Download Mapping quantitative trait loci in oligogenic models

Biostatistics (2001), 2, 2, pp. 147–162
Printed in Great Britain
Mapping quantitative trait loci in oligogenic
models
HSIU-KHUERN TANG, D. SIEGMUND†
Department of Statistics, 390 Serra Mall, Sequoia Hall, Stanford University, Stanford, CA
94305-4065, USA
Email: [email protected]
S UMMARY
We discuss strategies for mapping quantitative trait loci with emphasis on certain issues of study design
that have recently received attention: e.g. genotyping only selected pedigrees and the comparative value
of large pedigrees versus sib pairs. We use a standard variance components model and a parametrization
of the genetic effects in which the ‘segregation’ parameters are locally orthogonal to the ‘linkage’
parameters. This permits simple explicit expressions for the expectation of the score statistic, which we
use to compare the power of different strategies. We also discuss robustness of the score statistic.
Keywords: Gene mapping; Genome scan; Quantitative trait; Variance components.
1. I NTRODUCTION
The goal of genetic mapping is to locate the genes affecting particular traits by analysis of the
correlation between phenotypic values and genetic markers distributed throughout the genome. The traits
can involve a 0–1 phenotype (e.g. human diseases) or can be based on quantitative measurement. One
expects relatives who have similar phenotypes to have similar genotypes at marker loci close to genetic
loci that influence the trait, while the markers behave stochastically according to the rules of Mendelian
inheritance at distant loci. Until recently, the theory and practice of mapping quantitative trait loci (QTLs)
in humans has been relatively undeveloped.
The purpose of this paper is to discuss, essentially from first principles, the statistical theory for
mapping QTLs in humans under the simplifying assumptions of an oligogenic model of inheritance and
completely informative genetic markers. Although this model has the deficiency of a strong assumption
of normality, its interpretability and its computational tractability in the incorporation of covariates,
multivariate phenotypes, interactions, etc. is sufficiently attractive to have encouraged substantial recent
development (e.g. Amos (1994); Fulker and Cardon (1994); Kruglyak and Lander (1995); Fulker and
Cherny (1996); Almasy and Blangero (1998); Page et al. (1998); Williams and Blangero (1999)). We try
insofar as possible to give explicit analytic accounts of a number of issues that have usually been treated
in the literature by numerical methods or by simulation, especially the relative values of large versus
small pedigrees (cf. Almasy and Blangero (1998); Page et al. (1998); Williams and Blangero (1999)) and
of genotyping selected versus random sibships (Risch and Zhang, 1995; Eaves and Meyer, 1994). In the
process we generalize the notion of a discordant sib pair to a discordant sibship.
In the following section we describe the model to be studied. We discuss sib pairs in Section 3. Sibships
of arbitrary size are the subject of Section 4. Selective genotyping is the subject of Section 5. In Section 6
† To whom correspondence should be addressed.
c Oxford University Press (2001)
148
H.-K. TANG AND D. S IEGMUND
we give a brief discussion of robustness of the components of variance model when a biallelic major gene
model is correct. Our main conclusions are discussed in Section 7.
To facilitate our calculations we introduce a parametrization in which the ‘segregation’ parameters
(those that can be estimated from segregation data) and ‘linkage’ parameters (those requiring data from
linked markers for their estimation) are orthogonal under the null hypothesis of no linkage (Cox and
Hinkley, 1974, p. 324). We also use the standard asymptotic framework of statistical large sample theory,
that the sample size N is large and the noncentrality parameter for a single observation is inversely
proportional to N 1/2 . These techniques allow us to compute score statistics, their asymptotic expectations,
and Fisher information matrices comparatively explicitly.
2. M ODEL
We assume Hardy–Weinberg equilibrium throughout. Suppose there is a QTL at the locus τ , which is
an unknown parameter. The phenotypic value Y is assumed to be given by
Y = µ + αU + αV + δU,V + e.
(2.1)
The mean value µ may be expanded as a linear model with only minor changes to what follows. The
parameter αa denotes the additive genetic effect of allele a at locus τ , δa,b the dominance deviation of
alleles a, b. A subscript U denotes the allele contributed by the mother while a subscript V refers to
the father. The phenotypic variance is σY2 = E[(Y − µ)2 ]. The variances of the additive and dominance
2 . Implicitly we expect that there are
effects associated with the QTL at τ are σ A2 = 2EαU2 and σ D2 = EδU,V
several QTLs, which may interact. For this paper we assume that other QTLs are on other chromosomes,
are in linkage equilibrium with and do not interact with the QTL at τ . Under these assumptions their
contribution to the phenotype Y can be assumed to be a part of the residual term e, which we assume is
uncorrelated with the other terms in (2.1) and has variance σe2 . Then σY2 = σ A2 + σ D2 + σe2 . The locusspecific heritability associated with the QTL at τ is h 2 = (σ A2 + σ D2 )/σY2 . The term ‘oligogenic’ of the title
is intended to suggest that at least one QTL has a substantial locus-specific heritability, say h 2 > 0.1, and
hence might be detectable by linkage analysis with reasonable sample sizes.
In order to have some idea of the magnitudes of different components of variance and their relations
to more directly interpretable genetic parameters, we shall occasionally be interested in the special case
of two alleles A1 and A2 with frequencies of p and q = 1 − p. Using a and −a for the genotypic values
of the homozygotes A1 A1 and A2 A2 , respectively, and d for the dominance deviation of the heterozygote,
we have the standard formulas:
σ A2 = 2 pq[a + (q − p)d]2 ,
σ D2 = 4 p 2 q 2 d 2 .
