age and that
genetic equilibrium holds at the disease locus. These strong assumptions
are apt to be violated in practice, but they may hold approximately. For
instance, if selection is weak and mating is nearly random, then the as-
sumption of genetic equilibrium may not be too damaging. Furthermore,
if by a certain age every person definitely does or does not contract the
disease, then we can restrict our attention to people beyond this cutoff age.
Now consider two non-inbred relatives i and j of type R. Given that per-
son i is affected, it is often possible to estimate empirically the conditional
probability K R =Pr(X j =1 | X i = 1) that j is affected also. The joint
probability of both i and j being affected is

KK R =Pr(X i =1,X j = 1) (6.2)
=E(X i X j ).
For a single-locus model, the covariance decomposition for two relatives
gives

E(X i X j )=Cov(X i ,X j )+ K 2
2
2
2
=2Φ ij σ +∆ 7ij σ + K . (6.3)
a d
An important index for discriminating between genetic models is the risk
ratio λ R = K R /K for a relative of type R. λ R measures the increased risk
of disease for the relative of an affected person compared to the population
prevalence. It follows from equations (6.2) and (6.3) that
2σ 2 σ 2
a d
λ R − 1= Φ R 2 +∆ 7R 2 . (6.4)
K K
In equation (6.4), Φ and ∆ 7 are subscripted by the relative type R. Table 6.1
lists some relative types and their corresponding values of λ R −1. Note that
parent–offspring pairs are first-degree relatives; half-siblings, grandparent–
grandchild, and uncle–niece pairs are typical second-degree relatives; and
first cousins are typical third-degree relatives.