Summary
Many methods have recently been proposed for efficient analysis of case-control studies of gene-environment interactions using a retrospective likelihood framework that exploits the natural assumption of gene-environment independence in the underlying population. However, for polygenic modelling of gene-environment interactions, which is a topic of increasing scientific interest, applications of retrospective methods have been limited due to a requirement in the literature for parametric modelling of the distribution of the genetic factors. We propose a general, computationally simple, semiparametric method for analysis of case-control studies that allows exploitation of the assumption of gene-environment independence without any further parametric modelling assumptions about the marginal distributions of any of the two sets of factors. The method relies on the key observation that an underlying efficient profile likelihood depends on the distribution of genetic factors only through certain expectation terms that can be evaluated empirically. We develop asymptotic inferential theory for the estimator and evaluate its numerical performance via simulation studies. An application of the method is presented.
Keywords: Case-control study, Gene-environment interaction, Genetic epidemiology, Pseudolikelihood, Retrospective study, Semiparametric method
1. Introduction
Recent genome-wide association studies indicate that complex diseases, such as cancers, diabetes and heart diseases, are in general extremely polygenic (Chatterjee et al., 2016; Fuchsberger et al., 2016). Genetic predisposition to a single disease may involve thousands of genetic variants; each of these may have a very small effect individually, but in combination they can explain substantial variation in risk in the underlying population. As discoveries from genome-wide association studies continue to enhance understanding of complex diseases, in the future it will be critical to elucidate how these genetic factors interact with environmental risk factors, in order to better understand disease mechanisms and to develop public health strategies for disease prevention.
Because of its sampling efficiency, the case-control design is widely popular for conducting studies of genetic associations and gene-environment interactions. A variety of analytical methods have been proposed to increase the efficiency of analysis of case-control data for studies of gene-environment interactions by exploiting an assumption of gene-environment independence in the underlying population. It has been shown that under the assumptions of gene-environment independence and rare disease, the interaction odds-ratio parameters of a logistic regression model can be estimated efficiently based on cases alone (Piegorsch et al., 1994). A general logistic regression model can be fitted to case-control data under the gene-environment independence assumption using a log-linear modelling framework (Umbach & Weinberg, 1997) or a semiparametric retrospective profile likelihood framework (Chatterjee & Carroll, 2005). More recently, the assumption of gene-environment independence has been exploited to propose a variety of powerful hypothesis testing methods for conducting genome-wide scans of gene-environment interactions (Mukherjee & Chatterjee, 2008; Murcray et al., 2009; Hsu et al., 2012; Mukherjee et al., 2012; Gauderman et al., 2013; Han et al., 2015).
We consider developing methods for efficient analysis of case-control studies for modelling gene-environment interactions that involve multiple genetic variants simultaneously. To develop parsimonious models for joint effects, many studies have focused on developing models for gene-environment interactions using underlying polygenic risk scores that could be defined by all known genetic variants associated with the disease (Meigs et al., 2008; Wacholder et al., 2010; Chatterjee et al., 2013; Dudbridge, 2013; Chatterjee et al., 2016). Further, to obtain improved biological insights and to enhance statistical power for detection, one may often wish to model gene-environment interactions using multiple variants within genomic regions and/or biologic pathways (Chatterjee et al., 2006; Jiao et al., 2013; Lin et al., 2013, 2015). In standard prospective logistic regression analysis, which conditions on both the genetic and the environmental risk factor status of the individuals, handling multiple genetic variants is relatively straightforward. In contrast, with so-called retrospective methods, which aim to exploit the assumption of gene-environment independence, the task becomes complicated because all currently existing methods require parametric modelling of the distribution of the genetic or environmental variables.
We propose a computationally simple method for fitting general logistic regression models to case-control data under the assumption of gene-environment independence, but without requiring any further modelling assumptions about the distributions of the genetic or environmental variables. We extend the Chatterjee–Carroll profile likelihood framework, which originally considered modelling gene-environment interactions using single genetic variants for which genotype status could be specified using parametric multinomial models. The new method relies on the observation that the profile likelihood itself can be estimated based on an empirical genotype distribution that is estimable from a case-control sample. We develop the asymptotic theory of the resulting estimator under a semiparametric inferential framework. Simulations and an example illustrate the properties of the new method.