(2.2)
Consider a pair of siblings satisfying the model (2.1). Recall that at any locus two relatives share
alleles identical by descent if they inherit the same alleles from a common ancestor. Two siblings can
share 2, 1 or 0 alleles identical by descent depending on whether they inherit the same alleles from both
mother and father, from one but not both, or from neither. Let ν = ν(τ ) denote the number of alleles
identical by descent at τ . Letting Yi denote the phenotypic value of the ith sibling (i = 1, 2), we have
(Fisher, 1918; Kempthorne, 1955)
Cov[Y1 , Y2 |ν] = σe2r + σ A2 ν/2 + σ D2 1{ν=2} ,
(2.3)
where r = corr(e1 , e2 ) accounts for the correlation between sibs arising from other QTLs and from a
shared environment.
Mapping quantitative trait loci in oligogenic models
149
Taking the expectation of (2.3), we find the unconditional covariance. Hence we can rewrite (2.3) as
Cov[Y1 , Y2 |ν] = Cov(Y1 , Y2 ) + [(σ A2 + σ D2 )/2](ν − 1) − (σ D2 /2)[1{ν=1} − 12 ],
(2.4)
so the terms involving ν have mean 0 and are uncorrelated.
In the following we will be interested in marker loci t distributed throughout the genome and ν(t) for
a sib pair as a stochastic process in t. For markers t1 and t2 on different chromosomes, ν(t1 ) and ν(t2 ) are
stochastically independent. For markers on the same chromosome Cov[ν(t1 ), ν(t2 )] = 2−1 [1−2φ], where
φ is a function of the recombination frequency. We find it convenient to assume that recombination follows
the Haldane model of no interference, so 1 − 2φ = exp(−4λ|t1 − t2 |). Here the marker location t denotes
genetic distance in centimorgans (cM) from a designated end of the chromosome, and λ = 0.01/cM.
3. L IKELIHOOD THEORY FOR SIB PAIRS
In this section we consider genome scans to detect the QTL that is explicitly modelled in (2.1),
although we acknowledge that several QTLs may contribute to the trait. Since the location τ of the QTL
is unknown, our statistic takes the form of a stochastic process indexed by markers at loci t distributed
throughout the genome. Large values of this process indicate the likelihood of a QTL located near to the
markers where the large values occur.
Initially we consider N pairs of siblings, and for simplicity we assume that markers are completely
informative, so the values of ν(t) are known with certainty. Consider a sample of N sib pairs with
phenotypes (Y11 , Y12 ), . . . , (Y N 1 , Y N 2 ) and data ν1 (ti ), . . . , ν N (ti ) from markers ti distributed throughout
the genome. Our basic modelling assumption is that (Yn1 , Yn2 ) are independent with a common
distribution, which conditional on ν = νn (τ ) is bivariate normal with common means µ, variances σY2 ,
and correlation (cf. (2.4))
ρν = ρ + {α0 (ν − 1) − δ0 [1{ν=1} − 12 ]}/σY2 ,
(3.1)
where
α0 = [σ A2 + σ D2 ]/2,
δ0 = σ D2 /2.
(3.2)
It is convenient to define Dn = (Yn1 − Yn2 )/21/2 and Sn = (Yn1 + Yn2 − 2µ)/21/2 . For simplicity we
assume initially that the parameters µ and σY2 are known and equal to 0 and 1, respectively. As we show
below, this has no effect on the asymptotic theory. We also make the working assumption that the QTL
τ is one of the ti (cf. Remark (iii) below), so the marginal log likelihood function, = (τ, α0 , δ0 , ρ), is
given by
= −2−1
[log(1 − ρν2n ) + Dn2 /(1 − ρνn ) + Sn2 /(1 + ρνn )].
(3.3)
n
Here νn = νn (τ ), and ρν is given by (3.1). This can be regarded as the conditional likelihood of the
phenotypic data given (νn , n = 1, . . . , N ) or as the unconditional joint log likelihood of the phenotypic
data and the (νn ). All expectations are taken with respect to the joint distribution.
Partial derivatives with respect to unknown parameters are denoted by appropriate subscripts. Let
Cn = Cn (α0 , δ0 , ρ) be defined by
Cn = ρνn /(1 − ρν2n ) + Sn2 /2(1 + ρνn )2 − Dn2 /2(1 − ρνn )2 .
(3.4)
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H.-K. TANG AND D. S IEGMUND
Regarding τ momentarily as known, we obtain the components of the efficient score:
α (τ ) =
[νn (τ ) − 1]Cn ,
(3.5)
n
δ (τ ) =
n
[ 12 − 1{νn (τ )=1} ]Cn ,
and
ρ =
(3.6)
Cn .
(3.7)
n
When α0 = 0 (hence also δ0 = 0), the three coordinates of the score vector are uncorrelated, so the
Fisher information matrix is diagonal, with easily computed entries. When it is assumed that there is no
dominance variance, so δ0 = 0, the score statistic for testing α0 = 0 at a putative QTL τ = t is
1/2
Z 1 (t) = α (t)/Iαα
,
(3.8)
+
where now Cn = Cn (0, 0, ρ̂) and ρ̂ = N −1
is the maximum likelihood estimator under
n Y1,n Y2,n
the null hypothesis. Since the true value of τ is unknown, to scan an entire genome for linkage, we can
use
max Z 1 (t),
(3.9)
t
where the max is taken over all marker loci. Linkage is detected whenever maxt Z 1 (t) b for a suitable
threshold b. Thresholds to control the false detection rate have been discussed by Feingold et al. (1993)
for markers equally spaced at distance 0. See also Lander and Kruglyak (1995) and Lander and
Schork (1994).
