Conditional likelihood score functions for mixed models in linkage analysis.

Abstract

In this paper, we develop a general strategy for linkage analysis, applicable for arbitrary pedigree structures and genetic models with one major gene, polygenes and shared environmental effects. Extending work of Whittemore (1996), McPeek (1999) and Hossjer (2003d), the efficient score statistic is computed from a conditional likelihood of marker data given phenotypes. The resulting semiparametric linkage analysis is very similar to nonparametric linkage based on affected individuals. The efficient score S depends not only on identical-by-descent sharing and phenotypes, but also on a few parameters chosen by the user. We focus on (1) weak penetrance models, where the major gene has a small effect and (2) rare disease models, where the major gene has a possibly strong effect but the disease causing allele is rare. We illustrate our results for a large class of genetic models with a multivariate Gaussian liability. This class incorporates one major gene, polygenes and shared environmental effects in the liability, and allows e.g. binary, Gaussian, Poisson distributed and life-length phenotypes. A detailed simulation study is conducted for Gaussian phenotypes. The performance of the two optimal score functions S(wpairs) and S(normdom) are investigated. The conclusion is that (i) inclusion of polygenic effects into the score function increases overall performance for a wide range of genetic models and (ii) score functions based on the rare disease assumption are slightly more powerful.

Full text

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Biostatistics (2005), 6,2, pp. 313–332
doi:10.1093/biostatistics/kxi012
Conditional likelihood score functions for mixed models
in linkage analysis
OLA HÖSSJER
Department of Mathematics, Stockholm University, S-106 91 Stockholm, Sweden
ola@math.su.se
SUMMARY
In this paper, we develop a general strategy for linkage analysis, applicable for arbitrary pedigree struc-
tures and genetic models with one major gene, polygenes and shared environmental effects. Extending
work of Whittemore (1996), McPeek (1999) and Hössjer (2003d), the efficient score statistic is computed
from a conditional likelihood of marker data given phenotypes. The resulting semiparametric linkage
analysis is very similar to nonparametric linkage based on affected individuals. The efficient score S de-
pends not only on identical-by-descent sharing and phenotypes, but also on a few parameters chosen by
the user. We focus on (1) weak penetrance models, where the major gene has a small effect and (2) rare
disease models, where the major gene has a possibly strong effect but the disease causing allele is rare.
We illustrate our results for a large class of genetic models with a multivariate Gaussian liability. This
class incorporates one major gene, polygenes and shared environmental effects in the liability, and allows
e.g. binary, Gaussian, Poisson distributed and life-length phenotypes. A detailed simulation study is con-
ducted for Gaussian phenotypes. The performance of the two optimal score functions Swpairsand Snormdom
are investigated. The conclusion is that (i) inclusion of polygenic effects into the score function increases
overall performance for a wide range of genetic models and (ii) score functions based on the rare disease
assumption are slightly more powerful.
Keywords: Conditional likelihood; Founder alleles; Linkage analysis; Mixed model; Score functions.
1. INTRODUCTION
The goal of linkage analysis is to find the position (locus) along the genome of a gene which causes or
increases the risk of development of a certain inheritable disease. Disease related quantities, so called
phenotypes, and DNA samples, are collected for a number of families with aggregation of the disease.
The DNA samples are typed at a large number of genetic markers, distributed throughout the genome.
Using information from the markers, loci are sought at which segregation of DNA is correlated with
the inheritance pattern of the disease. Since each individual’s phenotype is a blurred observation of the
two copies of the disease gene (disease alleles) that he/she carries, DNA transmission is correlated with
phenotype segregation in close vicinity of the disease locus.
If genetic model parameters (disease allele frequencies and penetrance parameters) are known, the lod
score of Morton (1955) can be computed at each locus. For complex diseases, the genetic model is rarely
known and alternative procedures have been proposed. One possibility is to regard the genetic model
parameters as nuisance parameters and optimize the lod score with respect to them at each locus. This is
the mod score approach of Risch (1984) and Clerget-Darpoux et al. (1986).
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314O. HÖSSJER
For binary traits, nonparametric linkage (NPL) is another method developed in situations when the
genetic model parameters are unknown. The affected-pedigree-member (APM) method is the most
commonly used version of NPL. A score function S is used which does not require specification of a
genetic model. Instead, it quantifies the extent to which the APMs share their alleles identical-by-descent
(IBD) from the same founder alleles, see e.g. Penrose (1935), Weeks and Lange (1988), Fimmers et al.
(1989), Whittemore and Halpern (1994) and Kruglyak et al. (1996).
