A review of the genetics of hypertension with a focus on gene-environment interactions

Abstract

Purpose of review

Here, we discuss the interpretation and modeling of gene-environment interactions in hypertension related phenotypes, with a focus on the necessary assumptions and possible challenges.

Recent findings

Recently, small cohort studies have discovered several novel genetic variants associated with hypertension-related phenotypes through modeling gene-environment interactions. Several consortia-based meta-analytic efforts have uncovered many novel genetic variants in hypertension without modeling interaction terms, giving promise to future meta-analytic efforts that incorporate gene-environment interactions.

Summary

Heritability studies and genome-wide association studies have established that hypertension, a prevalent cardiovascular disease, has a genetic component that may be modulated by the environment (such as lifestyle factors). This review includes a discussion of known genetic associations for hypertension/blood pressure, including those resulting from the incorporation of gene-environmental interaction modeling.

1 Introduction

Cardiovascular disease (CVD) is the leading cause of death worldwide, and the burden is expected to increase in the coming years [1]. Further, CVD is considered to be largely preventable, and while prevention and mitigation efforts are leading to declines in CVD mortality in high income countries, the incidence of and mortality from CVD continues to increase in low- to middle-income countries [2]. In part, increasing rates of CVD mortality in low- to middle-income countries is a function of longer life expectancy resulting from decreasing prevalence of infectious disease [3]. This, combined with high smoking rates [4], increasing obesity rates, and relatively poor access to treatment [3], combine to increase CVD mortality in low- to middle-income countries. Hypertension (HTN) is characterized by high systolic blood pressure (SBP) and/or diastolic blood pressure (DBP), and is a major causal factor in heart failure, stroke, renal disease, and cardiovascular death [5]. A seminal study in BP research characterized the relationships between blood pressure and risk of CVD mortality and stroke as continuous, consistent, and independent of other risk factors; between the ages 40–69 years, CVD risk doubles for every 20 mmHg increase in SBP or 10 mmHg increase in DBP above 115/76 mmHg [6]. For 2015, the American Heart Association reported that high blood pressure (BP) was the leading factor in 40.6% of CVD and 34.7% ischemic heart disease attributable deaths in the US [7]. HTN incidence has been shown to differ among ancestral groups, as evidenced by comparisons between Caucasian and Hispanic American as well as Caucasian and African American groups [8, 9]. Between 2003 and 2013, the overall death rate attributable to HTN in the United States increased 14.4% and 1.7% in non-Hispanic Caucasians and Hispanics respectively, and declined 9.1%, and non-Hispanic African Americans [7].

CVD, as most human diseases, arises from a complex interaction between the environmental and genetic factors. Genome-wide association studies (GWAS) have identified a large number of genetic factors in a variety of diseases, including HTN, as reviewed by Zheng et al. [10]. In BP, family and twin studies have yielded heritability estimates in the ranges of 48% to 60% (SBP) and 34% to 67% (DBP) [11, 12, 13]. As of 2011, identified genetic loci from GWAS accounted for less than 2.5% of the phenotypic variance in BP studies [14], which has modestly increased to ≈3.5% with more recent GWAS [15, 16, 17]. Overestimation of heritability may occur in twin studies due to violations of shared environment assumptions, and poor phenotyping practices in control cohorts, failure to account for epistasis, GxE interactions, and other non-genetic sources of phenotype modulation may lead to underestimation of heritability in GWAS, all of which may contribute to the large difference in heritability estimates. Nonetheless, it is widely assumed that many of the causal loci are as of yet unidentified, a phenomenon commonly referred to as “missing heritability” [18, 19, 20]. Whereas GWAS have primarily focused on identifying genetic factors through modeling main effects of common SNPs, a focus on modeling gene-environment (GxE) and gene-gene (GxG) interaction may identify additional causal factors in human disease [18]. BP is known to be modulated by a variety of lifestyle factors, such as diet [21], exercise [22], and smoking [23], as well as other factors such as age [24, 25] and obesity [26]. In GWAS for HTN-related phenotypes, significant GxE interactions have been identified with alcohol consumption [27], body mass index (BMI) [28], smoking [29, 30], education levels [31], and sodium intake [46] which are all modifiable through lifestyle changes. These findings suggest that further investigation into GxE interaction may identify causal genetic loci that contribute to missing heritability [33].