To allow for the possibility that δ0 > 0, we can define
1/2
(3.10)
[Z 12 (t) + Z 22 (t)]1/2 ,
(3.11)
Z 2 (t) = δ (t)/Iδδ .
and use the two degree of freedom statistic
constrained by the relation that 0 δ0 α0 (cf. (3.2)). Dupuis and Siegmund (2000) discuss this
possibility in the context of sib pair analysis of qualitative traits.
R EMARKS (i) For unknown σY2 , µ, and ρ one uses µ̂ = (2N )−1 n (Y1,n + Y2,n ), σ̂Y2 =
+ 2
(2N )−1 n [(Y1,n − µ̂)2 + (Y2,n − µ̂)2 ] and ρ̂ = N −1
n (Y1,n − µ̂)(Y2,n − µ̂) /σ̂Y . The asymptotic
results are unchanged since the scores for the ‘segregation’ parameters σY2 , µ, ρ are uncorrelated with
those for the ‘linkage’ parameters α0 , δ0 under α0 = δ0 = 0 and asymptotically for local alternatives.
(ii) The components of variance analysis uses the relation for normally distributed X that the variance
of X 2 is twice the square of the variance of X , and hence it is not robust to violations of the assumed
normality. A simple device to obtain a more robust test would be to refer the statistic given above to
its conditional distribution given (C1 , . . . , C N ), where Cn = Cn (0, 0, ρ̂). This would make the type I
error probability nonparametric with respect to the distribution of the phenotypes while maintaining full
asymptotic efficiency if the normality hypothesis is satisfied. Asymptotically it amounts to replacing (3.8)
and (3.10) by
1/2
1/2
Z 1 (t) = α (t)/
Cn2
,
Z 2 (t) = δ (t)/
Cn2
,
(3.12)
n
n
Mapping quantitative trait loci in oligogenic models
151
Fig. 1. Expected value of the score statistic: QTL location is τ , expectation at τ is ζ , detection threshold is b, and the
× denote simulated values at equally spaced marker loci.
which has the effect of using fourth moments of the phenotypic values to estimate variability.
(iii) We have assumed completely informative markers in order to simplify the analysis and have made
the working assumption that the QTL τ is one of the markers. If either of these assumptions fails to be
true, the likelihood function involves a mixture based on the conditional distribution of νn (τ ) given the
marker data, say Mn , in the nth family. A convenient representation for the likelihood function is
E 0 [exp((τ, α0 , δ0 , ρ) − (0, 0, ρ))|M, Y ],
(3.13)
where M = (M1 , . . . , M N ), Y = (Y11 , Y21 , . . . , Y1N , Y2N ), is given by (3.3), and the subscript 0
denotes that the expectation is computed under the assumption that α0 = δ0 = 0. For the case of
partially informative markers, on the evidence of simulations, Fulker and Cherny (1996) report very good
success with the test to detect an additive effect that simply replaces νn in (3.3) by ν̂n = E 0 (νn |Mn , Yn ).
Equation (3.5) provides an explanation for the Fulker–Cherny results, since the efficient score for testing
α0 = 0 is precisely (3.5) with Cn = Cn (0, 0, ρ̂) and with νn replaced by ν̂n . To prove this, differentiate
the logarithm of (3.13) with respect to α0 and evaluate the result at α0 = δ0 = 0. See Teng and Siegmund
(1998) for a discussion of the impact of marker informativeness and intermarker distance on the power to
detect linkage. The same calculation gives the efficient score when the QTL τ is located between markers.
In principle one might extend the max in (3.9) from marker loci to all loci t, but in most cases there seems
to be very little power gained by this device (cf. Darvasi et al. (1993); Dupuis and Siegmund (1999)), so
we do not pursue the more complicated analysis.
Display (4.8) derived below gives an expression for the noncentrality ξ = E[Z 1 (τ )]. At a marker t
linked to τ , E[Z 1 (t)] = ξ exp[−4λ|t − τ |]. These relations are illustrated in Figure 1, where ξ > b, so
the probability of detection is large. The asymptotic squared noncentrality of (3.11) at the QTL τ is given
by (4.8) with α02 /2 replaced by α02 /2 + δ02 /4.
In Table 1 we have used (4.8) to evaluate the sample size necessary to achieve 90% power for the
score statistic (3.8) and various values of α0 and ρ. We consider a genome scan at 1 cM and assume a
genome of 23 chromosomes of average length 140 cM. This yields a detection threshold of b ≈ 3.91 (cf.
Feingold et al. (1993) or the Appendix). We assume the trait locus is at zero recombination distance from
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H.-K. TANG AND D. S IEGMUND
Table 1. Number of sib pairs for 90% power. The
overall correlation between two siblings is ρ, and
the locus-specific heritability is 2α0 /σY2 . For = 1,
the QTL is assumed to lie at a marker; a 2% larger
sample size is required for a QTL midway between
two flanking markers. For = 10, the QTL is
assumed to lie midway between two markers
ρ
0.25
α0 /σY2
0.25
0.15
0.10
0.05
N ( = 1)
662
1838
4136
16544
N ( = 10)
780
2168
4878
19512
0.40
0.25
0.15
0.10
0.05
487
1352
3041
12166
574
1594
3587
14348
the nearest marker. For 90% power the approximation of Feingold et al. (1993) given in the Appendix,
leads to the value of N that makes the squared noncentrality parameter given by (4.8) about equal to 25.