The NPL method can be extended to more general phenotypes by first noting that the lod score
is equivalent to a conditional likelihood L of DNA marker data (MD) given phenotypes (Whittemore,
1996). By differentiating log L with respect to genetic model parameters a score function S is obtained.
Whittemore (1996) showed that allele sharing score functions S used in NPL can be used to construct an
appropriate L. The opposite route is to start from L, based on a biologically based genetic model with
disease allele frequencies and penetrance parameters, and then to compute S. McPeek (1999) showed, for
binary traits, arbitrary pedigree structures and affected-only phenotypes, that several allele sharing score
functions S could be derived in this way. McPeek’s work was generalized in Hössjer (2003d) to arbi-
trary monogenic models, e.g. Gaussian, Cox proportional hazards and logistic regression models. The
resulting score functions S depend on IBD allele sharing in the pedigree, the observed phenotypes and
a small number of parameters which have to be specified by the user. This approach was referred to as
semiparametric linkage (SPL) in Hössjer (2003d).
The purpose of this paper is to extend the work of McPeek (1999) and Hössjer (2003d) to mixed gen-
etic models with one major susceptibility locus and polygenes/shared environmental effects. In Section
2, we define basic genetic concepts. In Section 3, we provide a general formulation of the efficient score
function approach. In Section 4, we define a broad class of genetic models which includes several existing
models based on Gaussian, binary, life-length or Poisson phenotypes as special cases. In Section 5, we
derive the efficient score functions S for weak penetrance models, where the major gene has a small effect
on the disease, and rare disease models, where the disease allele is rare. A simulation study in Section
6 for the Gaussian mixed model reveals that more powerful and robust procedures are obtained when
polygenic effects are included in the score function S. Some conclusions and further recommendations
are given in Section 7. In Section A of supplementary data available at Biostatistics online, henceforth
denoted HS, we show that the SPL approach is asymptotically equivalent to mod scores under the null
hypothesis of no linkage. In Section B of HS, we derive an orthogonal decomposition of functions of
several genotypes. This is used in Section 5.1 to derive weak penetrance optimal score functions, but we
believe it to be of independent interest. For instance, the classical variance components decomposition of
genetic variance (Fisher, 1918; Kempthorne, 1955) can be obtained from these expansions. Finally, the
proofs are collected in Section C of HS.
2. BASIC GENETIC CONCEPTS
Consider a pedigree with n individuals of which f are founders (without ancestors in the pedigree) and
n − f nonfounders. Assume that two forms of the disease gene exist—the normal allele (0) and the
disease allele (1). Each individual has a pair of alleles (genotype), of which one is inherited from the
father and one from the mother. The genotypes of all pedigree members can be collected into a vector
G = (G1,...,Gn) = (a1,...,a2n), where Gk= (a2k−1,a2k) is the genotype of the kth individual, with
a2k−1and a2kthe paternally and maternally transmitted alleles, respectively. We will use the convention
that founders are numbered k = 1,..., f . Since the gene of interest is unknown, we do not observe
G, but rather a vector of disease phenotypes Y = (Y1,...,Yn). Here Ykis the phenotype of the kth
individual, a quantity related to the disease. This could be binary (affected/unaffected) or a quantitative
variable such as insulin concentration, body mass index, etc. Some Yk may also represent unknown
phenotypes.
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Alleles are transmitted from parents to children according to Mendel’s law of segregation. At a certain
locus t, allele transmission can be summarized through the inheritance vector, introduced by Donnelly
(1983). It is defined as v(t) = (v1(t),...,vm(t)), where m = 2(n − f ) is the number of meioses and
vk(t) equals 0 or 1 depending on whether a grandpaternal or grandmaternal allele was transmitted during
the kth meiosis. A priori, grandpaternal and grandmaternal alleles are equally likely to be transmitted
during each meiosis, so that
P(v(t) = w) = 2−m
for all binary vectors w of length m.
If τ is the disease locus, the phenotype vector Y will be correlated to v(τ) if the genetic component is
strong enough. It is standard in linkage analysis to look at the conditional distribution
(2.1)
w −→ P(v(τ) = w|Y),
(2.2)
which differs from the prior (2.1). Moreover, (2.2) is unaffected by the way we sampled the pedigree
(ascertainment). The stronger the genetic component at τ is, the stronger is the discrepancy between
P(v(τ)|Y) and the uniform distribution, because of co-inheritance of phenotypes and DNA at τ. Even at
loci around τ, the conditional distribution of the inheritance vector given phenotypes differs from (2.1),
although the amount of co-inheritance decays with the genetic distance from the disease locus. This
is because of occurrence of so called crossovers—random points along the chromosome where, during
meioses, segregation switches between grandmaternal and grandpaternal transmission.