The remainder of this review is organized as follows: Section 2 presents a discussion of interaction, including common definitions of interaction, rationales for modeling interaction, and limitations in modeling GxE interaction. Section 3 briefly reviews selected methodology employing interaction that has been used to identify genetic factors in HTN and HTN-related phenotypes. Section 4 summarizes the genetic factors in HTN and HTN-related phenotypes, and focuses further on associations identified using models that account for GxE interactions. Section 5 ends with a discussion of future direction for GxE studies and a conclusion.

2 Interaction and accounting for interaction through statistical modeling

Although the term “interaction” is prevalent in GWAS and the statistical literature, the definition and interpretation of interaction is contextual. Here, we discuss possible interpretations of interaction in the contexts of GWAS and statistics, then discuss how these interpretations coincide. A series of rationales is given for investigation into GxE interactions, and we follow with a commentary on the difficulties surrounding modeling GxE interaction terms in GWAS.

In an oft-cited review of GxE interaction in GWAS, Thomas [34] presents five definitions of interaction, given in Table 1. Of the five, biological interactions are the assumed to account for a portion of the missing heritability. Using a more general definition, biological interaction occurs when the effect of one factor on the process driving a phenotypic response is affected by the presence or magnitude of another factor [24]. Qualitative and quantitative interactions are forms of statistical interaction, as mentioned by Thomas [34], and public health synergy is collapsible into statistical interaction using the definition of statistical interaction given by Cox [35]. In all cases where statistical interaction is modeled, it should be assumed that biological interaction exists, even if the biological mechanism is not understood. Ottman [36] gives five plausible models for environmental and genetic factors to cooperatively influence disease etiology, of which four could be appropriately modeled by the definition of statistical interaction given above. Un-modeled interactions, in the case of GxE, GxG, or gene-covariate interactions, may contribute to confounding if left unincorporated. In general, the statistical literature tends to focus on modeling interaction terms as a nuisance, while the genetic epidemiology literature models interaction to increase power in finding disease relevant genetic loci [33]. In the context of statistical analysis, erring on the side of overfitting rather than underfitting with respect to interaction terms is recommended when interaction is believed to exist [35]. In cases where GxE interaction is biologically feasible, modeling these interactions can aid in accounting for departures from independence between the genetic and environmental factors of interest.

Table 1.

Definitions of GxE interaction, as given by Thomas [34]

Biological interaction	An established biological relationship in which some environmental factor triggers a response in people that have a susceptible genotype.
Quantitative interaction	An interaction where effects of one factor go in the same direction at different levels of the other, but differ in magnitude
Qualitative interaction	An interaction in which the presence of factor A’s effect only occurs in the presence/absence of factor B
Public health synergy	When phenotype measurements are attributable to a number of risk factors, but phenotype realization is not a function of the sum of the risk factors
Statistical interaction	Any departure from y = v1(x1) + v2(x2) when modeling response y using factors and [9]

Hunter [37] gives a series of rationales for studying GxE interactions in GWAS, which is presented in its entirety in Table 2. Thomas [34] also describes the goal of incorporating GxE interaction terms in GWAS is to use interactions – not discover interactions, but use interactions – to “discover novel genes that act synergistically with other factors” in modeling disease etiology, which intersects with the first of the rationales given by Hunter [37] in Table 2. The resulting findings from modeling GxE interactions in GWAS focus on presenting results in light of the remaining four rationales. The genetic associations discovered in GWAS are often used to drive bioinformatics research in finding the biological pathways mentioned in rationales three and four. In cases where the biological pathways for genes associated with GWAS identified genetic loci are understood, findings are presented in light of the processes underlying the biological pathways. Incorporating the interaction term into modeling can make finding a sufficiently explanatory relationship between the phenotypic response and model possible. Ultimately, the goal of GWAS is to drive personalized medicine, which is reflected in the fifth and final rationale presented.

Table 2.