The required sample size would increase by about 2% if the QTL is midway between markers. We have
also included the sample size required for 90% power when markers are spaced at 10 cM and the QTL
is midway between markers. The detection threshold decreases to approximately b = 3.6; but for a QTL
midway between markers, we need an approximately 16–18% increase in sample size compared to the
1 cM spacing.
Simulations indicate that these approximations are very accurate. An interesting comparison is
available from Page et al. (1998), who have estimated the sample size required to have a probability of
0.9 for the LOD score evaluated at a single marker to exceed the conventional level of 3, or equivalently
b = 3.72 on the normal scale, which we are using here. Although this is in principle quite different
from the situation that arises when multiple, closely spaced markers are used, in the special case of 1 cM
spacing and the QTL exactly at a marker, by a numerical accident their definition also leads to the value
of N that makes the squared noncentrality parameter equal to (3.72 + 1.28)2 = 25. Hence our analytic
approximations can be compared with the results in Table 4 of Page et al. (1998), which were based on
5 000 000 simulations of an additive model with one biallelic major gene and a polygenic component. The
agreement is generally excellent.
If the true mode of inheritance is roughly additive, the two-dimensional statistic based on (3.11),
which under the conditions given above requires a threshold of about b = 4.11 (Dupuis and Siegmund,
2000), has less power than the corresponding one-dimensional statistic. Some numerical experimentation
indicates that (3.11) is more efficient than (3.8) only for a fairly rare recessive allele, which must have a
large effect for the QTL to be detectable with a reasonable sample size. Hence the simpler one-dimensional
statistic would seem adequate in most situations.
4. L IKELIHOOD THEORY FOR SIBSHIPS OF ARBITRARY SIZE
Starting with Blackwelder and Elston (1982), a number of authors have observed on the basis of
simulations that sibships of size s provide considerably more power than sib pairs, perhaps as much as
s(s − 1)/2 independent sib pairs. See also Page et al. (1998); Williams and Blangero (1999). Suppose
Mapping quantitative trait loci in oligogenic models
153
we have a sample of N sibships, each of size s. We index sibs within a sibship by i and j and sibships
by n = 1, . . . , N . The subscript n is often suppressed in our notation. For ease of exposition we again
assume that µ = 0 and σY2 = 1. Let νi j (t) denote the number of alleles shared identical by descent at the
marker locus t by the ith and jth sibs in the nth sibship. Let Aν denote the s × s matrix with entries νi j − 1
for i = j and zeros along the diagonal. Let ν = E(Y Y |Aν ). The log likelihood for a single QTL at τ is
= (τ, α0 , δ0 , ρ) given by
= −2−1
N
{log |ν | + trν−1 Y Y },
(4.1)
n=1
where ν = ν(τ ).
Recall that if G is a nonsingular matrix depending on x, then ∂ log |G|/∂ x = tr(G −1 ∂G/∂ x) and
−1
∂G /∂ x = −G −1 ∂G/∂ x G −1 . By differentiation of (4.1) we obtain the score equations
α = 2−1
{−tr(ν−1 Aν ) + tr(ν−1 Aν ν−1 Y Y )}
(4.2)
n
and
ρ = 2−1
n
{−tr(ν−1 B) + tr(ν−1 Bν−1 Y Y )},
where B = ∂ν /∂ρ = 11 − I . We omit the similar expression for δ (cf. (3.6)). Also
−αα =
{−2−1 tr(ν−1 Aν ν−1 Aν ) + tr(ν−1 Aν ν−1 Aν ν−1 Y Y )}.
(4.3)
(4.4)
n
Similar expressions for ρρ , αρ , δδ , etc. are easily obtained.
Let E 0 denote expectation under the hypothesis that α0 = 0 (hence also δ0 = 0) and let =
E 0 (Y Y ) = (1 − ρ)I + ρ11 . It is easy to see that
Iαα = E 0 (−αα ) = (N /2)trE 0 ( −1 Aν −1 Aν )
(4.5)
and
Iαρ = E 0 (α ρ ) = 0.
Hence the asymptotic noncentrality of the score statistic
1/2
Z t = α (t, 0, 0, ρ̂)/Iαα
(0, 0, ρ̂)
(4.6)
can be evaluated by taking the expectation of (4.6) with ρ̂ replaced by ρ.
From (3.1) and (4.2) we obtain for t = τ
E[α (τ, 0, 0, ρ)] = (N /2)α0 trE( −1 Aν −1 Aν ).
Hence by (4.5) the noncentrality of (4.6) is
α0 {(N /2)trE 0 ( −1 Aν −1 Aν )}1/2 .
To evaluate (4.7), we first observe that
−1 = K ρ {[1 + (s − 1)ρ]I − ρ11 },
(4.7)
154
H.-K. TANG AND D. S IEGMUND
where K ρ = {(1 − ρ)[1 + (s − 1)ρ]}−1 . See, for example, Rao (1973, p. 67). It is easy to see that
νi j = ν ji , and for i < j the νi j are pairwise independent whenever they differ for at least one subscript.
Hence E A2ν = [(s−1)/2]I . By straightforward and somewhat tedious algebra we find that E(Aν 11 Aν ) =
(s/2 − 1)I + (1/2)11 . Combining these results, we obtain
trE 0 (
−1
Aν ) =
2
K ρ2
s
{[1 + (s − 2)ρ]2 + ρ 2 },
2
which can be substituted into (4.7) to obtain the square of the asymptotic noncentrality parameter for N
sibships of size s:
s {[1 + (s − 2)ρ]2 + ρ 2 }
ξ =N 4
.
