Moderating the heritability of body mass index by age and sex with genomic data

Abstract

Environmental contexts may increase or decrease the heritability of a phenotype. Or equivalently, people with some genotypes may be more (or less) sensitive to environmental differences. Such genetic sensitivity to environmental contexts is called gene-environment interaction (GxE). While GxE has been robustly detected in twin and model organism studies for numerous phenotypes, there is a lingering perception that existing genome-wide GxE methods struggle to identify substantively significant and replicable interactions. We propose a novel method for examining GxE heritability using genetic marginal effects from genome-wide GxE analyses and Linkage Disequilibrium Score Regression (LDSC). We demonstrate the effectiveness of our method for body mass index (BMI) using biological sex (binary) and age (continuous) as moderators. Using the same procedures for both binary and continuous moderators, we detect robust evidence for GxE. Our results are consistent with findings from twin GxE studies of BMI and are more sensitive to environmental moderation than other LDSC-based methods. We conclude that BMI heritability is substantially more sensitive to variation in sex and age than is currently appreciated. Extending this method to other phenotype-moderator combinations has the potential to reveal GxE across numerous outcomes and moderators.

Introduction

Gene-environment interaction (GxE) plays a critical role in many complex behaviors, with body mass index (BMI) providing an exceptional model phenotype. Organisms, including humans, may thrive or struggle in an environment based on their genetic characteristics. Differences in environmental contexts can alter an individual’s phenotypic characteristics for a range of complex traits by modifying the relationships between genes and behaviors. For example, genetic variants that contribute to BMI in males and females may diverge, or aging may suppress gene action. Such differences in either the magnitude of heritability (i.e., the proportion of additive genetic variation in a phenotype) or genetic associations for individual variants reflect differential sensitivity to the environment (Mather & Jinks, 1982). Genetic sensitivity to environmental context is the hallmark of GxE. From a statistical perspective, GxE is detected when environmental differences amplify or suppress associations between individual genomic variants and a phenotype, having a corresponding impact on heritability. Thus, both genetic associations with a phenotype at the level of individual variants and the broad heritability of the phenotype depend on the environmental context. In this paper, we present a novel method for detecting differences in the single nucleotide polymorphism (SNP) heritability (h²_SNP) at different levels of a moderating environment using marginal genetic effects (Verhulst et al., 2021) and Linkage Disequilibrium Score Regression (LDSC; Bulik-Sullivan et al., 2015). We demonstrate the effectiveness of our method using BMI data from the UK Biobank as a model phenotype.

Our work integrates two overlapping theoretical perspectives that characterize two different types of GxE. The first framework was advanced by Durvasula & Price (2025) based on observation from human genomic data. The second is based on twin GxE studies (Neale et al., 2006; Purcell, 2002). Both frameworks identify two primary types of GxE: 1) imperfect genetic correlation or Qualitative GxE; and 2) varying genetic variance or Quantitative GxE. Importantly, these two classes of GxE are not mutually exclusive, as moderators may have both qualitative and quantitative effects at different loci; nor are they exhaustive, as other patterns of GxE may exist.

Imperfect genetic correlation (Durvasaula & Price, 2025) or Qualitative GxE (Neale et al., 2006) describes instances of GxE where the environmental context fundamentally changes the genetic architecture of the phenotype. Colloquially, different genetic factors contribute to the phenotype in different environments, driving the genetic correlations (r_G) for the phenotype at different levels of the environment below unity (r_G < 1). For example, imagine that most SNPs are invariant across sex, but a subset of the variants are only associated with the outcome in males and another subset in females. The sex-specific variants would contribute to the respective genetic variances (or heritability), but not to the genetic covariance. Thus, the genetic covariance (and hence r_G) would be driven by the sex-invariant associations. The twin literature refers to this phenomenon as qualitative sex limitation (Eaves et al., 1978; Neale et al., 2006), but it can easily be extended to other moderators. Quantitative GxE, or varying genetic variance, exists when the moderator amplifies or suppresses the effect for some (or all) genetic associations: the same variants are involved, but the genetic associations are stronger in some environments. Intuitively, the environment acts as a volume dial that can amplify (turn up) or suppress (turn down) genetic associations at a particular locus.

Durvasula & Price also refer to proportional amplification, which occurs when genetic variants account for the same proportion of variance in the phenotype across the moderating environment (i.e., the single nucleotide polymorphism (SNP) based heritability or h²_SNP is the same), but the variance of the phenotype is larger at some level of the environment. A parallel concept from the twin literature is scalar GxE, where an environmental factor amplifies the phenotypic variance so that each individual variance component is proportionally affected (Neale et al., 2006). It is difficult to definitively classify proportional amplification as a type of GxE, as the effects are likely driven by heteroskedasticity.