Hunter’s Rationales for studying GxE interactions in GWAS [37]

Obtain a better estimate of the population-attributable risk for genetic and environmental risk factors by accounting for their joint interactions

Strengthen the associations between environmental factors and diseases by examining these factors in genetically susceptible individuals

Help to dissect disease mechanisms in humans by using information on susceptibility (and resistance) genes to focus on the biological pathways that are most relevant to that disease, and the environmental factors that are most relevant to the pathways

Use the information on biological pathways to design new preventative and therapeutic strategies

Offer tailored preventative advice that is based on the knowledge that an individual carries susceptibility or resistance alleles

Appropriately modeling interaction from a statistical standpoint is difficult, and modeling interaction in the context of GWAS presents additional complexity. Much of the difficulty in appropriately modeling interaction in GWAS comes in light of the generally small sample sizes relative to the large number of possible combinations of factors in observational studies [38]. Further exacerbating this problem is the possibility that minor allele frequencies are small, that the disease prevalence is low, or both [33, 38]. Confounding issues stemming from failing to model non-GxE interactions may arise, and will be discussed briefly in the conclusion. Outside of practical modeling concerns, some researchers believe that modeling interaction in the GxE or GxG context is inherently problematic. Some “environmental” factors, such as alcohol intake in studies concerning alcohol metabolism, may be driven by the causal gene in question and should not be used as covariates [39]. This suggests that additional factors related to the phenotype of interest, especially genetic factors that could be considered intrinsic, may be better incorporated through methodologies that model pleiotropy [40]. Furthermore, because the genotype frequency of the marker found in linkage disequilibrium and the true causal locus likely differ, the imputed genotype frequency at an identified marker is not an unbiased estimator of the frequency at the causal locus, leading to unknown misclassification probabilities [40]. Misclassification or measurement error for environmental and nuisance factors may also exist in the data. While it has been shown analytically [41] and through simulation [42] that misclassification in environmental factors biases the interaction term toward zero, these studies assume independence in the prevalence of genetic and environmental factors in the population, and results may not hold when this assumption is violated [40].

3 Common methodologies used to account for interaction in GWAS modeling

Although the GWAS literature that do not account for interaction is diverse, these methodologies are well reviewed and do not fit the scope of this writing. The literature concerning the modeling of interaction in GWAS is also diverse, but we focus on methods that appear to be successful in uncovering genome-wide significant (GWS) SNP associations in BP-related traits. More specifically, we compare the assumptions, perceived notions of power and type I error rates, and discuss how violations of these assumptions may affect type I error rates. For all tests discussed, researchers mitigate multiple testing issues and control type I error rates with a restrictive definition of GWS by setting the target type I error rate at α=5×10⁻⁸., and studies often further control for false positives by accounting for population substructures through genomic control methods [43].

3.1 Tests involving a single cohort, or multiple cohorts with fully available data

Applications of the general linear model, or GLM, are appropriate to test for association between genomic markers and the phenotype of interest. In analyses where complete data for all individuals is readily available, a GLM can be fit to the full data. These tests can account for any confounding factor that may arise through modeling of nuisance factors. Unfortunately, these tests are limited by the complexities that arise from analyzing massive cohorts, and require full access to raw data from all cohorts, which is often unattainable. While a number of tests could arise from the use of GLMs in the GWAS GxE literature, we focus on the joint 2 degree of freedom likelihood ratio test (referred to as J2DF) and the Wald type 1 degree of freedom interaction effect test (referred to as 1DF). In applying a GLM, assumptions are imposed on the error structure of the resulting model. Typically, population substructures within the errors resulting from a GLM fit will inflate type I error rates [43]. For a more formal review of the mechanics surrounding these hypothesis tests and test statistics, see Appendix A.1.

3.2 Meta-analytic tests involving multiple cohorts

Studies concerning the detection of causal genetic loci are demanding regardless of the inclusion of interaction effects, which further compounds the issue [44]. In addition, fitting GLMs to datasets large enough to allow for the discovery of significant interaction terms may be computationally difficult, and in many cases, researchers do not have access to full cohort data. Meta-analytic procedures allow researchers to incorporate results from study-specific analyses to implement more powerful tests than what could be achieved through the use of a single cohort analysis; this does not require access to the individual level data used in each of the contributing cohorts.