2σY 2 {(1 − ρ)[1 + (s − 1)ρ]}2
2
α02
(4.8)
Although (4.8) increases rapidly with s, there are dependencies among the νi j within a sibship, so
α has a skewed distribution when s > 2, and hence a larger threshold is required to maintain a fixed
false positive error rate. In addition, standard asymptotic theory to the effect that the asymptotic variance
of the score statistic for small, positive α0 is effectively the same as when α0 = 0 is not a reasonable
approximation for large s (hence relatively small sample sizes). Consequently, the increase in power with
increasing s is less than is suggested by considering only the increase in the asymptotic noncentrality
parameter. Numerical examples are given in Table 2, for which we used a more refined asymptotic
analysis, which we describe in an Appendix and have spot checked for accuracy by simulations. (Even
for s = 2 there is a small discrepancy between the sample sizes in Tables 1 and 2 because of the different
approximations used.) We see from
Table 2 that, except for large s or large α0 , a sibship of size s turns
out to be roughly as powerful as 2s independent sib pairs.
For small sibships Williams and Blangero have also obtained the noncentrality parameter (4.8) and
have computed sample sizes directly from (4.8) without the corrections mentioned in the preceding
paragraph. These can be misleading, especially if carried out for larger s. For example, for the third
row of Table 2 their method would yield N = 100; for the sixth row it would yield N = 12. Our more
precise asymptotic analysis seems to be consistent with the simulations of Page et al. (1998).
Exactly as for sib pairs, we can obtain a distribution-free false positive rate if we consider the
conditional distribution of α given the phenotypic values. Asymptotically that means that α should
be standardized by {E 0 [2α |Y1 , . . . , Y N ]}1/2 , which is given by the square root of a sum of terms of the
form
E 0 {[−tr( −1 Aν ) + Y −1 Aν −1 Y ]2 | Y }
s 2 µ22
sµ4
4sµ3 Ȳ
2s(s − 3)µ2 Ȳ 2
−
+
+
(1 − ρ)4
[1 + (s − 1)ρ](1 − ρ)3
[1 + (s − 1)ρ]2 (1 − ρ)2
(1 − ρ)4
2sρµ2
s(s − 1){ρ[1 + (s − 1)ρ] + (1 − ρ)Ȳ 2 }2
−
+
,
3
[1 + (s − 1)ρ](1 − ρ)
[1 + (s − 1)ρ]4 (1 − ρ)2
=−
(4.9)
where µk = s −1 (Yi − Ȳ )k for k = 2, 3, 4. In the following section we find (4.9) useful for a completely
different purpose.
Our methods can be adapted to study extended pedigrees, although it is more difficult to obtain explicit
analytic results. An exception is the case of nuclear families consisting of parents and their children. In
addition to the inter-sib correlation ρ, we let ρ p denote the parental correlation (due to shared environment
only) and ρ̃ = (σ A2 /2 + r̃ σe2 )/σY2 the parent–sib correlation. For N nuclear families, each containing s
Mapping quantitative trait loci in oligogenic models
155
Table 2. Sample sizes required for 90% power. Sibships of size s; sib
pair correlation is ρ; sample size is N ; threshold is b; noncentrality
is ξ ; rel. eff. is the number of independent sib pairs needed to have
the same power as one sibship of size s; see the text for definitions
of other parameters
ρ
0.25
α0 /σY2
0.25
s
2
3
4
5
6
10
b
3.91
4.02
4.11
4.19
4.26
4.48
ξ
5.05
5.34
5.60
5.86
6.10
7.00
N
675
237
125
80
56
23
σ02
1.13
1.52
1.97
2.48
3.05
5.86
σ12
1.08
1.28
1.48
1.68
1.89
2.74
rel. eff.
1.0
2.8
5.4
8.5
12
30
0.25
0.125
2
3
4
5
6
10
3.91
3.97
4.02
4.07
4.12
4.27
5.01
5.15
5.29
5.43
5.57
6.07
2 658
881
445
272
186
69
1.03
1.20
1.39
1.60
1.81
2.84
1.02
1.11
1.19
1.28
1.38
1.75
1.0
3.0
6.0
9.8
14
39
0.25
0.0625
2
3
4
5
6
10
3.91
3.94
3.97
4.00
4.03
4.12
4.99
5.07
5.14
5.21
5.29
5.55
10 546
3 417
1 679
1 001
670
230
1.01
1.09
1.17
1.26
1.36
1.77
1.01
1.04
1.09
1.13
1.17
1.34
1.0
3.1
6.3
11
16
46
0.40
0.15
2
3
4
5
6
10
3.91
3.99
4.07
4.13
4.19
4.38
5.02
5.23
5.44
5.63
5.83
6.54
1 363
446
225
137
95
37
1.07
1.33
1.64
1.98
2.36
4.18
1.05
1.17
1.31
1.46
1.61
2.21
1.0
3.1
6.1
10
15
38
0.40
0.125
2
3
4
5
6
10
3.91
3.98
4.04
4.10
4.16
4.33
5.01
5.19
5.36
5.53
5.70
6.34
1 954
632
314
191
130
49
1.05
1.26
1.50
1.78
2.07
3.46
1.03
1.14
1.25
1.36
1.48
1.97
1.0
3.1
6.2
10
15
40
sibs, the squared noncentrality equals
N α02 s [1 − 2ρ̃ 2 /(1 + ρ p ) + (s − 2)(ρ − 2ρ̃ 2 /(1 + ρ p )]2 + (ρ − 2ρ̃ 2 /(1 + ρ p ))2
.