2. Model, method and theory
2.1. Background, model and method
In the following, we use notation similar to that in Chatterjee & Carroll (2005). We will denote disease status, genetic
information and environmental risk factors by 
,
and 
,
respectively. Here 
may correspond to a complex multivariate
genotype associated with multiple genetic variants or to a continuous polygenic risk score
that is defined a priori based on known associations of the genetic variants with the
disease. We assume that the risk of the disease given genetic and environmental factors in
the underlying population can be specified using a model of the form
 |
(1) |
where 
is the logistic
distribution function and 
is a parametrically specified
function that defines a model for the joint effect of 
and
on the logistic-risk scale. The goal of the
gene-environment interaction study is to make inference on the parameters
in (1), including interaction parameters.
Let 
denote the joint distribution of
and 
in the
underlying population. The key assumption that genetic factors, 
, and
environmental factors, 
, are independently distributed in the
underlying population can be mathematically stated as
 |
where 
and 
denote the underlying marginal distributions of 
and
, respectively. In the Supplementary Material we discuss
how to weaken this assumption by suitable conditioning on additional stratification
factors. In contrast to the existing literature, here we assume that the marginal
distributions 
and 
are both completely unspecified.
We consider a population-based case-control study, in which 
are
sampled independently from individuals with the disease, called cases, and those without
the disease, called controls. Suppose there are 
cases and
controls. Standard prospective logistic
regression analysis, which is equivalent to maximum likelihood estimation when
is allowed to be completely
unspecified, yields consistent estimates of 
(Prentice & Pyke, 1979).
The retrospective likelihood is the probability of observing the genetic and environmental variables, given the subject’s disease status. Under gene-environment independence in the underlying population, the retrospective likelihood is
 |
Let 
and 
represent the density or mass
functions of 
and 
,
respectively. The retrospective likelihood is
 |
(2) |
Chatterjee & Carroll (2005) profiled out
by treating it as discrete on
the set of distinct observed values 
of
with probabilities
, and then
maximizing (2) over
, leading
eventually to the semiparametric profile likelihood described as follows. Define
,
where 
is defined
as the probability of the disease in the underlying population. Define
.
Also let
 |
Then, with this notation, the semiparametric profile likelihood is
 |
(3) |
While the representation in (3) does
not involve the unknown density of 
, it does involve the
unknown density of 
. This is a major reason that methods in
the current literature specify a parametric distribution for 
. Our
aim in this paper is to dispense with the need to give a parametric form for the
distribution function of 
, so that analysis can be performed with
respect to potentially complex multivariate genotype data for which parametric modelling
can be difficult and cumbersome.
Here is our key insight, which we discuss first in the context that
is known or at least can be estimated
well. For case-control studies that are conducted within well-defined populations,
relevant probabilities of the disease can be ascertained using population-based disease
registries. When case-control studies are conducted by the sampling of subjects within a
larger cohort study, the probability of the disease in the underlying population can be
estimated using the disease incidence rate observed in the cohort.
Our key insight in treating the distribution of 
as
nonparametric concerns the term in the denominator of (3), defined as
 |
This is simply the expectation, in the source population, of 
; that is,
,
where the subscript pop emphasizes that the expectation is in the source population, not
in the case-control study. However, crucially,
 |
(4) |
Of course, 
is unknown, but we estimate it
unbiasedly and nonparametrically by
 |
(5) |
In the Supplementary Material,
we show that 
is an unbiased estimate
of 
which is
-consistent, and that it is
asymptotically normally distributed.
Ignoring the leading term 
in (3), which is not estimated, and taking logarithms leads us to an
estimated loglikelihood in 
across the data as
 |
(6) |
Define 
and similarly for 
. Then the
estimated score function, a type of estimated estimating equation, is
 |
(7) |
Define
 |
which is the profile loglikelihood score function when the
distribution of 
is known. Since the profile loglikelihood
score of Chatterjee & Carroll (2005) would have
mean zero if the distribution of 
were known, it follows
that
 |
(8) |
where the expectation in (8) is taken in the case-control study, not in the source population. Thus, since
and
converge in
probability to 
and 
, respectively, a
consistent estimate of 
can be obtained by solving
. This estimate
, which maximizes the
semiparametric pseudolikelihood (6),
will be referred to as the semiparametric pseudolikelihood estimator.
2.2. Rare diseases when 
is unknown
When the probability of disease in the source population is unknown, one can invoke a
rare disease assumption which is often reasonable for case-control studies (Piegorsch et al., 1994; Modan et al., 2001; Epstein & Satten,
2003; Zhao et al., 2003; Lin & Zeng, 2006; Kwee et al., 2007). If we assume that 
, then
,
and the expectation involved in the calculation of 
can be evaluated based on only
the sample of controls, with 
. In this case, the estimates of
converge not to
itself but to
, the solution to (8) with 
.
