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identity matrix. The parameter σ is referred to as a variance compo-
nent. For this choice of Υ, environmental contributions are uncorrelated
among pedigree members. Note that environmental contributions include
trait measurement errors. To represent shared environments within a pedi-
gree, it is useful to define a household indicator matrix H =(h ij )with
entries 8. The Polygenic Model 2 e
1 i and j are in the same household

h ij =
0 otherwise.
A reasonable covariance model incorporating both household and random
2
2
effects is Υ = σ H + σ I, giving an overall covariance matrix Ω for Y of
h e
2
2
2
2
Ω= 2σ Φ+ σ ∆ 7 + σ H + σ I.
a d h e
This last representation suggests studying the general model
r
2
Ω= σ Γ k , (8.1)
k
k=1
2
where the variance components σ are nonnegative and the matrices Γ k are
k
known covariance matrices. Since measurement error will enter almost all
models, at least one of the Γ k should equal I. For convenience, we assume
Γ r = I.
8.2 Maximum Likelihood Estimation by Scoring
2
The mean components µ 1 ,...,µ p and the variance components σ ,...,σ 2
1 r
appear as parameters in the multivariate normal loglikelihood
m 1 1 t −1
L(γ)= − ln 2π − ln det Ω − (y − Aµ) Ω (y − Aµ) (8.2)
2 2 2
for the observed data Y = y [15, 17, 21, 23]. In equation (8.2), det Ω
2 t
2
denotes the determinant of Ω, and γ =(µ 1 ,...,µ p ,σ ,... ,σ ) denotes the
1
r
parameters collected into a column vector. Because Γ r = I, Ω is nonsingular
2
whenever σ > 0.
r
To implement the scoring algorithm for maximum likelihood estimation
of γ, we need the loglikelihood L(γ), score dL(γ), and expected informa-
tion J(γ) over all the pedigrees in a sample. Because these quantities add
for independent pedigrees, it suffices to consider a single pedigree. In de-
riving the score and expected information for a single pedigree, we could
use the general results presented in Chapter 3 for exponential families of
distributions. It is more illuminating to proceed directly after reviewing the
following facts from linear algebra and calculus: