Introduction

Complex diseases are caused by a combination of genetic and environmental factors, creating a challenge for understanding the disease mechanisms. Understanding the interplay between genes and environmental factors is important, as genes do not operate in isolation but rather in complex networks and pathways influenced by environmental factors. In addition to providing insights into disease etiology, exploiting gene-environment (G-E) interaction can help discover novel susceptibility loci for complex diseases, where genetic effects are modified and masked by the effects of environmental factors. Therefore, evaluating the main effects of a gene without considering its interaction with environmental factors can miss true association signals [1,2,3]. From a public health perspective, G-E interaction is useful because findings based on interactions can help develop strategies for targeted intervention; conducting an intervention focusing on a subset of the population identified by G-E interactions can provide efficiency in disease prevention [4•, 5•].

Although G-E interaction has various meanings in epidemiology, it can be generally defined as a joint effect of genetic and environmental risk factors that cannot be explained by their separate marginal effects [6]. The recent advent of new technologies has made a massive amount of genetic data available, and various statistical methods have been developed to analyze genetic data and to identify G-E interactions. These methods include approaches that exploit additional assumptions such as G-E independence to improve power such as case-only analysis, retrospective likelihood-based analysis as well as empirical based estimators, methods that incorporate alternative disease risk models such as additive models, and tests for identifying interactions between rare variants and exposures based on exome or whole genome sequencing data. Various software packages have been also developed, which can be used to apply newly developed statistical methods for detecting G-E interactions.

The purpose of this article is to introduce recently developed statistical methods for evaluating G-E interactions across various complex diseases. While several study designs are available for examining G-E interactions such as prospective cohort studies, case-control studies, and family studies designs, we will focus on case-control studies that are mostly commonly used for genome-wide association studies (GWAS). In this review, we will first discuss the statistical models for joint effects of genetic and environmental factors and then introduce various statistical inferences methods under these models such as methods based on prospective and retrospective likelihoods as well as empirical base type approaches. We will then introduce statistical approaches that test for genetic associations in the presence of G-E interactions, various methods for two-stage analyses for GWAS, and methods for identifying interactions between rare variants and environmental exposures. We finalize this article with current challenges and future directions for analyzing G-E interactions.

Statistical Models for GxE and Interpretations of Interactions

There are several disease risk models for the joint effects of G and E, and interpretations of G-E interactions depend on the underlying disease risk models. A multiplicative model is one of the most commonly used models via logistic regression: logit (Pr(D = 1|G, E)) = β0 + β G G + β E E, where G is a genotype of a single nucleotide polymorphism (SNP), E is an environmental risk factor, and D is the disease status. Depending on the assumed genetic model, G can be coded for an additive genetic model (i.e., the number of the variant allele), dominant model (i.e., 1 for variant allele carriers and 0 for non-carriers), or recessive model (i.e., 1 if one carries two copies of the variant allele and 0 otherwise). A departure from this model is called a multiplicative interaction, which can be tested by H0 : β GE  = 0 in the following saturated model:

$$ \mathrm{logit}\ \left(\Pr \left(D=\left.1\right|G,E\right)\right)={\beta}_0+{\beta}_GG+{\beta}_EE+{\beta}_{GE} GE $$
(1)

Assuming binary factors for both G and E, a 2 × 2 table for a disease risk for each combination of G and E values can be constructed based on this model (see Table 1). Assuming a rare disease (i.e., relative risks can be approximated by odds ratios), “no multiplicative interaction” implies that the genetic effects measured by the ratios of the risks (e.g., \( \frac{R_{10}}{R_{00}} \) for E = 0) are the same across different exposure levels with the null hypothesis of \( {H}_0:\frac{R_{10}}{R_{00}}=\frac{R_{10}}{R_{01}} \). On the other hand, an additive model is shown as logit (Pr(D = 1|G, E)) = b0 + b G G + b E E, where the effects of G and E are additive on the disease risk scale, but not on the logit scale. An additive interaction is defined by the departure from this model, which implies that the genetic effects measured as the differences of absolute risks (e.g., R10 – R00 for E = 0) vary by exposure levels with the corresponding null hypothesis of H0 : R10 – R00 = R11 – R01. A number of researchers have shown that conceptual models for biologic interactions translate to the presence of interaction on the additive scale and not necessarily on the multiplicative scale [7]. In public health, evaluation of risk differences and additive interactions is directly relevant to problems such as whether it is beneficial to target individuals for intervention for an exposure based on genetic susceptibility [2, 8]. In addition to these multiplicative and additive models, there are some other non-standard models discussed in the literature [9, 10], including a liability threshold model, where the effects of G and E are additive on the probit scale [9].

