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Retrospective analysis of haplotype-based case–control studies

under a flexible model for gene–environment association

YI-HAU CHEN,

Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, People’s Republic of China

NILANJAN CHATTERJEE*, and

Biostatistics Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, 6120

Executive Boulevard, EPS 8038, Rockville, MD 20852, USA, chattern@mail.nih.gov

RAYMOND J. CARROLL

Department of Statistics, Texas A&M University, TAMU 3143, College Station, TX 77843-3143, USA

Summary

Genetic epidemiologic studies often involve investigation of the association of a disease with a

genomic region in terms of the underlying haplotypes, that is the combination of alleles at multiple

loci along homologous chromosomes. In this article, we consider the problem of estimating

haplotype–environment interactions from case–control studies when some of the environmental

exposures themselves may be influenced by genetic susceptibility. We specify the distribution of the

diplotypes (haplotype pair) given environmental exposures for the underlying population based on

a novel semiparametric model that allows haplotypes to be potentially related with environmental

exposures, while allowing the marginal distribution of the diplotypes to maintain certain population

genetics constraints such as Hardy–Weinberg equilibrium. The marginal distribution of the

environmental exposures is allowed to remain completely nonparametric. We develop a

semiparametric estimating equation methodology and related asymptotic theory for estimation of the

disease odds ratios associated with the haplotypes, environmental exposures, and their interactions,

parameters that characterize haplotype–environment associations and the marginal haplotype

frequencies. The problem of phase ambiguity of genotype data is handled using a suitable

expectation–maximization algorithm. We study the finite-sample performance of the proposed

methodology using simulated data. An application of the methodology is illustrated using a case–

control study of colorectal adenoma, designed to investigate how the smoking-related risk of

colorectal adenoma can be modified by “NAT2,” a smoking-metabolism gene that may potentially

influence susceptibility to smoking itself.

Keywords

Case-control studies; EM algorithm; Gene-environment interactions; Haplotype; Semiparametric

methods

1. Introduction

Genetic epidemiologic studies often involve investigation of the association between a disease

and a candidate genomic region of biologic interest. Typically, in such studies, genotype

*To whom correspondence should be addressed.

Conflict of Interest: None declared.

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Published in final edited form as:

Biostatistics. 2008 January ; 9(1): 81–99. doi:10.1093/biostatistics/kxm011.

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information is obtained on multiple loci that are known to harbor genetic variations within the

region of interest. An increasingly popular approach for analysis of such multilocus genetic

data are haplotype-based regression methods, where the effect of a genomic region on disease

risk is modeled through “haplotypes,” the combinations of alleles (gene variants) at multiple

loci along individual homologous chromosomes. It is believed that association analysis based

on haplotypes, which can efficiently capture inter-loci interactions as well as “indirect

association” due to “linkage disequilibrium” of the haplotypes with unobserved causal variant

(s), can be more powerful than more traditional locus-by-locus methods (Schaid, 2004).

A technical problem for haplotype-based regression analysis is that in traditional epidemiologic

studies, the haplotype information for the study subjects is not directly observable. Instead,

locus-specific genotype data are observed, which contain information on the pair of alleles a

subject carries on his/her pair of homologous chromosomes at each of the individual loci but

does not provide the “phase information,” that is which combinations of alleles appear across

multiple loci along the individual chromosomes. In general, the genotype data of a subject will

be phase ambiguous whenever the subject is heterozygous at 2 or more loci. Statistically, the

lack of phase information can be viewed as a special missing data problem.