[1 − 2ρ̃ 2 /(1 + ρ p ) + (s − 1)(ρ − 2ρ̃ 2 /(1 + ρ p ))]2 (1 − ρ)2
2σY2 2
This is somewhat larger than (4.8) for the siblings alone. For a completely additive trait with ρ = 0.25,
for two sibs the squared noncentrality with parents included is 15% larger than without parents. For five
sibs it is only 7% larger.
156
H.-K. TANG AND D. S IEGMUND
5. S ELECTIVE GENOTYPING
In some cases it may be relatively easy and inexpensive to phenotype individuals. When the cost of
phenotyping is indeed small compared to the cost of genotyping, it is possible to achieve an increase in
power by genotyping only sib pairs with particularly favourable phenotypes. See, for example, Risch and
Zhang (1995), who recommend using sib pairs with discordant phenotypes, and Xu et al. (1999) for an
application. In this section we study Risch and Zhang’s suggestion. We begin by considering instead of
α the simpler statistic suggested by Risch and Zhang,
(ν − 1)/(N /2)1/2 ,
(5.1)
where the summation extends over all genotyped sib pairs. Also assume for simplicity that µ = 0, σY2 = 1.
For a given sib pair, by Bayes’ formula one can easily show that E(ν − 1|D, S) is given by a fraction,
the numerator of which equals
P(ν = i)(i − 1)ϕ[S/(1 + ρi )1/2 ]ϕ[D/(1 − ρi )1/2 ]/(1 − ρi2 )1/2
while the denominator is a similar expression without the factor (i − 1). For small values of α0 , we see
from the first term of a Taylor series expansion that
E(ν − 1|D, S) ∼ α0 E 0 (ν − 1)2 [ρ/(1 − ρ 2 ) + S 2 /2(1 + ρ)2 − D 2 /2(1 − ρ)2 ].
(5.2)
It is evident from (5.2) that sib pairs with large values of D are particularly informative. If we genotype
only those sib pairs whose phenotypes satisfy |D| > t, for each genotyped pair we have at the QTL an
asymptotic noncentrality of
21/2 E(ν − 1||D| > t) ∼ −[α0 /2 · 21/2 (1 − ρ)]t ϕ(t )/{1 − (t )},
(5.3)
where t = t/(1 − ρ)1/2 .
For a numerical example suppose that ρ = 0.25 (corresponding to a heritability of 0.50 for a purely
additive trait) and t = 1.96. Then (5.3) equals −2.16α0 , while the noncentrality of a random sib pair is
0.78α0 . Hence only about one-eighth as many discordant sib pairs need be genotyped as random sib pairs.
On the other hand, roughly 20 random sib pairs must be phenotyped to find one discordant sib pair.
Other methods of selecting the sib pairs to be genotyped can be handled by modifications of the
preceding argument. For the situation described above, the preferred definition of Risch and Zhang (1995)
is about 82% as efficient as our suggestion. The case of concordant sib pairs, defined by max(Y1 , Y2 ) < −t
or min(Y1 , Y2 ) > t, can be treated similarly. Now the value of t corresponding to the preceding examples
is 1.19, and the asymptotic noncentrality is 1.49α0 .
Instead of the ad hoc statistic (5.1), one might consider the score statistic (3.8). Now the unknown
nuisance parameter ρ (and in general µ and σY also) must be estimated. This poses no problem if,
as we assume, the sib pairs to be genotyped are selected from a random sample of sib pairs of known
phenotypes, which are available to estimate the nuisance parameters. This will typically be a very large
sample, ensuring that the nuisance parameters are estimated accurately. By some routine Taylor series
expansions one sees that for purposes of asymptotic analysis one can regard them as known. Before
proceeding, it is worth noting, however, that the situation would be quite different if the sib pairs are
ascertained through their phenotypes, so the natural estimates of nuisance parameters are biased, perhaps
severely. See, for example, Beaty and Liang (1987) for ascertainment corrections.
An advantage of the score statistic over (5.1) is that it generalizes naturally to the case of larger
sibships. For the score statistic (4.2), the arguments given above show that
N
E(α |Y1 , . . . , Y N ) ∼ (α0 /4)n=1
E 0 {[−tr( −1 Aν ) + Y −1 Aν −1 Y ]2 |Y }.
(5.4)
Mapping quantitative trait loci in oligogenic models
157
We define a discordant sibship to be one in which the squared norm of the (s − 1)-dimensional vector
of orthogonal contrasts exceeds a threshold t. To facilitate the analysis of (5.4) we make the (Helmert)
orthogonal transformation from Y = (Y1 , . . . , Ys ) defined by Z s = (Y1 + · · · + Ys )/s 1/2 = s 1/2 Ȳ , and
for i = 1, . . . , s − 1 Z i = [(i + 1)i]−1/2 [ ij=1 Y j − iYi+1 ]. Under the probability P0 , Z 1 , . . . , Z s are
independent and normally distributed with Var0 Z s = 1+(s −1)ρ, and Var0 Z i = 1−ρ, i = 1, . . . , s −1.
The variables Z 1 , . . . , Z s−1 are the orthogonal contrasts in the Y , so our definition of discordant is that
2
Z 12 + · · · + Z s−1
> t.