In meta-analysis concerning synthesis of uncorrelated regression slopes, inverse variance weighted least squares approaches are common. Ordinary and weighted least squares approaches are inappropriate to model correlated regression slopes and may not result in consistent estimates. Becker and Wu [45] present a robust generalized least squares meta-analytic approach to estimate possibly correlated population regression coefficients given summaries of study-specific analyses. This technique was later applied to meta-analysis of GWAS results [33]. As with all meta-analytic approaches, we must assume that the results from the k studies incorporated are representative of the true processes in the population of interest, and are not pruned to the most significant results via publication bias. This assumption is somewhat safe in the realm of genetic epidemiology, where the aim of the meta-analysis is to bolster power through the incorporation of results from multiple cohorts performing agnostic GWAS. This methodology also assumes that either a) each of the k studies considered accounted for the same nuisance parameters, or b) that although the k studies did not incorporate the same nuisance parameters, the resulting parameter estimates are unbiased representations of the true slopes of interest. The validity of the latter assumption is generally corroborated, so long as modeling techniques in the individual studies are the same; studies have found little to no evidence of bias in the synthesized slopes of interest as a result of the estimation of additional nuisance parameters in the considered studies [46, 47], while analyses synthesizing results from different models show that the type of model, as well as the difference in nuisance parameters, will affect the meta-analytic slope estimates [48]. We focus on the joint Wald type 2 degree of freedom test (referred to as JMA2DF), which tests the genetic main and GxE interaction effects, and the Wald type 1 degree of freedom interaction effect test (referred to as MA1DF). For a more formal review of the mechanics surrounding these hypothesis tests and test statistics, see Appendix A.2.

4 Significant findings in the HTN literature

Although the above is primarily focused on incorporation of GxE interaction terms into models evaluating the effect of allelic presence at genetic loci of interest, major findings concerning causal genetic loci have been accomplished without modeling GxE interaction terms. Online Resource 1 gives a summary of GWS findings from studies that did not incorporate GxE interaction, including the gene, cytogenetic or molecular location, associated phenotype, lead/tag SNP (a representative SNP in an area of the genome that exhibits high linkage disequilibrium), and the ancestral groups. A short summary of all HTN GWAS GWS findings is given in Table 3.

Table 3.

GWAS studies identifying HTN loci

Paper	Loci Identified (known and novel, not unique)
Adeyemo et al., 2009 [49]	3
Cho et al., 2009 [50]	1
Ehret et al., 2016 [15]	66
Fox et al., 2011 [14]	3
Franceschini et al., 2013 [51]	5
Ganesh et al., 2013 [52]	12
Ganesh et al., 2014 [53]	19
Ho et al., 2011 [54]	10
Hoffman et al. 2017 [55]	316
Hong et al., 2010 [56]	3
Hong et al., 2012 [57]	4
ICBP, 2011 [58]	28
Johnson et al., 2011a [59]	1
Johnson et al., 2011b [60]	8
Kato et al., 2011 [61]	15
Kelly et al., 2013 [62]	4
Levy et al., 2009 [63]	10
Lin et al., 2011 [64]	3
Liu et al., 2016 [17]	21
Lu et al., 2014 [65]	17
Newton-Cheh et al., 2009 [66]	7
Org et al., 2009 [67]	1
Padmanabhan et al, 2010 [68]	1
Qi et al., 2014 [69]	3
Salvi et al., 2012 [70]	1
Simino et al., 2014 [25]	20
Sung et al., 2014 [29]	1
Surendan et al., 2016 [16]	49
Tabara et al., 2010 [71]	4
Takeuchi et al., 2010 [72]	6
Tomazewski et al., 2010 [73]	1
Tragante et al., 2014 [74]	55
Wain et al., 2011 [75]	7
Wang et al., 2009 [76]	1
Wang et al., 2013 [77]	1
Zhu et al., 2011 [78]	1
Total (Non-interaction)	708
Basson et al., 2014 [31]	5
Basson et al., 2015 [30]	13
Li et al., 2016 [32]	3
Simino et al., 2013 [27]	1
Simino et al., 2014 [25]	9
Sung et al., 2014 [29]	7
Total (Interaction)	38
Grand Total	746