Recently, a variety of methods have been developed for haplotype-based analysis of case–

control data using the logistic regression model (Zhao and others, 2003; Lake and others,

2003; Epstein and Satten, 2003; Satten and Epstein, 2004; Spinka and others, 2005; Lin and

Zeng, 2006; Chatterjee and others, 2006). Two classes of methods, namely, “prospective” and

“retrospective” have evolved. Prospective methods ignore the retrospective nature of the case–

control design. In the classical setting, without any missing data, justification of prospective

analysis of case–control data relies on the well-known result about the equivalence of

prospective and retrospective likelihoods under a semiparametric model that allows the

distribution of the underlying covariates to remain completely nonparametric (Andersen,

1970; Prentice and Pyke, 1979). Even with missing data, the equivalence of the prospective

and retrospective likelihood may hold, provided the covariate distribution is allowed to remain

unrestricted (Roeder and others, 1996). For haplotype-based genetic analysis, however,

complete nonparametric treatment of the covariates, including haplotypes, may not be possible

due to intrinsic identifiability issues for the phase-ambiguous genotype data (Epstein and

Satten, 2003). Thus, in this setting, the proper retrospective analysis of case–control data

requires special attention.

An attractive feature of the retrospective likelihood is that it can enhance efficiency of case–

control analysis by directly incorporating certain type of covariate distributional constraints

that are natural for genetic epidemiologic studies. The assumptions of Hardy–Weinberg

equilibrium (HWE) and gene–environment independence are 2 prime examples of such

constraints. The HWE model, which specifies simple relationships between “allele” and

“genotype” frequencies at a given chromosomal locus or between haplotype and diplotype

(pair of haplotypes on homologous chromosomes) frequencies across multiple loci, is a natural

law for a random mating large stable population. Often, it is also natural to assume that a

subject’s genetic susceptibility, a factor which is determined at birth, is independent of his/her

subsequent environmental exposures. However, if these assumptions are violated in some

situations, then retrospective methods can produce serious bias in odds ratio estimates (see,

e.g. Satten and Epstein, 2004; Chatterjee and Carroll, 2005; Spinka and others, 2005). Thus,

there is a need for alternative flexible models for specifying the joint distribution of genetic

and environmental covariates that could be used to assess the sensitivity of the retrospective

methods to underlying assumptions as well as to develop alternative robust methods.

Both Satten and Epstein (2004) and Lin and Zeng (2006) have described retrospective

maximum likelihood analysis of case–control data under flexible population genetics models

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that can relax the HWE assumption. Moreover, Lin and Zeng considered a model that allows

the conditional distribution of environmental exposure given unphased genotypes to remain

completely nonparametric, but they assumed conditional independence between haplotypes

and the environmental factors given the unphased genotypes. If, however, haplotypes are the

underlying biologic units through which a mechanism of gene is determined, then it is more

natural to allow for direct association between haplotypes and environmental exposures.

Moreover, if such association could exist, then quantifying the association between haplotypes

and certain type of environmental exposures, such as lifestyle and behaviorial factors, would

be of scientific interest.

In this article, we propose methods for retrospective analysis of case–control data using a novel

model for the gene–environment distribution that can account for direct association between

haplotypes and environmental exposures. The model is developed in Section 2. We assume a

standard logistic regression model to specify the disease risk conditional on diplotypes and

environmental exposures. In addition, we assume a polytomous logistic regression model for

specifying the population distribution of the diplotypes conditional on the environmental

exposures, with the intercept parameters of the model specified in such a way that the

“marginal” distribution of the diplotypes can follow certain population genetic constraints such

as HWE. Moreover, by exploiting the equivalence of prospective and retrospective odds ratios

under the polytomous regression model, we further incorporate certain constraints on the

diplotype-exposure odds ratio parameters that could reflect specific “mode of effects” for the

haplotypes. We allow the marginal distribution of the environmental exposure to remain

completely nonparametric.

Under the proposed modeling framework, we then describe in Section 3 a “semiparametric”

estimating equation method for inference about the finite-dimensional parameters of interest,

namely the disease odds ratios, haplotype frequencies, and haplotype-exposure odds ratios.

We develop a suitable expectation-maximization (EM) algorithm to account for the phase-

ambiguity problem. We study asymptotic theory of the proposed estimator under the

underlying semiparametric setting.