Integrating Linkage Score Disequilibrium Regression and Genetic Marginal Effects to Study GxE Using SNP-Based Data

The LDSC method can be extended to test for both qualitative and quantitative GxE. Standard LDSC regresses the χ² values for SNP associations identified by a genome wide association study (GWAS) onto a score that captures the known LD (i.e., correlations between the variants) in the sample population. The slope of this regression indexes the h²_SNP of the phenotype (Bulik-Sullivan et al., 2015). Because LDSC does not require variants to exceed the genome-wide significance threshold, it is an extremely useful tool for interrogating highly polygenic phenotypes with numerous effect sizes that may not be genome-wide significant. Extending LDSC to test for GxE requires GWAS summary statistics for different levels of the moderating environment. These summary statistics can be obtained by conducting stratified GWAS or by conducting GxE GWAS and calculating genetic marginal effects. While stratified GWAS are a straightforward extension of standard GWAS, subdividing the sample can reduce statistical power. For continuous moderators, median or tertiary splits may obscure subtle differences between different levels of the moderator. By contrast, conducting a GxE GWAS uses the entire sample, and genetic marginal effects can be calculated for any level of the moderator.

Genetic marginal effects simplify the interpretation of GxE GWAS results and can be interpreted in almost the same way as standard GWAS summary statistics, just at a specific level of the moderating environment (Verhulst et al., 2021). Specifically, a genetic marginal effect captures the rate at which the outcome is expected to change based on a one allele increase in the SNP at a specific level of the environment. In GxE GWAS, because the genetic association for each SNP may depend on the level of the environment, it is necessary to integrate the main effects and the interaction effects into a single interpretable value. Mathematically, this is as simple as taking the first derivative of the regression equation with respect to the SNP (See Eqs. 2, 3 & 4 in the method section). These genetic marginal effects (at characteristic levels of the moderator) can be directly used by LDSC to estimate heritability and cross-trait genetic correlations.

With stratified GWAS summary statistics or genetic marginal effects, qualitative GxE (imperfect genetic correlation) can be identified by using cross-trait LDSC at different levels of the moderator. If qualitative GxE exists, the genetic correlations between the GWAS summary statistics at different levels of the environment would be less than 1. Alternatively, using the same stratified GWAS summary statistics or genetic marginal effects, quantitative GxE (varying genetic variance) can be identified by estimating h²_SNP at different levels of the moderating environment and demonstrating that the heritability is different at one level of the moderator relative to another.

Existing Gene-Environment Interaction Methods

Several experimental and statistical methods have been developed to identify GxE (Dick, 2011; Herrera-Luis et al., 2024). Model organism studies identify genetic sensitivity to environmental characteristics across numerous environmental moderators, genomic variants, and phenotypes (Cooper & Zubek, 1958; Fisher et al., 1932; W. Huang et al., 2020; Kang et al., 2014; Yadesa, 2022) by strictly manipulating both the genotypes (via selectively breeding organisms with differential genetic strains of known effect) and environments (via careful experimental manipulations). While powerful, such methods often do not transfer to humans due to a mixture of practical and ethical concerns for many phenotypes. Twin and family studies of GxE also find widespread evidence of GxE (Dick, 2011; Verhulst & Hatemi, 2013), focusing on differences in the heritability of a phenotype across an environmental spectrum (Della Bella & Lucchini, 2015; Turkheimer et al., 2003; Young-Wolff et al., 2011). However, twin results cannot identify specific actionable alleles that are sensitive to environmental variation. Candidate GxE (cGxE) studies selected genes and environments based on a priori expectations of moderation. While moderately successful for some phenotypes including BMI (Reddon et al., 2016), cGxE studies often failed to replicate due to a confluence of study design factors such as small sample sizes, lower statistical power, and failing to adequately account for polygenicity and multiple testing (Culverhouse et al., 2018; Dick et al., 2015; Duncan & Keller, 2011). Nevertheless, these methods provide a framework for extending GxE to GWAS data.

To date, three general classes of methods have been developed to study GxE in large GWAS datasets. The first class of methods use PRS constructed using regression weights from well powered linear GWAS studies (Colodro-Conde, Couvy-Duchesne, Whitfield, et al., 2018; Colodro-Conde, Couvy-Duchesne, Zhu, et al., 2018; Durvasula & Price, 2025). In PRS by environment interaction (PRSxE) studies, a PRS is regressed upon a phenotype in the target sample, along with the environment and the interaction between the PRS and the environment. As PRS are constructed from standard (unmoderated) GWAS summary statistics, such techniques imply that the SNPs that contribute to the PRS interact with the environmental moderator in the same direction and proportional to their effect sizes in the original GWAS. While possible, it is unlikely that all SNPs both interact with the environment and do so in a consistent manner. Accordingly, PRSxE methods may provide biased estimates in some situations. To address some of these limitations, novel PRS-environment interaction models have been extended to actively take GxE in the discovery sample into account, such as interaction polygenic risk score analysis (iPRS) (Tang et al., 2022) or GxE PRS (Jayasinghe et al., 2023). However, as PRS methods require two independent samples, data requirements remain a barrier for many researchers, and harmonizing both phenotypes and moderators across samples may induce heterogeneity, leading to the potential underestimation of GxE.

