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8. The Polygenic Model

1
1

1
(8.15)
Cov(X k ,X l ).
+
Var(X l )+
1 −
2
4
2n − 1
In the absence of inbreeding, Cov(X k ,X l ) = 0, and one can argue induc-
tively that Var(X i )=2n. Indeed, the induction hypothesis Var(X k )=2n
and Var(X l )=2n and the recurrence (8.15) imply that Var(X i )= 2n.
Summarizing the situation for a non-inbred pedigree, all means reduce
to E(X i ) = 0 and all variances to Var(X i )=2n. All covariances either
are 0 or are governed by the recurrence (8.13). Thus, insofar as first and
second moments are concerned, the hypergeometric polygenic model ex-
2
actly mimics the inheritance of a fully additive polygenic trait (σ =0)
d
with mean 0 and variance 2n. This resemblance and the empirical calcula-
tions carried out by Elston, Fernando, and Stricker [11, 34] for the mixed
model suggest that the hypergeometric polygenic model is a computation-
ally efficient substitute for the polygenic model. A pleasing aspect of this
substitution is that all computations can be performed via a version of the
Elston-Stewart algorithm featured in Chapter 7.
8.10 Application to Risk Prediction
For a simple numerical application to the polygenic threshold model, con-
sider the pedigree of Figure 8.2. In this pedigree, darkened individuals
are afflicted by a hypothetical disease with a prevalence of .01 and a
heritability of .75. We approximate the polygenic liability to disease
1
of person i in the pedigree by the sum Z i = σ a ( √ X i )+ σ e Y i , where
2n
X i is determined by the hypergeometric polygenic model with 2n poly-
2
genes; the Y i are independent, standard normal deviates; σ = .75; and
a
2
2
2
2
σ = .25. The ratio σ /(σ +σ ) is by definition the heritability of each Z i .
a
e
a
e
Given that each Z i follows an approximate standard normal distribution,
the liability threshold of 2.326 is determined by the prevalence condition
2
1 " ∞ −z /2
√ e dz = .01.
2π 2.326
The individuals represented by ♦ marks in Figure 8.2 are unborn, po-
tential children. Table 8.3 gives the conditional probabilities that these
children will be afflicted with the disease. The recurrence risks recorded
evidently stabilize at about 35 percent, 12 percent, and 6 percent as the
number of polygenes 2n →∞. Under the alternative hypothesis of an
autosomal dominant mode of disease, these risks are 1/2, 1/2, and 0, re-
spectively. In counseling families such as this one, where risks are strongly
model dependent, one should obviously exercise caution.