In Section 4, we assess the finite-sample performance of the proposed estimator based on case–

control data that were simulated utilizing haplotype patterns and frequencies obtained from a

real study. In Section 5, we apply the proposed methodology to a case–control study of

colorectal adenoma to investigate whether certain haplotypes in the smoking metabolism gene,

NAT2, could modify smoking-related risk of colorectal adenoma and whether the same

haplotypes could influence an individual’s susceptibility to smoking as well. Section 6 contains

concluding remarks. All technical details are in an appendix. A SAS macro is available from

the Web site http://www.stat.sinica.edu.tw/yhchen/download.htm.

2. Notations and proposed model

For haplotype-based studies, the underlying genetic covariate for a subject is defined by

“diplotypes,” that is, the 2 haplotypes the individual carries in his/her pair of homologous

chromosomes, where each haplotype is the combination of alleles at the loci of interest along

an individual chromosome. Following the notation developed in Spinka and others (2005), let

the diplotype status for a subject be Hdi = (H1, H2), where H1 and H2 denote the constituent

haplotypes. We assume that there are J possible haplotypes indexed by hj for j = 1, …, J. The

diplotypes are then indexed by , j1 = 1, … j1, j2 = 1, …, j2. The diplotype data,

however, is not directly observable. Instead, for each subject, the multilocus genotype data

G is observed, which contains information on the pair of alleles the individual carries at each

individual locus but does not provide the phase information, that is which combination of alleles

appears along each of the individual chromosomes. Thus, the same genotype data G could be

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consistent with multiple diplotypes. We will denote (G) to be the set of all possible diplotypes

that are consistent with the genotype data G.

Given the diplotype data Hdi and a set of environmental covariate X, we assume that the risk

of the disease is given by the logistic regression model

(2.1)

for some known function m(·, β1). Often one further imposes structural assumptions on the

odds ratio parameters β1 by modeling the effect of the diplotypes through constituent

haplotypes according to a “dominant,” “additive,” or “recessive” mode of effect (Wallenstein

and others, 1998). For example, a logistic regression model which assumes an additive effect

for each copy of a haplotype corresponds to

(2.2)

where βX is the main effect of X, βhjk is the main effect of haplotype hjk, k = 1, 2, and βhjk X is

the interaction effect of X with haplotype hjk, k = 1, 2. Such modeling may be necessary due

to identifiability considerations (Epstein and Satten, 2003) and is desirable when the effects of

the haplotypes themselves are of direct scientific interest.

Unlike Spinka and others (2005), who assumed independence of Hdi and X, we assume a

general polytomous logistic regression for the conditional distribution of Hdi given X:

(2.3)

where

between Hdi and X through the regression parameters γ1j1j2. Let γ0 and γ1 denote the vectorized

forms for the parameters γ0 j1j2 and γ1j1j2. Let qhap(hdi|x, γ0, γ1) denote pr(Hdi = hdi|X = x) as

defined by model (2.3). We allow the marginal distribution of X, denoted by F(x), to remain

completely unspecified. If Hdi were directly observable, then, in principle, no further

assumptions are necessary, and one can estimate γ0 and γ1 together with the odds ratio

parameters of the disease risk using the profile likelihood approach developed by Chatterjee

and Carroll (2005). In the presence of phase ambiguity, however, the diplotypes being not

directly observable, further constraints on the parameters γ0 and γ1 are needed for the purpose

of identifiability. In the following, we show how certain natural genetic models can be used to

impose these constraints.

is a chosen reference diplotype. Observe that model (2.3) allows association

Given that genetic susceptibility may influence environmental exposures and not vice versa,

for causal interpretation of parameters it is more natural to consider a model for the

environmental exposures given the diplotypes. However, the odds ratios associated with the

distributions [X|H] and [H|X] being the same, the parameters in γ1 can be interpreted as

measures of “diplotype effects” on the distribution of exposure. Thus, it is natural to specify

the γ1 parameters according to certain mode of effects of the underlying haplotypes. For

example, assuming an additive effect for the haplotypes, one can write γ1 j1j2 = γ1, j1 + γ1, j2,

which allows the diplotype effects to be determined by a reduced set of “haplotype effect”

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parameters γ1, j; in this case, γ1 would denote the vectorized form for the parameters γ1, j.