The second class of genome-wide GxE methods use a genetic relatedness (i.e., GREML) mixed modeling approach to estimate SNP-based heritability. GREML partitions variance into homogeneous (i.e., additive genetic variance or V_G), heterogeneous (i.e., GxE variance or V_GxE), and residual variation (Dahl et al., 2020; Ni et al., 2019). When V_G, V_E and V_GxE are simultaneously included in GREML analyses, the genetic and environmental main effects (V_G and V_E) may absorb some of the GxE variance due collinearity between the effects, resulting in an overestimate of the main effects and an underestimation of the contribution of GxE. Importantly, the relatively small proportions of variance attributed to GxE in extant analyses may underestimate true GxE processes (Dahl et al., 2020; Shin & Lee, 2021). In addition, the computational intensity of GREML methods limits the scalability of the method for biobank-sized samples.

The final class of genome-wide GxE methods uses LDSC to estimate the heritability of the interaction coefficient under the assumption that the interaction coefficient uniquely indexes GxE (Shin & Lee, 2021). Estimating the h²_SNP from genome-wide interaction coefficients is intuitively appealing. However, because the interpretation of interaction coefficients depends on the genetic or environmental main effects, failing to integrate the main effects into GxE h²_SNP estimates makes any interpretation incomplete. Accordingly, h²_SNP estimates that focus on interaction coefficients is analogous to random effects of the SNPs (i.e., the variation around the SNP main effects) and should be interpreted with caution.

GxE in Body Mass Index (BMI)

For a practical application of our method to real data, we analyze data for body mass index (BMI). BMI is a measure of an individual’s body fat, captured by the ratio of their weight and height (BMI = kg/m²). As BMI is approximately normally distributed, it is an excellent model phenotype for methodological applications (Hall & Cole, 2006; Nuttall, 2015). While other, more invasive methods, of measuring body fat may have more medical validity, BMI is a simple, broadly used indicator of obesity. From a genetic perspective, twin studies find that BMI is highly heritable (Hakala et al., 1999; Horn et al., 2015; Locke et al., 2015) and genomic variants have been identified in both candidate gene (Rask-Andersen et al., 2017; Reddon et al., 2016; Yeo, 2017) and GWAS . Nevertheless, an individual’s BMI varies across time and is sensitive to a variety of lifestyle factors which amplify or suppress its genetic architecture. Notably, BMI varies across sex and age (the focus of the current paper), as well as diet, exercise, and numerous other health related factors.

Studies examining GxE for BMI report mixed results (Hakala et al., 1999; Horn et al., 2015; Pallister et al., 2014; Schousboe et al., 2004). Some twin studies report activity levels, smoking, eating habits, and nutrition intake moderate the genetic risk for BMI (Hakala et al., 1999; Horn et al., 2015), while others have been unable to replicate the findings (Schousboe et al., 2004). cGxE studies of BMI identified multiple genes that are sensitive to eating habits, physical activity levels, age, sex, psychosocial factors, smoking, and alcohol consumption (Reddon et al., 2016), but many do not replicate (Duncan & Keller, 2011). Finally, genome-wide GxE heritability studies of BMI have also found mixed results that, when successful, tend to stratify GWAS data into discrete subgroups to identify novel SNPs associated with BMI based on their interactions with age (younger or older than 50 years) and sex (Winkler et al., 2015). Finally, analyses conducted by the GIANT consortium identified genomic moderation in BMI for smoking behaviors (Justice et al., 2017) and physical activity (Graff et al., 2017) stratifying the analyses by high vs. low levels of the moderators.

While the heritability of BMI is potentially moderated by a wide range of lifestyles, behaviors, and contexts, we focus our analytical demonstration on sex and age. Previous research has found significant, though partially inconsistent, evidence of GxE for both sex and age. Twin studies consistently find that BMI is more heritable in females (Schousboe et al., 2004) and cGxE studies find stronger associations in females for BMI with key genes, such as MC4R (Reddon et al., 2016). However, GxE for age is inconsistent. While twin studies (Korkeila et al., 1991; Min et al., 2013) and stratified young/old molecular analyses (Reddon et al., 2016; Winkler et al., 2015) find that the heritability of BMI declines with age, genome-wide GxE h²_SNP methods find limited evidence of moderation (Poveda et al., 2017; Shin & Lee, 2021). This is somewhat expected given the small effect sizes in GREML and LDSC methods (Di Scipio et al., 2023; Poveda et al., 2017; Shin & Lee, 2021). As we treat sex as a binary variable and age as continuous, these variables serve as prototypical classes of moderators for examining GxE that can easily be extended to other binary or continuous moderators.

Methods

Our approach requires two preliminary steps before estimating h²_SNP. Specifically, it is necessary to obtain summary statistics from a GxE GWAS and then calculate genetic marginal effects and effective sample sizes at characteristic values of the moderator to estimate h²_SNP for the outcome. Because the necessary functions for conducting moderated GWAS and calculating marginal genetic effects are already included in GW-SEM (Pritikin et al., 2021; Verhulst et al., 2017), and functions to estimate LDSC heritability are available in both python (Bulik-Sullivan et al., 2015) and R (Grotzinger et al., 2019), it is unnecessary to release a new statistical package.