In practice we do not observe the process v(·), but MD from all or a subset of the pedigree members
give incomplete information of it. As we will see in the next section, up to a multiplicative constant
the conditional probability Pt,θ(MD|Y) serves as an approximation of (2.2). Here, θ is the set of ge-
netic model parameters (disease allele frequencies and penetrance parameters) and t corresponds to the
hypothesis t = τ.
3. CONDITIONAL LIKELIHOOD AND SCORE FUNCTIONS
Consider a chromosome of length tmaxcentimorgans (cM) with at most one disease locus τ located along
[0,tmax]. The hypothesis testing problem is
H0: τ is located on another chromosome
H1: τ ∈ [0,tmax].
The unconditional likelihood requires knowledge of MD, Y and the sampling scheme. Since the latter
is often (more or less) unknown, it is common in linkage analysis to consider the conditional likelihood
of MD given Y. The reason is that nuisance parameters involved in the ascertainment scheme are not
included in the conditional likelihood, see e.g. Ewens and Shute (1986). If v = v(τ) is the inheritance
vector at the disease locus, the conditional likelihood can be written as
?
= 2mP(MD)
where?
and genetic distances between the markers. These are assumed to be known in linkage analysis. Since
2mP(MD) is independent of t and θ, it is a fixed constant which we drop. Hence,
?
Pt,θ(MD|Y) =
v
Pt(MD|v)Pθ(v|Y)
?
wPt(MD|v = w)Pθ(v = w|Y). In the last equality of (3.1)
v
Pt(v|MD)Pθ(v|Y),
(3.1)
vPt(MD|v)Pθ(v|Y) is short for?
we applied Bayes’ rule and P(v) = 2−m. The factor P(MD) depends on marker allele frequencies
L(t,θ; MD) =
v
Pt(v|MD)Pθ(v|Y) = Et(Pθ(v|Y)|MD)
(3.2)
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316O. HÖSSJER
is our conditional likelihood. Notice that we only include MD as an argument of L, because we condition
on the phenotype vector Y and consider it as fixed. Kruglyak et al. (1996) noticed that the conditional
inheritance distribution Pt(v|MD) could be used both for NPL and parametric linkage analysis based on
lod scores.
In linkage analysis, t is the structural parameter of interest, whereas θ contains nuisance parameters.
The lod score was originally formulated as the base ten logarithm of a likelihood ratio, but it is a linear
transformation of, and hence equivalent to, Z(t) = Z(t; θ) = log L(t,θ), as noted by Whittemore (1996).
We regard Z(t) as a local test statistic for testing H0against the simple alternative hypothesis τ = t. When
testing H0against H1, the global test statistic
Zmax= sup
t∈?
Z(t)
(3.3)
is used instead, where ? = [0,tmax] or, more generally, ? may consist of several chromosomes. The null
hypothesis is rejected when Zmaxexceeds a given threshold T, which depends on the chosen significance
level, ? and θ.
When θ is unknown, the profile conditional likelihood Z(t;ˆθ(t)) = supθZ(t; θ) can be computed at
each t and H0is rejected when maxtZ(t;ˆθ(t)) exceeds a given threshold. This procedure is equivalent
to mod scores. The mod score is computationally demanding, especially for larger pedigrees, although a
faster version is obtained by replacing the original L(t,θ), based on penetrance and disease allele param-
eters, by a simplified one with only one parameter, see Whittemore (1996) and Kong and Cox (1997).
For quantitative phenotypes, it is common to use variance components techniques, see e.g. Amos
(1994) and Almasy and Blangero (1998). These can be interpreted as an approximate version of the mod
score, where the original L(t,θ), based on penetrance and disease allele parameters, has been replaced by
a simplified one, which is a ratio of two multivariate normal densities.
In this paper, we will not maximize with respect to θ at each locus. Instead, we take a local approach
and assume that {θε} is a one-dimensional trajectory of genetic model parameters such that θ0corresponds
to no genetic effect at the disease locus, i.e. Pθ0(v|Y) = 2−m. In other words, under θ0the prior distribu-
tion of v equals the posterior distribution v|Y. Our objective is to compute a likelihood score function by
differentiating the log conditional likelihood w.r.t. ε at each locus t.