Typically, except when the sample size is very large and hence standard errors are
unusually small, the small possible bias of the rare disease approximation is of little
consequence and coverage probabilities of confidence intervals remain near nominal; see §
3 for examples. The asymptotic theory of § 2.3 below is then unchanged.
In the Supplementary Material, we show that the score and the Hessian take simple forms in this case, and that the Hessian is negative semidefinite. Computation is thus very efficient.
2.3. Asymptotic theory
To state the asymptotic results, we first make the definitions
 |
In addition, define 
, 
,
and
.
We use the notational convention that for arbitrary functions 
,
.
Also, we use the convention that
 |
Define
 |
Finally, define 
.
Theorem 1.
Suppose that
, where
, and that
is known. Then
Therefore, since the
are independent and
, as
,
in distribution, where
In § 2.2, when 
is
unknown and the disease is relatively rare, the same result holds upon setting
.
3. Simulations
3.1. Overview
In our simulations, 
and the value of 
is binary with population frequency
. There are either three or five
correlated single nucleotide polymorphisms within a region; we report on the latter case,
but the results for the former case are similar. Each single nucleotide polymorphism takes
on the values 0, 1 or 2 following a trinomial distribution that follows the Hardy–Weinberg
equilibrium, i.e., the 
th component of 
equals
0, 1, 2 with probabilities 
,
respectively. The values of the 
are described
below.
To generate correlation among the single nucleotide polymorphisms, we first generated a
3- or 5-variate multivariate normal variate, with mean 0 and standard deviation 1, and a
correlation matrix with correlation between the 
th and
th components being
, where
. After generating these
random variables, we trichotomized them with appropriate thresholds so that the
frequencies of 0, 1 and 2 matched those specified by the allele frequency
and Hardy–Weinberg equilibrium.
In both simulations, the logistic intercept 
was chosen so
that the population disease rate 
.
However additional simulations with 
yielded very similar results in terms of coverage, efficiency gains, and unbiasedness. See
§ 3.3 and the Supplementary Material for a
discussion of additional simulations. In the simulation reported here,
,
,
and 
.
Here 
.
3.2. Results
The standard error estimators used in our simulation were based on the asymptotic theory described in Theorem 1; we also used the bootstrap and obtained very similar results. The appropriate bootstrap in a case-control study is to resample the cases and controls separately, thus maintaining the sample sizes for each.
The simulation results are presented in Table 1.
Our semiparametric pseudolikelihood estimator shows little bias and has coverage
percentages near the nominal level. Both with a rare disease approximation and with
known, our semiparametric
pseudolikelihood estimator achieves approximately a 25% increase in mean squared error
efficiency over ordinary logistic regression for the main effects in both
and 
.
Table 1.
Results of 
simulations as described
in
3: mean bias, coverage probabilities
of a 
nominal confidence interval,
and mean squared error efficiency of our semiparametric pseudolikelihood estimator
compared with ordinary logistic regression; the simulationsc were performed with
cases and
controls
|
|
|
|
|
|
|
|
|
|
|
|
|---|---|---|---|---|---|---|---|---|---|---|---|
| True | 0.18 | 0.18 | 0.00 | 0.18 | 0.00 | 0.41 | 0.26 | 0.00 | 0.00 | 0.26 | 0.00 |
| Logistic: 1000 cases | |||||||||||
| Bias | 0.00 | 0.01 | 0.00 | 0.01 | –0.01 | 0.01 | 0.01 | –0.01 | 0.00 | 0.00 | 0.01 |
| CI (%) | 94.3 | 95.2 | 95.7 | 95.1 | 94.7 | 94.6 | 94.9 | 94.2 | 94.5 | 96.0 | 94.2 |
| SPMLE Rare: 1000 cases | |||||||||||
| Bias | 0.01 | 0.00 | 0.00 | 0.02 | –0.01 | 0.02 | –0.02 | –0.01 | 0.01 | –0.02 | 0.01 |
| CI (%) | 95.2 | 95.4 | 96.4 | 95.8 | 95.3 | 95.1 | 95.4 | 94.8 | 96.1 | 95.5 | 94.9 |
| Avg MSE Eff | All  :
1 28 |
: 1 26 |
All  :
2 18 |
||||||||
SPMLE  known:
1000 cases | |||||||||||
| Bias | 0.00 | 0.00 | 0.00 | 0.01 | –0.01 | 0.01 | 0.00 | –0.01 | 0.01 | –0.01 | 0.01 |
| CI (%) | 95.1 | 95.5 | 96.4 | 95.8 | 95.0 | 95.5 | 95.6 | 94.6 | 95.9 | 95.2 | 94.5 |
| Avg MSE Eff | All  :
1 28 |
All  :
1 28 |
All  :
2 07 |
||||||||
Logistic, ordinary logistic regression; SPMLE Rare, our estimator using the rare
disease approximation with unknown 
(§ 2.2); SPMLE 
known, our estimator when
is known in the source
population (§ 2.1); CI, coverage of a
nominal 95% confidence interval, calculated using the asymptotic standard error;
Avg MSE Eff, mean squared error efficiency of our method compared to logistic
regression averaged over 
(All
), over 
(All 
) or over all
interactions (All
).