Moderated Genome-Wide Association Study (moderated GWAS).

At present, appropriate summary statistics to conduct moderated LDSC analyses are relatively rare, meaning researchers may need to conduct the GxE GWAS themselves. The moderated GWAS model is an extension of the standard GWAS. The standard GWAS regression model is:

$$Y_i={\widehat{\beta }}_0+{\widehat{\beta }}_{1}SN{P}_{ij}+\widehat{\gamma }Covariate{s}_i+{ϵ}_i$$ Eq. 1

where $Y_i$ is the outcome phenotype for the i^th person, SNP_ij is the j^th genetic variant for the i^th person, and Covariates_i are the standard GWAS covariates for person_i, such as sex, age, and genetic ancestry principal components. Thus, ${\widehat{\beta }}_1$ is the estimate of the genetic association (which is passed to LDSC), while ${\widehat{\beta }}_0$ is the intercept, and $\widehat{\gamma }$ is a vector of regression coefficients corresponding to the included covariates.

The GxE GWAS model extends the standard GWAS model by adding an interaction between the environment and each variant, as well as explicitly including the environmental factor (Verhulst et al., 2021):

$$Y_i={\widehat{\beta }}_0+{\widehat{\beta }}_{1}SN{P}_{ij}+{\widehat{\beta }}_{2}En{v}_i+{\widehat{\beta }}_{3}SN{P}_{ij}xEn{v}_i+\widehat{\gamma }Covariate{s}_i+{ϵ}_i$$ Eq. 2

In the GxE GWAS model, the association between both the genetic and environmental variables and the phenotype depend on the interaction effect, ${\widehat{\beta }}_3$. The coefficients ${\widehat{\beta }}_1$ and ${\widehat{\beta }}_2$ index the genetic and environmental moderator associations, respectively. Importantly, ${\widehat{\beta }}_3$ magnifies (or reduces) these associations. Thus, while ${\widehat{\beta }}_3$ provides a test of whether the effect of a SNP on the phenotype varies at different levels of the environment, it does not incorporate the baseline association. As we focus on sex and age as moderators, the explicit inclusion of the environment in Eq 2 may seem redundant as both variables are routinely included in GWAS as covariates. However, other moderators may not be routinely included as standard GWAS covariates to avoid introducing collider bias. Importantly, excluding the main effect of the moderator from the GxE GWAS may bias ${\widehat{\beta }}_3$ and threaten the validity of the results. Because the interaction parameter is difficult to interpret directly, it is advantageous to calculate genetic marginal effects to examine the genetic association at specific levels of an environmental moderator.

Calculating Genetic Marginal Effects.

The interpretation of genetic marginal effects is very similar to stratified GWAS (a separate GWAS for each level of the moderator). For continuous moderators, fine-grained stratified GWAS analyses may reduce the sample size for any specific level of the moderator (e.g. age in single year bins) to such an extent that it would negate the possibility of detecting reliable genomic signals. Because genetic marginal effects can be calculated for multiple values of a moderator from a single set of GxE GWAS summary statistics, they are effective for continuous, ordinal, or binary moderators (Verhulst et al., 2021). For example, it is possible to calculate genetic marginal effects for observed categories of discrete moderators, critical diagnostic thresholds for medical traits used as moderators, or multiple notable thresholds in the distribution of a continuous moderating variable. In a standard GWAS model, the genetic marginal effect of the SNP is the regression coefficient, ${\widehat{\beta }}_1$. In the GxE GWAS model, the genetic marginal effect is a function of both genetic and environmental factors (Verhulst et al., 2021). To calculate genetic marginal effects $\left({\widehat{\beta }}_{ME}\right)$, we take the first derivative of the GxE GWAS model with respect to the SNP, leaving:

$${\widehat{\beta }}_{ME}={\widehat{\beta }}_1+{\widehat{\beta }}_{3}Env$$ Eq. 3

We can then use this equation to calculate genetic marginal effects for any value of the environment by inserting a characteristic value for Env on a genome-wide basis.

We can calculate the standard errors of the genetic marginal effects (SE_ME) using parameters from the variance-covariance (vcov) matrix of the moderated GWAS model:

$${\text{SE}}_{\text{ME}}=\sqrt{{\sigma }_{{\beta }_1}^2+{\sigma }_{{\beta }_3}^2{\text{Env}}^2+2{\sigma }_{\left({\beta }_1,{\beta }_3\right)}\mathrm{Env}}$$ Eq. 4

inserting the same value of Env that was used to calculate the corresponding genetic marginal effects. After calculating the marginal effect and the standard error, the z-statistic and p-value are easily calculated for use in LDSC. This process is then repeated for each SNP. This process is automated in GW-SEM (Pritikin et al., 2021), which stores the ${\sigma }_{\left({\beta }_1,{\beta }_3\right)}$ covariance statistic necessary to calculate the standard error of the marginal effects.

Adjusted Sample Size.