Similarly, other commonly used models, such as dominant or recessive models, could be used

to impose natural constraints on the γ1 parameters in model (2.3). We also observe that the

parametric model (2.3), combined with the non-parametric distribution F(x), imposes a

semiparametric model on the distribution of [X|H] with a density

This class of semiparametric models includes the parametric submodel where X|Hdi = hdi

follows a multivariate normal distribution with mean μhdi and common variance–covariance

matrix Σ. In this case, it is easy to see that

shift in the mean of the distribution of X due to differences in the diplotypes.

, which is a measure of the

The parameter γ0 in model (2.3) defines the population diplotype frequencies for a baseline

value of the exposure X. It is common to use population genetics models, such as HWE, to

specify a relationship between diplotype and haplotype frequencies. However, observe that if

the diplotypes can influence certain environmental exposures, then the frequencies of the

diplotypes within exposure categories may not follow the HWE constraints although the

underlying population, as a whole, may be in HWE. Thus, the population-level marginal

haplotype-pair distribution is assumed to follow HWE and is characterized by the parameters

θ = (θ2, …, θJ) so that

(2.4)

where h1 denotes the chosen reference haplotype and θ1 = 0. Let

be the marginal frequency for the diplotype hdi. Recall that in the proposed model, γ0 is defined

as an implicit function of γ1, θ, and F(x) through the relationship

(2.5)

Note that F is left unspecified, and hence the model propoised is semiparametric.

3. Semiparametric estimating equation inference

3.1 Estimation with known haplotypes

In what follows, where there can be no confusion, we will write h for hdi.

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Let ℋ(x) = exp(x)/{1+ exp(x)} be the logistic distribution function. Write the risk model

probability as

Recall that qhap(h|X, γ0, γ1) = pr(Hdi = h|X; γ0, γ1) is the conditional model of Hdi given X that

is specified as in (2.3).

To start with, consider the ideal case that the phase information is known so that Hdi is observed.

Since F is treated nonparametrically, assume that F is discrete and has mass δk at xk, k = 1, …,

K, where {x1, …, xK} are the distinct values of X that are observed in the case–control sample.

Let ndkh be the number of subjects in the sample with (D = d, X = xk, Hdi = h). Ignoring the

dependence of γ0 on F tentatively, the log-likelihood of the case–control data can then be

written as

Maximizing l with respect to δ for fixed values of ω = (β, γ0, γ1) then leads to

(3.1)

and the profile log-likelihood

(3.2)

where

and

with ℬ = (β0, β1, κ)T and κ = β0 + log(n1/n0) − log{pr(D = 1)/pr(D = 0)}. The calculation is

similar to that in Chatterjee and Carroll (2005).

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As noted by Chatterjee and Carroll (2005), the parameter β0 is separable from κ and hence is

theoretically identifiable. In practice, however, there is usually little information about β0

available in the observed data, and hence the information matrix is nearly singular. One way

to bypass this problem is to use external information on the disease prevalence pr(D = 1), while

another way is to use the rare-disease approximation when the disease is rare. The estimation

method described below can be applied to both the 2 cases of pr(D = 1) being known and the

rare-disease approximation being made, with suitable definitions on ℬ and S(d, h, x, ℬ, γ0,

γ1). When pr(D = 1) is known, κ depends on β0 only, hence here we define ℬ = (β0, β1)T. When

the disease is rare so that

(3.3)

we have

Note that β0 does not appear in this expression, and hence we define ℬ = (κ, β1)T.

Our goal is to estimate the parameters (ℬ, θ, γ1) based on the profile log-likelihood (3.2), where

γ0 is defined as an implicit function of (θ, γ1, F) through (2.5), and we write γ0 = (θ, γ1, F).

