Page 167 - Applied Probability

8. The Polygenic Model
152
are hard to interpret. This suggests transforming the asymptotic variances
and covariances based on the expansion
min{i,j,f} min{i,j,f}
ˆ ˆ
ˆ ˆ
ˆ

Var(λ ij )=
k=1 Cov(δ ik δ jk , δ il δ jl ). (8.10)
l=1
The delta method of large sample theory uses the linearizations
ˆ ˆ ˆ ˆ
δ ik δ jk ≈ δ ik δ jk + δ ik (δ jk − δ jk )+ δ jk (δ ik − δ ik )
ˆ
ˆ
ˆ ˆ δ il δ jl + δ il (δ jl − δ jl )+ δ jl (δ il − δ il )
δ il δ jl ≈
to approximate
ˆ ˆ
ˆ ˆ
Cov(δ ik δ jk , δ il δ jl )
ˆ ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
≈ δ ik δ il ˆ σ jk,jl + δ ik δ jl ˆ σ jk,il + δ jk δ il ˆ σ ik,jl + δ jk δ jl ˆ σ ik,il , (8.11)
ˆ
ˆ
where ˆ σ jk,jl is the estimated covariance of δ jk and δ jl and so forth. Inverting
the observed information matrix produces the estimated covariances, which
can be substituted in the expansion (8.10) of the asymptotic variance.
8.8 A QTL Example
We now examine data submitted by the Collaborative Study on the Ge-
netics of Alcoholism (COGA) to the Eleventh Genetic Analysis Workshop.
COGA investigators at six American sites conducted a genome scan of al-
coholism and related risk factors on 105 pedigrees containing 1214 people.
The relevant risk factors in our case are platelet activity levels of the en-
zyme monoamine oxidase B (MAOB) and auditory and visual event related
potentials (ERPs). ERPs are complex brain waves indicative of cognitive
brain activity in response to certain stimuli such as light or sound. P300 is
one component of these waves that shows an amplitude reduction in recov-
ering alcoholics and relatives of alcoholics at risk for developing alcoholism.
As a followup to the positive linkage findings of various workshop par-
ticipants, [2, 18, 25, 31] we undertake here a trivariate analysis of MAOB
activity and two ERP measurements on the z area of the scalp, the Pz
and Cz leads of P300 amplitude. Two families have been excluded in this
analysis, one with MAOB levels more than 10 standard deviations from
the mean and one with questionable genotyping results. All three traits are
adjusted for sex and the Pz and Cz leads for age. Figure 8.1 depicts three
location score curves, the lower one with one QTL factor and the upper two
with two and three QTL factors, respectively. These curves are defined by
log 10 L d/L, where L d is the maximum likelihood of the multivariate normal
model with the QTL at position d, and L is the maximum likelihood of the
multivariate normal model omitting the QTL.