Users must supply accurate sample sizes to estimate h²_SNP using LDSC to avoid over- or under-estimating heritability, but the total sample size for the GxE GWAS analysis does not reflect the sample size for a marginal effect. Thus, an adjusted sample size must be calculated for each marginal effect before estimating h²_SNP. Adjusted sample size is calculated based on the assumption that the standard error of a parameter estimate is equal to the ratio of the standard distribution of the parameter and the square root of the sample size. Using the standard error of the genetic marginal effects for the mean level of the moderator (which has the highest level of statistical precision) and the overall sample size from the GWAS which is the adjusted sample size at the mean of the moderator, we calculate the standard deviation (SD) of the marginal effect.

$$SD=\sqrt{N}xS{E}_{ME.MEAN}$$ Eq. 5

Then, assuming the SD is homoscedastic, we can calculate the adjusted sample size, N_adj, for a marginal effect at any level of the moderator by solving for N, using the calculated SE_ME.

$$N_{adj}={\left(\frac{SD}{S{E}_{ME}}\right)}^2$$ Eq. 6

N_adj can then be used calculate h²_SNP.

Estimating Moderated h²_SNP.

Standard LDSC uses GWAS summary statistics to calculate h²_SNP. Marginal genetic effects have similar properties to standard GWAS summary statistics, just at a specific level of the moderator, leaving the interpretation of h²_SNP minimally affected. After 1) conducting the GxE GWAS or obtaining GxE GWAS summary statistics, 2) calculating genetic marginal effects, and 3) calculating the adjusted sample size for the marginal effects, users will have all the necessary information to use LDSC to estimate h²_SNP (Bulik-Sullivan et al., 2015). As genetic marginal effects can be interpreted in the same ways as other GWAS summary statistics, h²_SNP for a particular value of a marginal effect can be interpreted in the same way as standard h²_SNP. More specifically, the interpretation of h²_SNP explicitly references the level of the environment used to calculate the marginal genetic effect.

Qualitative GxE: Identifying differences in cross-trait genetic correlations between different levels of the moderating environment.

The null hypothesis for qualitative GxE is r_G = 1 across the entire range of the moderating environment. If different genetic factors contribute to phenotypic variation at different levels of the environment, r_G will be less than 1. Thus, qualitative GxE can be tested with cross-trait LDSC using genetic marginal effects for different levels of the moderator.

Quantitative GxE: Identifying differences in h²_SNP at different levels of the moderating environment.

After estimating heritability, we calculate a pooled standard error to test for differences between h²_SNP estimates at difference levels of the moderator. We estimate the pooled standard error using the equation:

$$S{E}_{pooled}=\sqrt{\left(\right[(N_a-1)*s{e}_a^2]+[(N_b-1)*s{e}_b^2\left]\right)/(N_a+N_b-K)}$$ Eq. 7

where N is the adjusted sample size for the level of the moderator, se_a and se_b are the standard errors of the h²_SNP estimates for the respective marginal effects, and K is the number of levels of the moderator being compared (i.e., 2 in most cases). We then subtract the difference in heritability for two levels of the moderator and divide by the pooled standard deviation. Using a critical value for the desired level of significance (e.g. 1.96 for a p-value threshold of 0.05), we compare the result of the difference divided by the pooled standard deviation to determine if the difference is significant.

Sample for the Demonstration Analyses.

We used data from the UK Biobank (application number 57923) to conduct the demonstration analyses. The UK Biobank is a large, phenotypically rich dataset, that assessed a range of variables from demographic categories to detailed health information (Allen et al., 2014). We used data from individuals who self-identified as white British, as well as other individuals with European ancestry based on ancestry principal components. We selected BMI as our outcome variable. Subsequently, we selected two prototypical moderators that influence BMI: sex and age (Link & Reue, 2017; Rawlik et al., 2016; Robinson et al., 2017). Descriptive statistics for the sample are presented in Table 1.

Table 1:

Descriptive statistics for the key variables in the GxE GWAS analysis of Body Mass Index (BMI).

	N	Mean BMI	(sd)
Full Sample
BMI	387984	27.38	(4.77)
Age‡	387984	57.26	(8.01)
Sex
Females	209427	26.99	(5.15)
Males	178557	27.84	(4.24)
Age
BMI (<50)	86845	26.92	(4.93)
BMI (50–60)	128973	27.45	(4.95)
BMI (>60)	172130	27.56	(4.53)

Note: BMI is calculated as the ratio of weight to height squared (BMI = kg/m²). The total sample size for both GxE GWAS analyses is 387,984, with 54.0 percent female. While the h²_SNP for age is presented at 40, 50, 60, and 70 in the results section, a tertiary split was conducted for younger than 50, between 50 and 60, and older than 60.

^‡For Age, the N, mean, and sd are the values for Age instead of BMI.