4.1 Findings relying on the incorporation of GxE interactions

A set of GWS findings from models incorporating interactions is given in Online Resource 2. Notably, this table is somewhat smaller, suggesting that this is a more recent direction of research which needs more effort. Also, some of the identified loci that were not novel may have been observed in studies that did not include interactions. It is notable that when findings of association were also detected in studies accounting for GxE interactions, these models lend great power to uncovering possibly causal genomic variants. An example of increased power from the inclusion of GxE terms in GWAS can be seen through the analysis of 6,889 participants from the Framingham Heart Study cohort by Sung et al. [29]. In Table 4, the findings overlap at the MECOM gene are displayed as well as cohort information for the studies involved. Although these findings have shortcomings, recall that our purpose is not to detect interaction, but rather detect novel loci by accounting for interaction, similar to the arguments made by Cox [35] and Thomas [34]. In this context, the interaction can be thought of more as a nuisance parameter, much like the covariates modeled.

Table 4.

Studies with Statistically Significant Findings at the MECOM Gene for HTN Related Phenotypes

Study (Interaction?)	Sample Size
ICBP, 2011 [58] (No)	200,000
Tragante et al., 2014 [74] (No)	87,736
Sung et al., 2014 [29] (Yes)	6,889
Ehret et al., 2016 [15] (No)	342,415
Surendan et al., 2016 [16] (No)	192,763

5 Conclusion and future direction

We have reviewed the definition, rationale, and interpretation of GxE interaction in the context of GWAS. The discussion includes limitations in select methodology, and relevant GWS findings within the HTN GWAS literature. Accounting for GxE interaction in GWAS can aid in the detection of new GWS genetic loci [34], which is shown through the examples in Online Resource 2 and Table 3. Additionally, although the mechanistic processes underlying GxE interactions is unknown, their existence is well accepted and studied [36], which gives credibility to the incorporation of GxE interaction terms in GWAS.

It is commonly known that even in the absence of a true effect, increases in the study sample size n for a given study leads to more significant p-values, which may cause a researcher to declare spurious results statistically significant. Through simulation, VanderWeele et al. [79] show that this phenomenon is more pronounced at smaller sample sizes as the effect size of an uncontrolled confounder increases. Keller [80] attributes this type I error rate inflation to either uncontrolled confounding involving environmental and/or un-modeled population substructures in GxE GWAS studies or improper model specification, adding that, of 45 reviewed studies with GWS findings in the GxE schizophrenia literature, only 12 were thus far replicated. Further, in situations where measurement error or misclassification exists in the genetic, phenotype, and/or environmental covariates, effect estimates may be inconsistent, resulting in lower power, or bias, which may increase type I error rates [81]. In light of the previous discussion regarding the assumption of a true biological interaction between the genetic factor and environmental covariates of interest, note that we implicitly assume independence of all possible un-modeled covariates and factors, interactions included. As meta-analytic approaches become more prevalent, increasing the sample size and power of GxE GWAS, this assumption will need to be given more attention to avoid inflating type I error rates. While misclassification of environmental factors has been shown to bias the interaction term toward zero, thus not contributing to type I error rate inflation [41, 42] these analyses assume independence of the genetic and environmental factors in the population conditional on the phenotype of interest, and these results may not hold when this independence assumption is violated [40]. Although research has not specifically been conducted to study type I error rates as a function of minor allele frequencies in GxE studies, inclusion of genetic variants with small minor allele frequencies is uncommon due to error rate inflation concerns [82]. While these concerns are echoed throughout the literature, it has been shown through simulation that inclusion of genetic loci with small minor allele frequencies does not increase type I error rates above expected levels [83].

Incorporation of GxE interaction terms into GWAS may enhance statistical power without increasing sample sizes by reducing error variance, which may facilitate personalized medicine approaches in healthcare and may help in the development of pharmaceutical treatment for complex diseases. Although arguments against interaction studies exist, replication in additional cohorts and biological explanations of gene expression can mitigate spurious findings. Investigation of pleiotropy and epistasis can yield further insights into the genetic underpinnings of complex diseases, but require much larger sample sizes. As the cost of sequencing decreases, these approaches may shed light on genetic variants that either work in parallel to influence disease, or contribute to multiple diseases.