Let Ω = (ℬ, γ1, γ0), Ω* = {ℬ, γ1, (θ, γ1, F)}, and Φ = (ℬ, γ1, θ). Let ℒΩ(·) and ℒΦ(·) be,

respectively, the derivatives of ℒ(·) with respect to Ω and Φ, and θ and γ1 the derivatives

of (·)with respect to and θ and γ1 We then have

Explicit expressions for γ1 and θ; are given in Appendix C. Also, the information matrix is

given by

where ℐΩΩ = E(−ℒΩΩ), with ℒΩΩ the second derivative of ℒ with respect to Ω; note that the

terms involving second derivatives of Ω* do not appear in the information matrix because E

(ℒΩ) = 0, which is a direct consequence of the Lemmas A.1 and A.2 in Appendix A. We

propose to obtain the estimate of Φ by solving the estimating equation

(3.4)

where we have substituted an estimate F̂ for F in (·); that is, for each fixed value of (θ, γ1),

we solve γ0 from

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(3.5)

One convenient choice of F̂ is the empirical estimate F̂emp, which is given by

for the case where pr(D = 1) is known, where F̂emp,1(x) and F̂emp,0(x) are the empirical

distributions of X in the case and control samples, and is given by F̂emp(x) = F̂emp,0(x) for the

case where the rare-disease assumption can be made. An alternative choice of F̂ (x) would be

the profile likelihood estimate (3.1). Numerical calculations not given here show that the latter

choice requires more computational efforts while yielding results very similar to those given

by the empirical estimate F̂emp.

3.2 Estimation with ambiguous haplotype data

Now, we turn to the more practical case where the haplotype data cannot be observed directly

and must be inferred from the unphased genotype data, that is, the haplotype information may

be subject to ambiguity. In this case, we apply an EM-like algorithm to the “complete data”

estimating equation (3.4). Let Gi denote the observed unphased genotype of subject i and

(Gi) the set of diplotypes that are consistent with Gi. When only Gi instead of

for each subject, we propose to obtain the estimate Φ̂ for Φ = (ℬ, γ1, θ) as the solution of the

weighted version of (3.4):

is observed

(3.6)

where using the short-hand notation that γ̂0 = (θ, γ1, F̂emp), the weights are given by

(3.7)

The limiting version of the weights is given as

(3.8)

Solving the estimating equation (3.6) can be implemented simply by an EM-like algorithm as

follows: starting with an initial value for Φ and hence an initial value for γ0, we

i.

calculate the weights {ω̂i} from (3.7) and

ii.

solve (3.6) to obtain an updated estimate of Φ using the weights {ω̂i} given in (i);

note that within this step we also need to solve (3.5) to obtain updated value of γ0.

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The algorithm is iterated between the 2 steps until convergence. Note that the weights {ω̂i} are

only used in solving Φ from (3.6) and are not required in solving γ0 from (3.5).

3.3 Asymptotic theory

Make the following series of definitions. Expectations denoted as Ecc(·) are taken under the

case–control sampling design, that is, for any random vector Y,

where μd = plim nd/n, d = 0, 1. Then, define

,

(3.9)

Note that the second derivative of Ω* does not appear in ℐ̄ since E(ℒ̄Φ) = E(ℒΦ) = 0, and the

last identity in (3.9) is given by Lemma A.3 in Appendix A.

Define p̂emp(Di) to be the mass of F̂emp(Xi), which is equal to

(D = d) is known and is equal to I (Di = 0)/n0 when the rare-disease approximation is used. Let

qhap(X, γ1, γ0) = {qhap(h|X, γ1, γ0)} be the vector collection over h of qhap(h|X, γ1, γ0) for all

diplotypes except the reference diplotype, and let qHWE(θ) be defined similarly. Define

if πd = pr

where ℐΩγ0 is the obvious submatrix of ℐΩΩ.

Theorem 3.1—Let

Suppose that Ecc{ℰ̄(·) ℰ̄T (·)} exists and the matrix ℐ̄ is invertible. Then, n1/2(Φ̂ − Φ) is

asymptotically normal with mean zero and covariance matrix

(3.10)

Remark 3.2—The asymptotic variance Γ̄ can be readily estimated by replacing each

component matrix with its empirical counterpart. Lemma A.3 gives useful expressions to

facilitate this computation.