Because GxE summary statistics were unavailable, we conducted GxE GWAS in GW-SEM (Pritikin et al., 2021). Age, sex, and the first ten genetic principal components were included as covariates, as were interactions between the moderator (i.e., sex or age depending on the model) and the first ten PCs (Keller, 2014). Following the GxE GWAS, genetic marginal effects, standard errors, z-statistics, and p-values were calculated. For the sex moderated GWAS, marginal effects were calculated for males and females, and for the age moderated GWAS, genetic marginal effects were calculated for ages 40, 50, 60, and 70. Finally, h²_SNP was estimated for each level for the genetic marginal effects. For all LDSC analyses, the precomputed LD scores for the European reference panel were used (available at: https://console.cloud.google.com/storage/browser/broad-alkesgroup-public-requester-pays). The procedures for estimating LDSC for each marginal effect follows the LDSC tutorial (https://github.com/bulik/ldsc).

Comparison with existing genome-wide GxE methods.

We compared the h²_SNP estimates from our model with GxEsum (Shin & Lee, 2021), another LDSC-based GxE method that focusing on the interaction coefficient $\left({\widehat{\beta }}_3\right)$. While the interpretation of GxE heritability from our method and GxEsum, because both methods use the same GxE GWAS summary statistics, the results are instructive.

Results

Gene-by-sex interactions for BMI (binary moderator).

We conducted a moderated GWAS of BMI treating biological sex as a binary moderator to illustrate how the h²_SNP of BMI differs between males and females. We then calculated genetic marginal effects and standard errors for females (moderator = 0) and males (moderator = 1). Overall, the genetic architecture of BMI appears broadly similar across sex, with genetic associations appearing at similar loci (see Figure SI1 for Manhattan Plots for males and females), with marked differences in the magnitude of the significance between females and males. Most notably, the genetic associations for females tend to be more statistically significant (particularly associations on chromosomes 1, 2, 3, and 18), suggesting genetic factors may play a larger role for females.

Figure 1 shows that the heritability of BMI is higher in females (Females-h²_SNP = 0.28, se = 0.01; Males-h²_SNP = 0.21, se = 0.01; p = 5.36 ×10⁻⁰⁶). This is consistent with quantitative GxE or varying genetic variance. Furthermore, the genetic correlation (r_G) between males and females for BMI was estimated to be 0.94 (se = 0.03, p = 0.04), implying that a slightly different set of genetic factors contributes to BMI depending on sex. This would be considered qualitative sex limitation or imperfect genetic correlation. The subtle differential genetic factors that contribute to BMI in males and females is consistent with known sexual dimorphisms in body composition and body fat percentage (Bredella, 2017). Furthermore, the h²_SNP estimates are significantly different for males and females using the pooled standard deviation calculation (see Eq.7). This implies a mixture of qualitative and quantitative sex limitation where a subset of the genetic variants that contribute to BMI are not shared across males and females, and the variants that are shared across sex have a larger impact on females.

Figure 1: h²_SNP of BMI moderated by biological sex.

h²_SNP for biological females and males with 95% confidence intervals represented by the grey bars.

Gene-by-Age Interactions for BMI (Continuous Moderator).

Like sex, we conducted a GxE GWAS of BMI using age as a continuous moderator and then calculated genetic marginal effects and standard errors for ages 40, 50, 60 and 70 (see Figure SI2 for Manhattan Plots). Consistent with quantitative GxE, we observe a significant decrease in the h²_SNP for BMI as people age (Figure 2). Again, while the genetic associations at each age identify similar loci, the level of statistical significance of genetic loci varies across age.

Figure 2: h²_SNP of BMI moderated by age.

h²_SNP for ages 40, 50, 60 and 70 with 95% confidence intervals represented by the purple shaded region.

Specifically, we observe the highest magnitude of h²_SNP at age 40 which declines steadily at older ages. The differences between h²_SNP are statistically significant between all ages. The h²_SNP estimates for the mean age (56 years of age) remain similar to previous h²_SNP (Locke et al., 2015; Marderstein et al., 2021; Yengo et al., 2018), and we see the same trend as other studies examining heritability across age, where it decreases as age increases (Figure 2) (Korkeila et al., 1991; Winkler et al., 2015). In contrast with sex, the rG_SNP across ages are not statistically different from unity (rG_SNP = 0.95–0.99, ps > 0.05). This suggests the same genetic factors contribute to BMI across the lifespan, while the magnitudes of their influence vary by age.

Comparing the current results with alternative genome-wide GxE methods.

When estimating h²_SNP with GxEsum from the interaction coefficient $\left({\widehat{\beta }}_3\right)$, h²_SNP estimates explain 0.9% (se = 0.001; p < 2.2 10e⁻¹⁶) of the variance of BMI with the interaction with sex and 0.2% for the interaction with age (se = 0.001; p < 0.046). While both GxE h²_SNP estimates are statistically significant, the magnitude of these estimates is smaller than the difference in the heritability estimates we observe based on the marginal effects. Specifically, the h²_SNP of BMI is 7% larger in females than males, and 11% larger at age 40 relative to age 70. Importantly, the magnitude of h²_SNP estimates of the interaction coefficients are comparable to previous h²_SNP estimates from the existing literature (Shin & Lee, 2021). While our method and GxEsum use the same summary statistics, and therefore seem superficially similar, the methodological differences may obscure direct and meaningful comparisons. Specifically, the interaction coefficient $\left({\widehat{\beta }}_3\right)$ is very difficult to interpret without reference to the genetic main effect $\left({\widehat{\beta }}_1\right)$. Accordingly, the interpretation of h²_SNP for the interaction coefficient is more interpretable as a random effect of the h²_SNP main effect, rather than as a GxE effect. This interpretation is very different from our proposed method where differences in h²_SNP can be directly interpreted.