Strikingly, the mean squared error efficiency of our semiparametric pseudolikelihood
estimators compared to ordinary logistic regression is approximately
for all the interaction terms,
thus demonstrating that our methods, which do not model the distribution of either
or 
,
achieve numerically significant increases in efficiency.
3.3. Additional simulations
The Supplementary Material
presents a series of additional simulations. These include the results of a simulation to
evaluate the robustness of our method with respect to misspecification of the population
disease rate; we found a surprising robustness with respect to disease rate
misspecification. Additionally, we performed simulations to examine the robustness of our
method with respect to violations of the gene-environment independence assumption. Those
simulation studies show that there will be bias in the estimates of gene-environment
interaction parameters for the specific single nucleotide polymorphisms that violate
gene-environment independence, but the average mean squared error for parameter estimates
across all the different single nucleotide polymorphisms could still be substantially
lower than that obtained from prospective logistic regression analysis. We also show in
the Supplementary Material how
to remove this bias when 
and 
are
independent conditional on a discrete stratification variable. Mukherjee & Chatterjee (2008) and Chen et al. (2009) show how to use empirical Bayes methods to provide additional
robustness with respect to violations of the gene-environment independence assumption.
4. Data analysis
In this section, we apply our method to a case-control study for breast cancer arising from
a large prospective cohort at the National Cancer Institute: the Prostate, Lung, Colorectal
and Ovarian cancer screening trial (Canzian et al.,
2010). The design of this study is described in detail by Prorok et al. (2000) and Hayes et al.
(2000). The cohort data consisted of 
women, of whom
3
56% developed breast cancer (Pfeiffer et al., 2013). The case-control study analysed
here consists of 753 controls and 658 cases. Although 
is
known in this population, we analyse the data both with 
known and with 
unknown but using a rare disease
approximation.
We had data available on genotypes for 21 single nucleotide polymorphisms that have been
previously associated with breast cancer based on large genome-wide association studies. The
polygenic risk score was defined by a weighted combination of the genotypes, with the
weights defined by log-odds-ratio coefficients reported in prior studies. We examined the
interaction of the polygenic risk score with age at menarche, 
, a known
risk factor for breast cancer, defined as the binary indicator of whether the age at
menarche exceeds 13 or not. We also adjust the model for age as a continuous variable,
denoted here by 
, so that the model fitted is
 |
Results when age was categorized as 
35, 35–40, 40–45,
75 were similar.
We also performed analyses to check the gene-environment independence assumption. Since
is binary, we ran a 
-test of
the polygenic risk score against the levels of 
, of course among the
controls only. The 
-value was 0
91,
indicating almost no genetic effect. We also ran chi-squared tests for the 21 individual
genes, finding no significant association after controlling the false discovery rate: the
minimum 
-value was 0
09.
In addition, we checked for correlation, known as linkage disequilibrium, between the 21
loci used to create the polygenic risk score and 32 loci that are known to influence age at
menarche (Elks et al., 2010). The data available to
us do not contain the necessary information to analyse linkage disequilibrium between the
two sets of loci.
Using phased haplotypes from subjects of European descent from the 1000 Genomes Project
(The 1000 Genomes Project Consortium, 2015) and
HapMap (Gibbs et al., 2003), no evidence of linkage
disequilibrium was found: the maximum 
was
0
1 and the minimum
-value was 0
85.
Finally, a 2014 study examined the relationship between age at menarche and 10 of the 21
single nucleotide polymorphisms used to create our polygenic risk score, none of which were
found to influence age at menarche (Andersen et al.,
2014).
Table 2 presents the results for the cases where
is unknown and known; as remarked upon
previously, the results are very similar. Because of the very different scales of the
variables, to provide a basis for comparison the variable age at baseline was standardized
to have mean zero and standard deviation one. In addition, we standardized some of the
coefficient estimates so that 
was multiplied by the standard
deviation of the polygenic risk score, and 
was multiplied
by the standard deviation of 
times the polygenic risk score.
Table 2.