Table 1 Disease risk for binary factors G and E, were R ij  = Pr(D = 1|G = i, E = j) for i, j = 0, 1
Full size table

Inferences: Methods for Testing G-E Interactions

Standard Prospective Likelihood-Based Approaches

Based on the models introduced in the previous section, several inference methods have been developed to test for G-E interactions. Standard analyses of case-control studies are typically based on a prospective likelihood of case-control data. While this approach does not take into account the retrospective nature of the sampling design, it is shown that such prospective treatment of case-control data is valid when there is no assumption made about the joint distribution of covariates, including genetic and environmental factors and other confounders [11]. This likelihood can be used for both additive and multiplicative models, and several studies have used this approach for evaluating G-E interactions for various complex diseases [12•, 13, 14]. Figueroa et al. conducted a genome-wide interaction study of smoking for bladder cancer risk by applying both multiplicative and additive interactions based on a prospective likelihood and a retrospective likelihood [12•]. They identified 10 significant SNPs that interact with smoking status (ever versus never smokers) for bladder cancer; these included rs1711973 that had an increased risk (OR = 1.34; 95% confidence interval (CI): 1.2–1.5) among never smokers (multiplicative interaction P = 6.38E-06) and rs12216499 that had a reduced risk (OR = 0.75; CI: 0.67–0.84) for ever-smokers (additive interaction P = 1.41E-06). Multiplicative interactions based on a prospective likelihood can be tested using any statistical software package (e.g., SAS, R, or Stata). For example, in R, the glm() function can be used for testing a multiplicative interaction using a logistic regression based on a prospective likelihood. An R package, CGEN (https://bioconductor.org/packages/release/bioc/html/CGEN.html) implements the methods for both additive and multiplicative interaction based on a prospective likelihood; the additive.test function can be used for performing an additive interaction test and snp.logistic and snp.score for conducting a multiplicative interaction test (see Supplemental Fig. 1).

Case-Only Design

In evaluating G-E interactions, there have been several approaches that assume that G and E are independent in the underlying population. This assumption is plausible because the genetic variation an individual receives from a parent is determined during meiosis, and hence is not affected by subsequent environmental exposures after birth. Genetic susceptibility is unlikely to influence various exogenous exposures such as environmental pollutants or occupation exposures with some exceptions, whereas this assumption can become questionable for endogenous exposures, such as biomarkers. The case-only design is one of the non-traditional methods that depend on an assumption of G–E independence in the underlying population, which can be used to test for multiplicative interactions [15]. In brief, under the assumption of G-E independence in the underlying population (i.e., controls), a multiplicative interaction test statistic becomes equivalent to testing the association between G and E among cases. This method has been applied to the analyses of G-E interaction for various complex diseases [16, 17]. Freedman et al. used a case-only interaction test to evaluate the interaction between two independent genes, FRMD3 and MYH9 for end-stage renal disease risk. Any standard statistical software can be used to conduct this test. For example, in R, a linear regression function (lm function) or generalized linear regression functions (glm function) can be used to evaluate an association between an environmental exposure and a genotype based on the data for controls. One major limitation of the case-only design is that while the case-only method has improved power over the traditional methods when G and E are independent in the underlying population, this method has an increased type I error if the independence assumption is violated [18]. In addition, the regression parameters for the main effects of G and E cannot be estimated using this method because the case-only test is only for evaluating a multiplicative interaction.