Discussion

We present a method to estimate moderated h²_SNP that highlights how GxE influences the proportion of phenotypic variation for BMI that can be detected at different levels of the two moderators: sex and age. Our method is easy to adapt to a variety of moderators and phenotypic outcomes and straightforward to interpret. We demonstrate that the h²_SNP contributing to BMI variance differs by sex (a binary moderator) and age (a continuous moderator). We detected large and significant effects of GxE on BMI for both sex and age. Consistent with existing evidence, BMI was more heritable in females than males(Link & Reue, 2017; Rawlik et al., 2016; Robinson et al., 2017), and the magnitude of heritability for BMI declined between 40 and 70 years of age (Korkeila et al., 1991; Winkler et al., 2015). These results highlight the benefits of our moderated h²_SNP approach relative to other LDSC-based GxE methods.

The interpretation of moderated h²_SNP follows from the interpretation of marginal genetic effects calculated from GxE GWAS summary statistics. In GWAS, the beta coefficient is the expected change in a dependent variable for each additional allele. Almost identically, the interpretation of a genetic marginal effects is the expected change in a dependent variable for each additional allele for a particular level of the moderator. Therefore, we can estimate h²_SNP for the specific level of the environment that was used to calculate the marginal genetic effect, making the interpretation of moderated h²_SNP straightforward. As LDSC is rapid, this procedure can be repeated for any value of the moderator. The direct interpretation of moderated h²_SNP simplifies the characterization of the interaction between genetic factors and potential moderators that contribute to a phenotype and reduces the likelihood of misinterpretation.

Sex as a significant moderator of BMI heritability.

In our first example, we used sex as the moderator, as there are well-established sex differences in BMI (Bredella, 2017). Male and female differences in BMI arise, in part, from differential genetic and biological factors that affect a variety of different anthropometric pathways such as fat storage, muscle development, and stature (Link & Reue, 2017; Randall et al., 2013; Rawlik et al., 2016; Rosa-Caldwell & Greene, 2019). We calculated two marginal effects: one for males and one for females. To examine quantitative GxE, we estimated the difference between h²_SNP for males and females. We found BMI was more heritable in females (Figure 1). To examine qualitative GxE, we conducted cross-trait LDSC to estimate r_G between males and females. Consistent with qualitative GxE, we found that r_G was significantly less than 1, indicating partially distinct genetic factors contribute to BMI in each sex.

Our results are similar to results from stratified GWAS studies which rely on related methods and assumptions (Randall et al., 2013; Rask-Andersen et al., 2019; Venkatesh et al., 2023; Winkler et al., 2015). Our results are further supported by cGxE studies, which have consistently found differences between males and females for BMI (Reddon et al., 2016). Importantly, research examining sex differences for related measures, like waist-to-hip ratio as a proxy for obesity and fat distribution, and metabolic traits, suggest more genetic loci are associated with these traits in females (Link & Reue, 2017). Given the age range of the participants, the female-specific results may have important implications for the genetic factors that influence BMI during perimenopause and menopause.

Age as a significant moderator of BMI heritability.

Our second example used continuous age as the moderator and, as such, is slightly more complicated. Because age is continuous, we calculated the marginal genetic effects at easily interpretable values: 40, 50, 60, and 70. The age range in the UK biobank data is approximately 40 to 70 years of age, and thus the marginal genetic effects reflect characteristic ages from the sample as people tend to think of age in decades. However, because the marginal genetic effects are calculated from GxE GWAS summary statistics, we could have calculated marginal genetic effects for every year of age between 40 and 70. The results suggest that the heritability of BMI decreases with age (Figure 2). Our results are broadly consistent with previous analyses that have found the heritability of adult BMI progressively decreases with age (Korkeila et al., 1991; Min et al., 2013) with stronger genetic associations in younger sample (Poveda et al., 2017; Reddon et al., 2016; Robinson et al., 2017; Winkler et al., 2015). In contrast with other recently developed methods for estimating genome-wide GxE and heritability (Poveda et al., 2017; Shin & Lee, 2021), our method identifies substantively meaningful differences in the magnitude of h²_SNP BMI across different ages (Figure 2). This is similar to the results of Robinson et al., (2017) who used GCI-GREML to identify genotype-age interactions in BMI, and Poveda et al., (2017), who used a maximum likelihood-based variance component decomposition.