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Remark 3.3—In our numerical experiments, the estimated covariance based on formula

(3.10) is very close to that based on the “naive” covariance estimate obtained by naively treating

the estimating equation (3.6) as a genuine score equation; namely, treating the F̂emp plugged

into (·) as the true covariate distribution F. In this case, by applying Proposition 1(ii) in

Chatterjee and Carroll (2005), the naive co-variance estimate can be obtained simply as the

empirical counterpart of the matrix ℐ̄−1 − ℐ̄−1Ψℐ−1, where

. Whether this naive estimate performs well in

general is unknown, and we suggest using the estimate based on (3.10).

4. Simulations

4.1 Finite-sample performance under correct model

In this section, we study the finite-sample performance of the proposed estimator using

simulated data generated under the proposed modeling framework. We simulated haplotypes

following published data (Epstein and Satten, 2003) on haplotype patterns and frequencies for

5 single-nucleotide polymorphisms (SNPs) in a putative susceptibility gene for diabetes (see

Table 1). The simulations involved a single environmental covariate X, assumed to follow a

standard normal distribution in the population. Given X, the diplotypes (haplotype pair) for an

individual were generated from a polytomous logistic regression of the form (2.3), where the

diplotype-specific odds ratios were further specified according to an additive model of the form

γ1j1j2= γ1, j1 + γ1, j2, where j1 and j2 denote the index for 14 haplotypes shown in Table 1. We

assume γ1,4 = γ1,5 = −0.4 and γ1,12 = 0.4, and all the other γ1, j = 0. The parameters γ0j1j2 ’s in

model (2.3) are then specified in such a way that the marginal diplotype distribution follow

HWE with haplotype frequencies given in Table 1.

For generating disease outcome, we chose the haplotype “01100” (j = 5) to be causal and used

the logistic model

where Z(5) denotes the number of the copies of the causal haplotype contained in Hdi. The true

value of the parameter vector (β0, βH, βX, βH X) was set to (−3.0, 0.2, 0.1, 0.3). A case–control

sample with 600 controls and 600 cases was then sampled. The results were based upon 1000

simulated data sets.

When analyzing the data, we only used the unphased genotype information. We did not assume

the causal haplotype to be known. Thus, in both the disease-risk model (2.1) and the diplotype-

frequency model (2.3), we choose the most common haplotype “10011” as a reference and

estimated a separate regression parameter for each of the non-referent haplotypes. Since rare

haplotypes may lead to unreliable estimates of the associated regression parameters, when

estimating β and γ1, rare haplotypes with frequency <1% are grouped into the reference

haplotype. The resulting 8-grouped haplotypes are labeled as h j′, j′ = 2, …, 8; see Table 1 for

details about how the haplotypes are grouped.

In each simulation, we obtain 2 sets of estimates from the proposed method, one using the rare-

disease approximation (3.3) and the other using the known value of the population disease

prevalence. Results shown in Table 2 show that both sets of estimates are essentially unbiased.

Also, the standard error estimates are in close agreement with the true values, and the coverage

probabilities are close to the nominal value (95%). As expected, the estimates for θ and γ1 are

generally more efficient using external information on the disease prevalence than when using

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the rare-disease approximation, but no such efficiency gain is observed for the parameters β

in the disease-risk model. Similar conclusion can be drawn from the simulations with a

Bernoulli covariate (success probability = 0.5), showing the applicability of the proposed

method to the categorical covariate. Detailed results for this latter set of simulations are

included in the supplementary material available at Biostatistics online.

4.2 Model robustness

Here, we consider a simulation study where we generate the data in such a way that the

polytomous model for diplotype frequencies may not exactly hold. The main goal is to give

an indication of the robustness of the estimate of the association parameters (β) from the

proposed method when the model for [Hdi|X] is misspecified.