Retrospective Likelihood Approach

To address the limitations of case-only approaches that can only test for multiplicative interactions (not for the main effects of G and E), Umbach and Weinberg (1997) generalized the case-only design idea to use a log-linear model based on case-control data. They showed the maximum-likelihood estimates for all parameters of a logistic regression model can be obtained using a log-linear model [19]. Along the same line, Chatterjee and Carroll developed a general method using a retrospective likelihood that exploits the G-E independence assumption to test for multiplicative interaction, but can use both cases and controls to estimate all of the parameters in a general logistic regression model [20]. Basically, this method employs a retrospective likelihood that explicitly models the conditional probability of G given E mediated by an association parameter θ that can be constrained to be zero when the G-E independence assumption holds. This likelihood can be used for testing both multiplicative and additive interactions; recently, Han et al. developed a likelihood ratio test that exploits the G-E independence assumption using a retrospective likelihood [21•]. Their numerical investigation of power suggests that the incorporation of the independence assumption can enhance the efficiency of the test for additive interaction by 2- to 2.5-fold. The multiplicative and additive interaction tests based on a retrospective likelihood are implemented in the CGEN R package. The function snp.score can be used for testing a multiplicative interaction and the additive.interaction function (with an argument indep = T) can be used for testing an additive interaction using the G-E independence assumption.

Empirical Bayes Type Approaches

Despite the power gain using methods that rely on the G-E independence assumption—such as the case-only, log-linear, and retrospective methods—they can cause a large type 1 error when the underlying assumption is violated [18]. To address this issue, an empirical Bayes type method was developed that uses a weighted average of the case-control and case-only estimators of the multiplicative interaction, which yields an acceptable trade-off between bias and efficiency [22••]. A stochastic framework is used to allow for uncertainty around the G-E independence assumption, which estimates the uncertainty parameter using data. The empirical-Bayes type estimator is provided as follows: \( {\widehat{\beta}}_{EB}=\frac{{\widehat{\sigma}}_{CC}^2}{\left({\widehat{\theta}}_{GC}^2+{\widehat{\sigma}}_{CC}^2\right)}{\widehat{\beta}}_{CO}+\frac{{\widehat{\sigma}}_{CC}^2}{\left({\widehat{\theta}}_{GC}^2+{\widehat{\sigma}}_{CC}^2\right)}{\widehat{\beta}}_{CC} \), where \( {\widehat{\beta}}_{CO} \) is a case-only estimator and \( {\widehat{\beta}}_{CC} \) is a case-control estimator for a multiplicative interaction, respectively. Here, \( {\widehat{\theta}}_{GE} \) is the measure of the G-E association among controls and \( {\widehat{\sigma}}_{CC}^2 \) is the estimated variance of the case-control estimator. The intuition is that \( {\widehat{\theta}}_{GE} \) is a measure of the bias of the case-only method, and the empirical Bayes method provides more weight to the case-control method when this bias is large. How much weight will be given is calibrated by \( {\widehat{\sigma}}_{CC}^2 \), which is the variance of the less efficient case-control estimator. If the G-E independence assumption is violated, i.e., true \( {\widehat{\theta}}_{GE}=0 \), then the empirical Bayes estimator will asymptotically behave the same as the case-control estimator. However, when G-E independence holds, the asymptotic weight for the empirical Bayes estimator will be non-zero for both case-control and case-only estimators and thus will have efficiency in between. A general approach for deriving empirical Bayes-type shrinkage estimators was also proposed for all of the parameters of a general logistic regression model [22••, 23], which is implemented in the CGEN R package. The empirical Bayes type estimator for an additive interaction was also developed in the general regression setting [24].

Testing for Genetic Association in the Presence of G-E Interaction

When identifying susceptibility loci for complex diseases, allowing for interactions to test for association could increase power when such interactions exist. It has been shown that a joint test of genetic association and interaction has robust performance over a wide range of underlying models [25•], although it could be less powerful than a marginal association test when there is no evidence of G-E interaction. Using the equation in (Eq. 1), the null hypothesis of the joint test is given as H0 : β G  = β GE  = 0, which has increased degrees of freedom compared to a marginal association test (i.e. H0 : β G  = 0) that can lead to a decrease in power when there is no interaction effect, i.e., β GE  = 0. Various likelihoods with or without the assumption of G-E independence can be used for joint tests. Recently, Hamza et al. conducted a genome-wide joint test for gene x coffee interaction for Parkinson’s disease and identified a novel susceptibility locus in the GRIN2A gene. In the gene, the T allele of the SNP rs4998386 is associated with a reduced risk among heavy coffee drinkers, whereas this variant has a minimal effect among light coffee drinkers [26•]. While a joint test can be powerful when the assumed interaction exists, the increased degrees of freedom of this test (versus a marginal association test) can lead to a reduced power when such interaction effects are relatively small or when these effects do not exist. A maximum score test was developed to overcome the potential loss of power of a joint test due to increased degrees of freedom [10]. This method provides a unified approach that integrates a class of disease risk models by maximizing over a class of score tests, each of which involves modified standard tests of genetic association through a weight function. This weight function reflects the potential heterogeneity of the genetic effects by levels of environmental exposures. Both joint test and maximum score test are implemented in the CGEN R package.

