Page 169 - Applied Probability

8. The Polygenic Model
154
8.9 The Hypergeometric Polygenic Model
Two elaborations of the polygenic model present substantial computational
difficulties. In the polygenic threshold model, a qualitative trait such
as the presence or absence of a birth defect is determined by an underly-
ing polygenically determined liability. If a person’s liability falls above a
fixed threshold, then the person possesses the trait [9, 10]; otherwise, he or
she does not. Likelihood evaluation under the polygenic threshold model
involves multivariate normal distribution functions and nasty numerical in-
tegrations [23]. In the mixed model, a quantitative trait is determined as
the sum of a polygenic contribution plus a major gene contribution [8, 27].
In this case, computational problems arise because the likelihood is a mix-
ture of numerous multivariate normal densities [28].
One strategy to overcome these computational barriers is to approximate
polygenic inheritance by segregation at a large, but finite, number of addi-
tive loci. In the finite polygenic model, the alleles at n symmetric loci
are termed polygenes and are categorized as positive or negative [11, 34].
Positive polygenes contribute +1 and negative polygenes −1 to a trait. If
positive and negative polygenes are equally frequent at each locus, then
the trait mean and variance for a random non-inbred person are 0 and 2n,
2
respectively. An arbitrary mean µ and variance σ for the trait X can be
σ
achieved by transforming X to √ X + µ. When the number of loci n is
2n
moderately large, X appropriately standardized is approximately normal.
Although the finite polygenic model is superficially attractive, it is defeated
n
by the 3 multilocus genotypes per person necessary to implement it.
If one is willing to allow nongenetic transmission, then the situation
can be salvaged by employing the hypergeometric polygenic model of
Cannings et al. [5]. In this model the 2n polygenes of a person exist in a
common pool that ignores separate loci. If we equate a person’s genotype
to the number of positive polygenes within it, then there are only 2n +1
n
possible genotypes. This is a major reduction from 3 . A gamete is gener-
ated in this model by randomly sampling without replacement n polygenes
from a parental pool of 2n polygenes. Thus, a person having i positive poly-
genes transmits a gamete having j positive polygenes with hypergeometric
probability
i 2n−i

j n−j
τ i→j = .
2n
n
Two independently generated gametes unite to form a child. To make this
hypergeometric polygenic model as similar as possible to the finite poly-
genic model, we finally postulate that all pedigree founders independently
share the binomial distribution
2n

2n 1
(8.12)
i 2