Following the argument of causality in Section 2, if [X|Hdi] follows a normal distribution with

constant variance, then the polytomous model is exact. So, to show a modest violation of the

polytomous model, for given diplotype we generate the data on X from

where the diplotype data are again generated from the distribution in Table 1, ε is a t-distribution

(d.f. = 3) truncated at ±5, λ4 = λ5 = −1.2, λ12 = 1.2, and all the other λj are 0. The disease status

data are generated from the same logistic model as in the previous simulation. The simulated

data on 600 cases and 600 controls are then analyzed with the proposed method, where the

analysis models for the disease risk, [Hdi|X], and the marginal diplotype distribution are

specified the same as those in the previous simulation. As a comparison, we also fit to the

simulated data a model with the haplotype–environment (H-X) independence assumption, i.e.

pr(Hdi|X) = pr(Hdi) = qHWE(Hdi), using the method proposed by Spinka and others (2005). The

rare-disease approximation is made when applying both the 2 methods.

The results shown in Table 3 reveal that, for the estimation of the association parameters β, the

proposed method may be quite robust to modest misspecification of the model for [Hdi|X]. On

the other hand, the estimates from the H-X independence method does result in substantial

bias, especially for parameters corresponding to haplotypes for which [X|Hdi] have nonzero

mean; for example, the estimate for the interaction parameter between h5 and X is severely

biased with the H-X independence method. The estimates for the marginal haplotype-

frequency parameters θ seem to be robust to misspecification of [Hdi|X] for both the 2 methods.

5. Case–control study of colorectal adenoma study, NAT2 haplotype, and

smoking

We illustrate the proposed modeling and estimating methodologies with an application to a

case–control study of colorectal adenoma, a precursor of colorectal cancer. The study involved

628 prevalent advanced adenoma cases and 635 gender-matched controls, selected from the

screening arm of the Prostate, Lung, Colorectal and Ovarian Cancer Screening Trial at the

National Cancer Institute, USA (Gohagan and others, 2000; Moslehi and others, 2006). One

of the main objectives of this study is to assess whether smoking-related risk of colorectal

adenoma may be modified by certain haplotypes in NAT2, a gene known to be important in

metabolism of smoking-related carcinogens. In addition, since NAT2 is involved in the

smoking metabolism pathway, potentially it can influence an individual’s addiction to

smoking. Thus, it was also of interest to identify potential haplotypes that could influence an

individual’s susceptibility to smoking.

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Genotype data were available on 6 SNPs. We initially applied the EM algorithm proposed by

Li and others (2003) for haplotype-frequency estimation to derive 7 common haplotypes with

estimated frequency greater than 0.5%, which are then included in our association analysis

with the most frequent haplotype served as the reference haplotype. Subjects were categorized

as “never,” “former,” or “current” smokers. We fit a logistic regression model (2.1) assuming

an additive effect for each haplotype other than the reference one; see (2.2). The haplotype–

environment interaction terms include only those for the haplotype “101010” with “Smk1” and

“Smk2,” 2 dummy variables for former and never smokers, because they are the only promising

interactions according to preliminary analysis. The disease-risk model was further adjusted for

“age,” recorded in years, and “gender.” A polytomous logistic regression (2.3) is specified for

the conditional distribution of diplotypes given the environmental covariates Smk1 and Smk2

with the marginal diplotype distribution being specified by the HWE constraints. The main

parameters of interest include the disease-haplotype odds ratio parameters β1, the haplotype–

environment odds ratio parameters γ1, and the marginal haplotype frequencies in the whole

population. The marginal distribution for the environmental covariates is left unspecified. For

estimation of regression parameters β and γ1, we grouped haplotypes with frequency less than

2% into the reference haplotype. The rare-disease approximation was made in deriving the

estimating equation, and the EM algorithm proposed in Section 3 is utilized to accommodate

the unphased genotype data.

Results from this application are displayed in Table 4. It is clear that current smokers can have

significantly elevated risk for colorectal adenoma relative to nonsmokers, adjusting for gender

and age. Relative to the reference haplotype “001100,” all the other haplotypes are associated

with reduced risk for colorectal adenoma, but the statistical evidence is not significant.

