Page 161 - Applied Probability

8. The Polygenic Model
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matrix E(−d L) also simplifies drastically when Ω = I. This fact can be
used to good advantage in a quasi-Newton search of the loglikelihood [29].
Although quasi-Newton methods explicitly require neither the observed nor
the expected information matrix, a good initial approximation to the ob-
2
served information matrix is crucial. Starting a search at σ =(0,... , 0, 1)
and approximating the observed information matrix by the expected infor- t
mation matrix fits in well with the quasi-Newton strategy.
The basic model just presented can be generalized in many useful ways
[21]. First, missing observations can be handled by deleting the appropri-
ate rows (and columns) of the observation vector y, the design matrix A,
and the covariance matrix Ω of a pedigree. Second, theoretical means and
covariances that depend nonlinearly on the parameters can be accommo-
2
dated. To be precise, suppose A(µ) is the mean vector and Ω(σ ) is the
covariance matrix of a given pedigree. Then any appearance of A ∂ µ in
∂µ i
the scoring algorithm is replaced by ∂ A(µ), and any appearance of Γ i is
∂µ i
2
replaced by ∂ 2 Ω(σ ). Third, as we shall see later in this chapter, covari-
∂σ
i
ance models for multivariate traits can be devised.
8.3 Application to Gc Measured Genotype Data
Human group specific component (Gc) is a transport protein for vitamin
D. The Gc locus determines qualitative variation in the Gc protein. An
interesting question is whether the genotypes at the Gc locus also influence
quantitative differences in plasma concentrations of the Gc protein. Daiger
et al. [6] collected relevant data on 31 identical twin pairs, 13 fraternal twin
pairs, and 45 unrelated controls. Gc concentrations and Gc genotypes are
available on all individuals. The three genotypes 1/1, 1/2, and 2/2 at the
Gc locus are distinguishable.
A reasonable model for these data involves p = 5 mean components
and r = 3 variance components [4]. The covariates are Gc genotype, sex,
and age. To accommodate these covariates requires mean components µ 1/1 ,
µ 1/2 , and µ 2/2 for the three genotypes, a male offset µ male to distinguish
males from females, and a regression coefficient µ age on age. With these
components, the expected trait value for a 35-year-old female with Gc
genotype 1/2 is µ 1/2 +35µ age, for instance. For a 15-year-old male with
genotype 2/2, the expected trait value is µ 2/2 + µ male +15µ age.
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2
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Instead of choosing the variance components σ , σ , and σ parameter-
a
e
d
izing the additive genetic variance, the dominance genetic variance, and
the random environmental variance, we can proceed somewhat differently
2
for these data. Let σ tot be the total trait variance, ρ ident the correlation
between identical twins, and ρ frat the correlation between fraternal twins.
Mathematically the two sets of parameters give the same model. Biologi-
cally, the interpretation of the second set of parameters is less rigid. Note