An expression for the expectation in (5.4) is given in (4.9). To find the asymptotic noncentrality
2
parameter, we evaluate the expectation of (4.9) given that Z 12 + · · · + Z s−1
> t. It is straightforward
s
k
to compute each term, except possibly those involving i=1 Yi for k = 3, 4. Since our definition of
discordance is symmetric in the Y , all terms in the sum have the same (conditional) expectation, and from
the inverse of the Helmert transformation, we see that Ys = −(s − 1)1/2 Z s−1 /s 1/2 + Z s /s 1/2 . Hence these
expectations are also readily evaluated, and we obtain the following expression:
2
E 0 {[−tr( −1 Aν ) + Y −1 Aν −1 Y ]2 | Z 12 + · · · + Z s−1
> t}
2
3
2
1
3(s − 1) 2t f s−1 (t )
(s − 3s + s + 3)ρ + s 2 − 3 2t f s−1 (t )
=
1
−
+
s(s + 1)
Fs−1 (t )
Fs−1 (t )
(1 − ρ)2
s[1 + (s − 1)ρ](1 − ρ)2
+
s(s − 1){[1 + (s − 2)ρ]2 + ρ 2 }
,
[1 + (s − 1)ρ]2 (1 − ρ)2
where t = t/(1 − ρ), and f s , Fs are respectively the density and right tail distribution functions of a χs2
variable.
Some examples of the efficiency gained by selective genotyping of the most discordant 10% of a
sample of sibships of size s are given in Figure 2. For small s the gain in efficiency compared to genotyping
random sibships is quite large; but as the size of the sibship increases the relative value of selective
genotyping decreases while, as we saw in the preceding section, the unconditional value of the sibship
increases. For example, a random sibship of size 4 is about as powerful as a selected sib pair from the
most discordant 10% of the population.
6. ROBUSTNESS
As we indicated above, by conditioning on the phenotypic values it is possible to make the statistics we
have considered nonparametric with respect to the false-positive error rate. In this section we make a brief
study of robustness of the power of these tests when a different model is assumed to be true—in particular
for the model having a major gene with two alleles and a normally distributed residual. For simplicity we
consider only sib pairs. The standardized ‘nonparametric’ version of the score statistic is α /[n Cn2 /2]1/2 .
Since the expected value of the numerator, E(α ), is computed without making distributional assumptions,
to evaluate the noncentrality parameter we need only evaluate E 0 C 2 , which under the assumption of
normality equals (1 + ρ 2 )/(1 − ρ 2 )2 .
Let = E 0 C 2 −(1+ρ 2 )/(1−ρ 2 )2 . Algebraic expressions for are straightforward, albeit somewhat
complicated for the case of a single biallelic QTL and normally distributed residual e. Table 3 contains
numerical examples of the percentage increase in E 0 C 2 , which is also the percentage increase in sample
size that would be required to maintain the same power as determined for the assumed components of
variance model. For the most part, the impact of using the components of variance when a biallelic major
gene model is correct has a negligible effect on the power, but the effect can be substantial in the case of
a rare recessive allele of large effect.
H.-K. TANG AND D. S IEGMUND
0
100
200
300
400
500
600
700
158
2
4
6
8
10
Fig. 2. Number of genotyped sibships for selected (proportion = 0.1) and unselected genotyping.
It is possible that a likelihood analysis of the correct model would produce a completely different and
more efficient statistic. To simplify the notation we assume there is no dominance deviation. Let gi denote
the indicator that the allele inherited from the ith parent is A1 . Then we can write Y = µ + a(g1 + g2 −
2 p) + e. Let p(g1 , g2 |ν) denote the conditional distribution of (g1 , g2 ) given ν. The likelihood function
for a single pedigree is the mixture
D − a(g1 − g2 )/21/2
p(g1 , g2 |ν)
S − a(g1 + g2 − 2 p)/21/2
g1 ,g2 2
ϕ
.
(6.1)
ϕ
σe (1 − r 2 )1/2
σe (1 + r )1/2
σe (1 − r )1/2
If we take the first two terms of the Taylor series expansion of (6.1) about a = 0, we obtain after some
calculation that (6.1)
ϕ[S/σe (1 + r )1/2 ]ϕ[D/σe (1 − r )1/2 ]
σe2 (1 − r 2 )1/2
2
2
D − σe (1 − r ) 1
S 2 − σe2 (1 + r ) 3
× 1 + α0
−
(ν
−
1)/2
+
+
(ν
−
1)/2
,
2
2
σe4 (1 − r )2
σe4 (1 + r )2
≈
(6.2)
where, as above, α0 = σ A2 /2 = pqa 2 .
The efficient score for testing α0 = 0 is the logarithmic derivative of the likelihood function evaluated
at α0 = 0. This is just the coefficient of α0 in (6.2). Any term not involving ν can be omitted, and unknown
parameters must be estimated, so after summing over all sib pairs the statistic becomes
D 2 − σ̂e2 (1 − r̂ )
S 2 − σ̂e2 (1 + r̂ )
(ν − 1) −
.
(6.3)
+
2σ̂e4 (1 − r̂ )2
2σˆe 4 (1 + r̂ )2
Mapping quantitative trait loci in oligogenic models
159
Table 3. Percentage increase in sample size for biallelic
major gene. The column headed % change gives the percentage increase in sample size for a biallelic major gene
(cf. (2.2)) relative to the assumed oligogenic model with
the same variance components. The major gene contributes
25% of the trait variance; its additive effect is a, dominance
deviation is d, and allele frequency is p; the sib correlation
is ρ
a
0.707
d
0
p
0.50
ρ
0.125
0.250
0.817
0
0.25
0.125
0.250
1.0
1.6
1.179
0
0.10
0.125
0.250
6.7
5.9
% change
−0.3
0.6
0.577
−0.577
0.50
0.104
0.208
2.0
2.8
1.033
−1.033
0.25
0.088
0.175
20.6
17.7
2.513
−2.513
0.10
0.074
0.148
137
116
0.504
0.504
0.25
0.116
0.232
−0.3
1.2
0.637
0.637
0.10
0.122
0.243
4.3
4.9
This is exactly of the form of the score statistic for the components of variance model, except that σ̂e2
appears in place of σ̂ 2Y and r̂ appears in place of ρ̂ (cf. (3.5) ff.). However, the estimates for these
parameters are calculated under the condition α0 = 0, and in spite of the difference in notation the
parameters, hence the estimates, are the same under that hypothesis. Thus the score statistics for the twoallele model with normal residuals and for our components of variance model are the same statistic. This
provides further evidence of the robustness of our components of variance test.
