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.
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 : 128 | : 126 | All : 218 | ||||||||
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 : 128 | All : 128 | All : 207 |
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 356% 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 091, 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 009. 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 01 and the minimum -value was 085. 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.
Logistic | ||||
Estimate | 0018 | 0297 | 0165 | 0124 |
Std err | 0054 | 0064 | 0132 | 0068 |
-value | ||||
SPMLE Rare | ||||
Estimate | 0024 | 0321 | 0175 | 0138 |
Std err (asymptotic) | 0054 | 0067 | 0134 | 0055 |
-value (asymptotic) | ||||
SPMLE known | ||||
Estimate | 0022 | 0313 | 0174 | 0141 |
Std err (asymptotic) | 0054 | 0065 | 0133 | 0055 |
-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 005 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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