Results of the analysis of the Prostate, Lung, Colorectal and Ovarian cancer screening trial data
|
|
|
|
|
|---|---|---|---|---|
| Logistic | ||||
| Estimate | 0 018 |
0 297 |
0 165 |
0 124 |
| Std err | 0 054 |
0 064 |
0 132 |
0 068 |
-value |
|
|
|
|
| SPMLE Rare | ||||
| Estimate | 0 024 |
0 321 |
0 175 |
0 138 |
| Std err (asymptotic) | 0 054 |
0 067 |
0 134 |
0 055 |
-value (asymptotic) |
|
|
|
|
SPMLE  known |
||||
| Estimate | 0 022 |
0 313 |
0 174 |
0 141 |
| Std err (asymptotic) | 0 054 |
0 065 |
0 133 |
0 055 |
-value (asymptotic) |
|
|
|
|
Logistic, ordinary logistic regression; SPMLE Rare, our method using the rare
disease approximation with unknown 
; SPMLE
known, our method when the
disease rate is known in the source population (
); Std err, the
asymptotic standard error estimate; 
, the main
effect for age; 
and
, the main effects for the
polygenic risk score (
) and the environmental variable
(age at menarche
13), respectively;
, the gene-environment
interaction.
As expected from the known association of the single nucleotide polymorphisms with risk of
breast cancer, the polygenic risk score was strongly associated with breast cancer status of
the women in the study. Standard logistic regression analysis reveals some evidence for
interaction of the polygenic risk score with age at menarche, but the result was not
statistically significant at the 0
05 level. When the
analysis was done under the gene-environment independence assumption, the evidence for
interaction appeared to be stronger.
The coefficient estimate for the interaction term is slightly larger for our semiparametric methods than for logistic regression. Also, the asymptotic standard error estimate of logistic regression is approximately 23% larger than that for our methods, indicating a variance increase of approximately 50%. Although not listed here, the bootstrap mentioned in § 3.2 has very similar standard error estimates. In that bootstrap, 33% of the time the logistic interaction estimate was actually greater than that of the disease-rate-known estimate.
5. Discussion and extensions
We have proposed a general method for using retrospective likelihoods to study gene-environment interactions involving multiple markers, an approach that does not require any distributional assumption on the multivariate genotype distribution. Sometimes, one may consider modelling multimarker gene-environment interactions using an underlying polygenic risk score, which is a weighted combination of numerous genetic markers where the weights are predetermined from previous association studies. In such situations, the polygenic risk score might be assumed to follow approximately a normal distribution in the underlying population, and the profile likelihood method of Chatterjee & Carroll (2005) can be used with appropriate modification by replacing the parametric multinomial distribution for a single nucleotide polymorphism genotype with a parametric normal distribution for the polygenic risk score; see also Chen et al. (2008) and Lin & Zeng (2009). In general, however, if one wishes to explore complex models for multivariate gene-environment interactions retaining separate parameters for distinct single nucleotide polymorphisms or for distinct genetic profiles defined by combinations of correlated single nucleotide polymorphisms, then one cannot avoid dealing with complex multivariate genotype distributions, something that is not easy to specify through parametric models.
Our methods are types of semiparametric plug-in estimators, and thus have certain features
in common with the work of Newey (1994), namely that
the profile likelihood has the nonparametric component 
in (4) that is estimated by (5). Generally, however, such plug-in estimators are not
semiparametric efficient. We believe it will be possible to create an efficient
semiparametric estimator by modifying the work of Ma
(2010); we are exploring this and its computational aspects, which may be
daunting.
Supplementary Material
Acknowledgement
Stalder and Asher should be considered joint first authors. Carroll is also Distinguished Professor at the University of Technology Sydney. Chatterjee is also Bloomberg Professor of Oncology at the Johns Hopkins University. Stalder was supported by a fellowship from the Fondation Ernest Boninchi. Ma was supported by the U.S. National Science Foundation and National Institute of Neurological Disorders and Stroke. Asher, Liang and Carroll were supported by the National Cancer Institute. Chatterjee’s research was partially funded through a Patient-Centered Outcomes Research Institute Award. The statements and opinions in this article are solely the responsibility of the authors and do not necessarily represent the views of the Patient-Centered Outcomes Research Institute, its Board of Governors or Methodology Committee.
Supplementary material
Supplementary Material available at Biometrika online contains proofs, skewness and kurtosis and Q-Q plots for the simulation in Table 1, a discussion of how to modify our methods to account for strata, results of additional simulations, and software written in R. The data used in § 4 are available from the National Cancer Institute via a data transfer agreement.
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