Two-Stage Analysis or G-E Interactions for Rare Variants

Several approaches have been proposed to conduct a two-stage analysis to improve the efficiency of detecting G-E interactions on a genome-wide scale [27,28,29]. In general, these methods suggest selecting a subset of SNPs based on the marginal effects of SNPs or G-E correlation tests in the first stage and conducting standard G-E interaction tests in the second stage, where the independence between the test statistics used in the two stages is required to provide a valid screening procedure. Applications of such methods are shown in a recent G-E analysis for colorectal cancer [30] that involves (i) a screening step based on marginal associations and gene-diet correlations and (ii) a testing step for multiplicative interactions. They identified a significant interaction between rs4143094 and processed meat consumption (OR = 1.17; p = 8.7E-09), which was consistently observed across studies. With the advent of high-throughput technologies, various statistical methods have been developed for identifying G-E interactions based on data for rare variants, generated by whole genome sequencing and exome sequencing [31,32,33,34,35]. A standard approach for this problem is a set-based G-E interaction framework that tests for an interaction between a set of rare variants and an environmental risk factor. Burden type tests [33, 36] and variance component tests [31] are available for analyzing G-E interaction in this framework. Some of these methods are implemented in the R packages rareGE (https://www.hsph.harvard.edu/han-chen/software/) and SIMreg (http://www4.stat.ncsu.edu/~jytzeng/software_simreg.php).

Software Available for G-E Analysis

There are several software packages that provide tools for conducting G-E interactions using the methods described in this review. The CGEN R package provides various functions that can conduct tests for multiplicative and additive interactions, joint tests, as well as maximum score tests under both prospective and retrospective likelihoods assuming the G-E independence assumption (https://bioconductor.org/packages/release/bioc/html/CGEN.html). The empirical Bayes type method for multiplicative interaction is also implemented in CGEN. The rareGE R package (https://www.hsph.harvard.edu/han-chen/software/) provides various functions for detecting G-E interaction as well as for testing the joint effect of a gene and G-E interaction under a set-based framework. The SIMreg R package (http://www4.stat.ncsu.edu/~jytzeng/software_simreg.php) offers functions for testing a set-based G-E interaction by using genetic similarity to aggregate information across SNPs, and incorporating adaptive weights depending on allele frequencies to accommodate rare and common variants. For calculating power for G-E interactions, the powerGWASinteraction R package is available (https://cran.r-project.org/web/packages/powerGWASinteraction/index.html), which includes a power calculation tool for four two-stage screening and testing procedures. Several studies compared the power of various G-E interaction tests including standard prospective likelihood approaches, case-only designs, retrospective likelihood methods, empirical Bayes-type estimators, and two-stage analyses [37, 38].

Challenges for G-E Analysis and Future Directions

There are several challenges of G-E interaction analysis. One main challenge is replication issues. While various GWAS findings of the main effects of SNPs have been replicated by independent studies for many complex diseases (http://www.ebi.ac.uk/gwas/), relatively few interactions have been reproduced. It is likely that the sample sizes of GWAS that have required measurements on environmental exposures are not yet adequate to reliably identify G-E interactions of modest magnitude. In addition, differences in the underlying distribution of environmental exposures across various studies as well as difficulties in accurately measuring environmental exposures can also lead to reduced power of detecting G-E interactions. While more powerful statistical methods for detecting interactions are helpful, ultimately studies with larger sample sizes are needed to identify interactions (e.g., through consortium-based studies) to achieve adequate power for G-E analysis. A reasonable goal for the future will be to at least identify parsimonious models that adequately describe the risks of diseases associated with a combination of genetic and environmental risk factors. The lack of reporting of interaction in current studies so far indicates that linear logistic models, i.e., multiplicative models, in general may be a good starting point for building models for evaluating the joint effects of genetic and environmental factors [39].