Added in proof: As of this revision, Warren et al. [84] have identified an additional 32 novel blood pressure loci, and further validated 75 loci discovered by Hoffman et al. [55].

Appendix: Formulation of relevant test statistics

A.1: Tests involving individual level cohort data

Let y denote a quantitative vector containing measurements representing the phenotype of interest. We specify the GLM

$$f(\mathit{y}\mid \mathit{X},\mathit{\beta })={\mathit{X}}_C{\mathit{\beta }}_C+{\mathit{x}}_G{\beta }_G+{\mathit{x}}_E{\beta }_E+{\mathit{x}}_G\odot {\mathit{x}}_E{\beta }_{G\times E}+\mathit{\epsilon },$$ (1)

where X_C and β_C are the design matrix and regression coefficients for the nuisance factors, x_G and β_G is the covariate vector and regression coefficient for the genetic locus of interest, x_E and β_E is the covariate vector and regression coefficient for the environmental variable of interest, x_G⊙x_E is the elementwise product of the genetic and environmental covariate vectors, β_G×E is the regression coefficient for the GxE interaction term, and ε is a multivariate normal random vector with mean 0 and covariance matrix Σ. Coding schemes for x_E depends on the study in question, as environmental factors may be discrete or continuous. x_G may be binary in the case of dominant alleles, or coded based on the number of copies of a reference allele at a locus of interest.

In analyses where all data for all individuals is readily available, (1) can be fit to the full data, usually with a set of assumptions on the distribution of ε. In particular, researchers will assume independence amongst subjects, restricting the covariance matrix Σ to the diagonal case, or choose to only model within family correlation, giving Σ a block diagonal structure.

In some cases, researchers choose to isolate the genetic and environmental factors of interest portrayed in (1) through a two step modeling process. Let

$$f(\mathit{y}\mid {\mathit{X}}_C,{\beta }_C)={\mathit{X}}_C{\beta }_C+\mathit{z},$$

where X_C and β_C are the design matrix and regression coefficients for the nuisance factors. Then, the model

$$f(\mathit{z}\mid {\mathit{x}}_G,{\mathit{x}}_E,{\beta }_G,{\beta }_E,{\beta }_{G\times E})={\mathit{x}}_G{\beta }_G+{\mathit{x}}_E{\beta }_E+{\mathit{x}}_G\odot {\mathit{x}}_E{\beta }_{G\times E}+\mathit{\epsilon }$$ (2)

is fit, where x_G, x_E, β_G, β_E, β_G,×E and ε are defined as in (1). While the model in (2) assumes that x_G, x_E, β_G, β_E, β_G×E|z is independent of y, which is likely unrealistic, (2) fits the coefficients of interest much more quickly than one through only fitting β_C once.

In both (1) and (2), parameter estimates β̂_G, β̂_E, and β̂_G×E, along with parameter variance estimates var(β̂_G), var(β̂_E) and var(β̂_G×E) are obtained through maximum likelihood methods. For the Wald type hypothesis test

$$\begin{array}{l}{H}_0:{\beta }_{G\times E}=0\\ H_A:{\beta }_{G\times E}\ne 0\end{array}$$

the test statistic

$$T_{G\times E}=\frac{{\left({\widehat{\beta }}_{G\times E}\right)}^2}{\mathit{var}({\widehat{\beta }}_{G\times E})}$$

asymptotically follows a ${\chi }_1^2(0)$ distribution under the null hypothesis. This is commonly referred to as the one degree of freedom interaction test, shortened to 1DF here. The ProbABEL software [85] is commonly used to carry out this test.

