Page 156 - Applied Probability

8
The Polygenic Model
8.1 Introduction

The standard polygenic model of biometrical genetics can be motivated
by considering a quantitative trait determined by a large number of loci
acting independently and additively [12]. In a pedigree of m people, let X k
i
k
be the contribution of locus k to person i. The trait value X i = k X for
i
t
person i forms part of a vector X =(X 1 ,... ,X m ) of trait values for the
pedigree. If the effects of the various loci are comparable, then the central
limit theorem implies that X follows an approximate multivariate normal
distribution. (See the references [19, 21] and Appendix B.) Furthermore,
k k
independence of the various loci implies Cov(X i ,X j )= Cov(X ,X ).
k i j
From our covariance decomposition for two non-inbred relatives at a single
locus, it follows that
2
2
Cov(X i ,X j )=2Φ ij σ +∆ 7ij σ ,
a
d
2
2
where σ and σ are the additive and dominance genetic variances summed
a d
over all participating loci. These covariances can be expressed collectively
2
2
in matrix notation as Var(X)= 2σ Φ+ σ ∆ 7 . Again it is convenient to
a d
assume that X has mean E(X)= 0. Although it is an article of faith that
the assumptions necessary for the central limit theorem actually hold for
any given trait, one can check multivariate normality empirically.
Environmental effects can be incorporated in this simple model by sup-
posing that the observed trait value for person i is the sum Y i = X i + Z i
of a genetic contribution X i and an environmental contribution Z i .Ifwe
t
assume that the random vector Z =(Z 1 ,...,Z m ) is uncorrelated with X
and follows a multivariate normal distribution with mean vector ν and co-
t
variance matrix Υ, then the trait vector Y =(Y 1 ,... ,Y m ) is multivariate
2
2
normal with mean E(Y )= ν and covariance Var(Y )= 2σ Φ+ σ ∆ 7 +Υ.
a
d
Different levels of environmental sophistication can be incorporated by
appropriately choosing ν and Υ. Typically ν is defined to be a linear func-
tion ν = Aµ of a parameter vector µ of p mean components. The m × p
design matrix A specifies the measured covariates determining this lin-
t
ear function. For instance, µ might be (µ f ,µ m ,α) , where µ f is the female
population mean at birth, µ m is the male population mean at birth, and
α is a regression coefficient on age. A row of A is then either (1, 0, age)
or (0, 1, age), depending on whether the corresponding person is female or
male.
2
Among the simplest possibilities for Υ is Υ = σ I, where I is the m × m
e