Assume first complete MD. It corresponds to an infinitely dense set of markers genotyped for all
pedigree members so that P(v = v(t)|MD) may be determined unambiguously at all loci. We let MDcompl
denote complete MD. The corresponding conditional likelihood is
L(t,θ; MDcompl) = Pθ(v = v(t)|Y).(3.4)
Define the score function
S(v) =dρlogPθε(v|Y)
dερ
????ε=0
,
(3.5)
where ρ is the smallest positive integer such that the right-hand side of (3.5) is nonzero for at least one v.
When ρ ? 2 the estimation problem is singular (with zero Fisher information) for ε at ε = 0. By means
of the reparametrization
? = ερ/ρ!,
S is interpreted as a (conditional) likelihood score function for ? at ? = 0. Originally, Whittemore (1996)
used ρ = 1 in her definition of S, but ρ = 2 is also possible for weak penetrance models, see McPeek
(1999) and Hössjer (2003d).
Our goal is to test ε = 0 against ε ?= 0 at each locus t, which can also be interpreted as testing H0
against τ = t at each t. Depending on the application, there may or may not be a sign constraint on ε.
In any case, the sign of ε is not of interest, and for this reason we use ? instead of ε as parameter when
(3.6)
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formulating a test statistic, also when ρ is even. The Fisher information for ? is Icompl= Eθ0(S2(v)|Y) =
2−m?
For incomplete MD, the observed conditional likelihood L(t,θ; MD) =
MD) is obtained by averaging the complete conditional likelihood, treating MDcomplas hidden data. Then
wS2(w) at ? = 0, with the sum ranging over all binary vectors w of length m. The square root of
the likelihood score statistic for testing ? = 0 against ? ?= 0 is Wcompl(t) = S(v(t))/√Icomplat locus t.
E(L(t,θ; MDcompl)|
W(t) =[dρlog L(t,θε; MD)/dερ]ε=0
√I(t)
=
√Icompl
√I(t)
E?Wcompl(t)|MD?,
(3.7)
where I(t) = Eθ0(E2(S(v(t))|MD)|Y) is the Fisher information. It depends on t in a way that reflects
positioning and informativity of markers. The outer expectation is taken over variations in MD and is
typicallycomputationallyinvolved. Itcanbecalculatedexactly(WhittemoreandHalpern, 1994), approxi-
mated as I(t) ≈ Icomplso that the first factor of the right-hand side of (3.7) vanishes (Kruglyak et al.,
1996) or approximated by a multiple imputation Monte Carlo algorithm (Clayton, 2001). Yet, another
method of handling incomplete MD was suggested by Kong and Cox (1997).
By definition of W(t) we have
Eθ0(W(t)) = 0,
Vθ0(W(t)) = 1,
(3.8)
where expectation is with respect to variations in MD.
So far, we have only discussed a single family. The extension to N pedigrees with mutually independ-
ent phenotype/MD is straightforward. We allow the pedigree structures to vary arbitrarily and index
quantitiesfortheithfamilywithi. Theoverallconditionallikelihood L(t,θ) =?N
?N
W(t) =[dρlogL(t,θε; MD)/dερ]ε=0
i=1Li(t,θ)isthensim-
ply the product of the familywise conditional likelihoods, and the total Fisher information is I(t) =
i=1Ii(t) at locus t. From this it follows that the linkage score function for N families can be written as
√I(t)
=
N
?
i=1
γi(t)Wi(t),
(3.9)
where Wi(t) is the score (3.7) for family i and γi(t) =
(McPeek, 1999). Since?N
The local SPL test statistic at locus t is defined as
?
Ii(t)/?N
j=1Ij(t) are the locally optimal weights
1γ2
i(t) = 1, (3.8) also holds for the total linkage score with N families.
Whittemore (1996) noticed that (3.9) includes APM methods as special case.
Z(t) =
?W(t)2,
W(t),
no sign constraint on ?,
? ? 0,
(3.10)
and the corresponding global test statistic for testing H0against H1is obtained by inserting Z(t) in (3.10)
into (3.3). Two separate definitions of Z(t) are needed, because ? = 0 is at the boundary of the parameter
space when ? ? 0. This is always the case when ρ is even and sometimes, depending on the applica-
tion, when ρ is odd (in that case ε ? 0 is imposed). A derivation of (3.10) is given in Section A of HS.
There, it is also shown that the SPL and mod score approaches are asymptotically equivalent under the
null hypothesis of no linkage when the genetic model parameters are restricted to a one-dimensional
trajectory {θε}.
For quantitative phenotypes, the SPL approach is also asymptotically equivalent to variance compon-
ent techniques when the parameters of the latter model are varied along a one-dimensional trajectory. This
is briefly discussed in Example 6 of Section 5.
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