Of course, given the discussion above regarding accounting for interaction rather than modeling it, the researcher may be more interested in tests that incorporate interaction into the whole model. Let L(β_G, β_G×E|Data) be the profile likelihood for the model proposed in (1) or (2). To test the hypothesis

$$\begin{array}{l}{H}_0:{\beta }_G,{\beta }_{G\times E}=0\\ H_A:{\beta }_{G}or{\beta }_{G\times E}\ne 0,\end{array}$$

we perform a likelihood ratio test, whose test statistic

$$\mathit{LRT}=-2log\left(\frac{L\left({\beta }_G={\widehat{\beta }}_G,{\beta }_{G\times E}={\widehat{\beta }}_{G\times E}\mid \mathit{Data}\right)}{L({\beta }_G=0,{\beta }_{G\times E}=0\mid \mathit{Data})}\right)$$

asymptotically follows a ${\chi }_2^2(0)$ distribution under the null hypothesis. This is commonly referred to as the joint two degree of freedom test, shortened to J2DF here. The ProbABEL software [85] may be used to carry this test out as well.

A.2: Meta-analytic approaches

Let β̂_G,i and β̂_G×E,i denote the regression coefficients associated with the genetic locus and gene by interaction effects estimated in the ith study in consideration. Then, for studies i = 1, …, k,

$$\mathit{b}=\left[\begin{array}{c}{\widehat{\beta }}_{G,1}\\ {\widehat{\beta }}_{G\times E,1}\\ ⋮\\ {\widehat{\beta }}_{G,k}\\ {\widehat{\beta }}_{G\times E,k}\end{array}\right],\widehat{\mathit{\sum }}=\left[\begin{array}{ccc}{\widehat{\mathit{\sum }}}_1& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \ddots & \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\widehat{\mathit{\sum }}}_k\end{array}\right],$$

where Σ̂_i = cov(β̂_G,i, β̂_G×E,i) is the covariance matrix for the regression coefficients from the ith study. We write the vector b as a linear combination of the parameters of interest ${\beta }_G^I$ and ${\beta }_{G\times E}^I$

$$b=\mathit{W}{\mathit{\beta }}^I+\mathit{e}=\left[\begin{array}{cc}1& 0\\ 0& 1\\ ⋮& ⋮\\ ⋮& ⋮\\ 1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{\beta }_G^I\\ {\beta }_{G\times E}^I\end{array}\right]+\mathit{e},$$

where e has a multivariate normal distribution with zero mean vector and covariance matrix Σ̂. Using the generalized estimating equations as presented by Becker and Wu [45] and the imposed inverse variance weighting scheme from our definition of Σ̂, we estimate β¹ as

$$\widehat{{\mathit{\beta }}^I}={({\mathit{W}}^T{\widehat{\mathit{\sum }}}^{-1}\mathit{W})}^{-1}{\mathit{W}}^T{\widehat{\mathit{\sum }}}^{-1}\mathit{b},$$

and the covariance matrix associated with the β¹estimate vector as

$$\mathit{Cov}(\widehat{{\mathit{\beta }}^I})={({\mathit{W}}^T{\widehat{\mathit{\sum }}}^{-1}\mathit{W})}^{-1}.$$

The Wald type hypothesis test

$$\begin{array}{l}{H}_0:{\mathit{\beta }}^I=\mathbf{0},\\ H_A:{\mathit{\beta }}^I\ne \mathbf{0},\end{array}$$

has test statistic

$$T_{\text{JMA}2\text{DF}}={\widehat{{\mathit{\beta }}^I}}^T\mathit{Cov}{(\widehat{{\mathit{\beta }}^I})}^{-1}\widehat{{\mathit{\beta }}^I},$$

which asymptotically follows a ${\chi }_2^2(0)$ distribution under the null hypothesis. Because $\mathit{var}({\widehat{\beta }}_{G\times E}^I)=\mathit{Cov}{(\widehat{{\mathit{\beta }}^I})}_{2,2}$, we can also test for a significant interaction term under the Wald type hypothesis test

$$\begin{array}{l}{H}_0:{\beta }_{G\times E}^I=0,\\ H_A:{\beta }_{G\times E}^I\ne 0,\end{array}$$

which has test statistic

$$T_{\text{MA}1\text{DF}}=\frac{{\left({\widehat{\beta }}_{G\times E}^I\right)}^2}{\mathit{var}({\widehat{\beta }}_{G\times E}^I)}$$

which asymptotically follows a ${\chi }_1^2(0)$ distribution under the null hypothesis.