7. D ISCUSSION
For an oligogenic model with normally distributed phenotypic data, we have introduced a parametrization that makes the ‘linkage’ parameters orthogonal to the ‘segregation’ parameters and hence allows us to
compute explicitly score statistics, Fisher information matrices, and noncentrality parameters in a number
of important special cases.
We have evaluated the asymptotic noncentrality parameter for sibships of arbitrary size, which
suggests what others
have observed as a result of simulations, that a sibship of size s can be roughly
as powerful as 2s independent sib pairs. Our more precise analysis shows that this assessment is overly
optimistic when s and α0 are large, but large sibships are, nevertheless, extremely valuable even in these
cases.
160
H.-K. TANG AND D. S IEGMUND
We have evaluated the power of genotyping a selected subset of sibships defined by their phenotypes,
which we select from a large random sample of sibships. The relative value of selective genotyping
decreases rapidly as the sibship size increases.
The Haseman–Elston regression statistic (Haseman and Elston, 1972) can be derived as a special
case of the calculations of Section 3. One ignores S1 , . . . , S N and starts from the likelihood function for
D1 , . . . , D N , then uses the robust version of the score statistic suggested in (3.12). Compared to the fully
efficient likelihood analysis, for moderate phenotypic correlation (0.25) between sibs, Haseman–Elston
regression is about 75% efficient when the mode of inheritance is additive. The modified Haseman–Elston
statistic (Elston et al., 2000) is more (less) efficient than the classical for small (large) correlation between
sibs. It also is about 75% efficient for moderate correlation. See also Teng (1996) and Wright (1997).
In sibships the dominance component of variance contributes to the noncentrality of the score statistic
designed to detect an additive component. Consequently, even if there is a large dominance component,
but we model only the additive components, our loss of efficiency is usually relatively modest. Even for
rare recessively acting alleles of relatively large effect, the loss rarely exceeds 10–20% of the sample size.
Based on a conditioning argument, we have suggested a modified statistic, which is nonparametric
under the hypothesis of no linkage and can be expected to be robust to moderate departures from normality.
We have briefly discussed robustness against a true model involving a (biallelic) major gene. It appears
that our model is robust if the allelic substitutions have small phenotypic effect, or modest effect but small
dominance deviation. An interesting case deserving more careful attention involves a major gene having
rare alleles of large effect.
We expect to discuss gene–gene and gene–environment interactions in a future paper.
ACKNOWLEDGEMENTS
This research was partly supported by NIH Grant HG-00848. The authors thank two referees and the
associate editor for their thoughtful suggestions.
A PPENDIX : BETTER APPROXIMATIONS FOR SIBSHIPS OF SIZE s
Because of the dependence among the νi j (which are pairwise independent), the null distribution of
α is skewed when the number of siblings is s 3. To deal with a similar problem involving qualitative
traits Tu and Siegmund (1999) suggested a p-value approximation that uses the third moment to correct
for skewness. Let −β be the one-sided derivative at 0 of Cov(Z 0 , Z t ), γ be N 1/2 times the third moment
of Z t under the hypothesis of no linkage, θ = [−1 + (1 + 2bγ /N 1/2 )1/2 ]/γ , and ν = ν[b(2β)1/2 ]
the special function defined by Siegmund (1985, p. 82). The following is a slight modification of the
approximation of Tu and Siegmund (1999): for a chromosome of length L with markers equally spaced
at distance ,
P0 { max Z i > b}
0i<L
−1/2
≈ [2π(1 + γ θ )]
{1/θ N 1/2 + νβ Lb} exp[−N θ 2 (1 + 2γ θ/3)/2].
Substantial calculation shows that the value of γ equals
[3/2] 3s {[1 + (s − 2)ρ]3 + (3s − 10)ρ 3 + 3ρ 2 }
.
{[ 2s /2][(1 + (s − 2)ρ)2 + ρ 2 ]}3/2
As a function of ρ this ratio is practically constant, so in evaluating the thresholds in Table 2 we have used
the value for ρ = 0.
Mapping quantitative trait loci in oligogenic models
161
To determine the sample sizes in Table 2, we have used suitable versions of the power approximations
provided by Feingold et al. (1993):
P(max Z k > b) ≈ 1 − ((b − ξ )/σ0 )
2ν
ν2
,
+ ϕ((b − ξ )/σ0 )
−
bσ0 /σ12 + (ξ − b)/σ0
2bσ0 /σ12 + (ξ − b)/σ0
where ν = ν[b(2β)1/2 /σ1 ]. This approximation, which is valid when there is a marker at the QTL τ ,
involves (i) the probability that the statistic Z τ is above the detection threshold b (or that the statistic
exceeds b at one or both of the two markers flanking τ in the case that τ is not itself a marker) and (ii) the
probability that the statistic is below the threshold at τ but Z t b at some nearby marker t. To implement
this approximation we require the mean and variance (σ02 ) of Z τ and the conditional mean and variance
(σ12 ) of Z t − Z τ given Z τ . See Tang (2000) for details.
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[Received March 6, 2000; revised June 28, 2000; accepted for publication June 29, 2000]