However, the significance of the interaction 101010 × Smk2 suggests that smoking-related

risk of adenoma was much reduced for carriers of the haplotype 101010 than non-carriers. The

finding is consistent with previous laboratory and epidemiologic studies that have identified

the haplotype 101010, known as “NAT2*4,” as a rapid metabolizer for smoking-related

carcinogens. The estimates for the parameter γ1 for the conditional diplotype distribution reveal

that the susceptibility to smoking seems not to be influenced by any haplotypes we considered.

Finally, the estimates for the marginal haplotype frequencies derived from the estimates of θ

are quite close to those obtained by the EM algorithm of Li and others (2003) applied to the

genotype data of the controls.

To check if the analysis is sensitive to model specification for the conditional distribution of

diplotypes given the environmental covariates, we further fit the model (2.3) with various

choices of the environmental covariates. The results (not shown) for the association parameters

β and the marginal haplotype frequencies are fairly consistent across the analyses.

6. Concluding remarks

The model we have proposed for gene–environment association is suitable when the underlying

haplotypes of a genomic region may causally influence the environmental exposure(s) under

study. The model, however, requires special treatment for environmental factors, such as

ethnicity or geographic region(s), which may be associated with the genomic region under

study, not because of any causal relationship but merely due to population stratification.

Suppose, in addition to the main environmental exposure X, there is a set of environmental

factors S which could be used to divide the underlying population into K strata that are likely

to be genetically heterogenous. In such a situation, a natural model for describing the

association between diplotypes Hdi and environmental factors W = (X, S) is given by

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(6.1)

where the stratum-specific intercept parameters γ0j1j2 (S) should be specified in such a way

that the diplotype frequencies, after marginalized over X, follow population genetics

constraints, such as HWE, within each stratum defined by S. The disease-risk model could be

also extended to include S as a risk factor. The proposed estimating equation methodology can

be easily modified to estimate the gene–environment interaction and association parameters

of interest under these extended models.

Acknowledgements

Chen’s research was supported by the National Science Council of the People’s Republic of China (NSC 95-2118-

M-001-022-MY3). Chatterjee’s research was supported by the Intramural Research Program of the National Cancer

Institute. Carroll’s research was supported by a grant from the National Cancer Institute (CA57030) and by the Texas

A&M Center for Environmental and Rural Health via a grant from the National Institute of Environmental Health

Sciences (P30-ES09106). A SAS macro is available from the Web site

http://www.stat.sinica.edu.tw/yhchen/download.html.

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APPENDIX A: BASIC LEMMAS

The following lemmas are required to derive the asymptotic distribution of the proposed

estimator. Lemma A.1 below is in fact Lemma 3 of Chatterjee and Carroll (2005).

Lemma A.1

Under the case–control sampling design where the total sample size n = n1 + n0 tends to infinity

but the sampling proportions for the cases and controls, that is, n1/n and n0/n, remain fixed at

μ1 and μ0, we have for any measurable function M(D, Hdi, X) of data (D, Hdi, X),

where E*(·|X) denotes the expectation with respect to the joint distribution of (D, Hdi) given

X defined by

(A.1)

and λ(x) = Σd Σh, S(d, h, x, ℬ, γ1, γ0)

Lemma A.2 below provides an explicit expression for the estimating function ℒ̄Φ(·).

Lemma A.2

Write

Then

(A.2)

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Proof

By definition,

and direct calculation yields

which proves the result.

Lemma A.3 provides explicit forms for the information matrices.

Lemma A.3

Let ℐΩΩ = Ecc{−ℒΩΩ(D, Hdi, X, Ω)}, where ℒΩΩ is the second derivative of ℒ(·) with

respective to Ω and ℐΩΩ = Ecc{−∂ℒΩ(·)/∂ΩT}. Then

Proof

The first identity has been given in Lemma 4 of Chatterjee and Carroll (2005). To show the

second identity, applying the chain rule we have

(A.3)

The first term of (A.3) equals

By the definition of ω(h, Ω) given in (3.8), it easy to see that

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