Notably, the estimates from GxEsum h²_SNP (which are based on the interaction coefficients) accounted for a statistically significant but small proportion of variance when compared with the differences in the h²_SNP observed from our method. The magnitude of the GxEsum estimates are comparable to other GxE estimates from similar methods across a range of traits (Di Scipio et al., 2023; Jayasinghe et al., 2023; Tang et al., 2022). As the GxEsum analyses used the same summary statistics as our method, the reason our method provides larger effects cannot be due to differences in the discovery GWAS. Instead, our conceptual approach differed from theirs. Specifically, we focus on the difference in h²_SNP estimates for different marginal effects, while GxEsum focuses on raw ${\widehat{\beta }}_3$ coefficients. Framing GxE in the context of differential h²_SNP rather than the heritability of ${\widehat{\beta }}_3$ produces larger, meaningful, and interpretable differences in the sensitivity of genetic factors to environmental variation. Further, we believe this interpretation offers a more intuitive explanation for the underlying GxE process.

Caveats and Considerations.

While our moderated h²_SNP method aids in understanding GxE in GWAS data, several factors must be properly understood to avoid erroneous interpretations and applications. Here we emphasize the choice of characteristic levels of the environmental moderator, the calculation of adjusted sample sizes for the marginal effects, the use of stratified GWAS results, and integrating gene-environment correlations into the understanding of GxE.

When calculating marginal genetic effects, it is possible to choose any value of the moderator, even if the value is nonsensical. Importantly, the marginal genetic effect at the mean of the moderator corresponds with estimates from standard (unmoderated) GWAS summary statistics, providing the maximum precision and sample size. As the value of the moderator diverges from its mean, the precision of a marginal genetic effect and the adjusted sample size decreases. For normally distributed moderators, the peak of the density of the moderator is the mean (or the maximum sample size) and as values diverge from the mean, the density, and hence the sample size, will decrease. Accordingly, an adjusted sample size must be calculated for each marginal effect before estimating h²_SNP, which accounts for the fact that observations take on different weight the more they deviate from the mean.

As proof-of-principle, when we compared the adjusted sample size for the sex-moderated analysis with the observed number of males and females, the numbers are quite similar (Observed: N_males = 178,557, N_females = 209,427; Mean adjusted sample size: N_males = 178,684, N_females = 209,577). The similarity between the observed and adjusted sample size is expected as marginal genetic effects for moderated regression analyses using binary moderators are analogous to stratified analyses. Accordingly, the difference between conducting LDSC on summary statistics from stratified GWAS and marginal genetic effects from GxE GWAS would be minimal. However, while not necessary for the current method, stratified GWAS analyses will not provide the interaction coefficient that indexes direct locus level tests between groups which would require additional analyses. For ordinal or continuous moderators, stratified GWAS may reduce the sample size to such an extent that the results lack the statistical power to detect any genetic signals whatsoever. Recoding such moderators into binary categories could provide a palliative solution, but the loss of power and precision from this approach may obscure the findings. As such, for continuous moderators, conducting GxE GWAS and calculating marginal genetic effects is the optimal strategy for detecting moderated h²_SNP.

Finally, if a moderator is heritable and correlated with the outcome, it is possible that gene-environment correlations may influence the interpretation of GxE if not integrated into the analysis. The impact of gene-environment correlation (rGE) on GxE has been described in detail for twin models (Purcell, 2002; Van Der Sluis et al., 2012) and GREML models (Dahl et al., 2020; Ni et al., 2019). The current model integrates gene-environment correlation into the estimation of the GxE GWAS by including a term for the main effect of the environment $\left({\widehat{\beta }}_2\right)$ in the moderated regression analysis (Eq. 2) for each SNP in the GxE GWAS. This will account for genetic variance in the outcome that is associated with the environmental main effect prior to estimating heritability. Accordingly, the observed heritability of the outcome may appear underestimated if vertical pleiotropy induces strong associations between the moderator and the outcome, as would be predicted by high levels of active gene-environment correlation. However, excluding the environmental main effect from the GxE GWAS will tend to upwardly bias the GxE coefficient and thus exaggerate differences in h²_SNP across levels of the moderator.

Conclusions

The use of genetic marginal effects to estimate moderated h²_SNP allows researchers to examine how environmental moderators alter the genetic architecture of a phenotype without conducting stratified GWAS. Our results identify differences in h²_SNP of BMI for both binary (i.e., sex) and continuous (i.e., age) moderators. However, our proposed method can be applied to GxE GWAS summary statistics for any moderating variable. Further, our method circumvents the need to conduct multiple stratified analyses that may reduce statistical power in the GWAS summary statistics and thus the h²_SNP estimates.

We view moderated h²_SNP as a starting point for refining the search for specific genetic variants that interact with moderators, allowing researchers to subsequently prioritize investing time, effort, and resources for collecting data and conducting analyses that pinpoint significant locus-level genetic interactions. For example, in samples where people disproportionately have moderators associated with higher phenotypic heritability (e.g., females and younger people in our analyses), we should expect stronger genetic associations for both individual variants and polygenic scores. However, this will likely only apply to quantitative GxE or varying genetic variance where the same variants have a stronger effect in the “environmentally risky” sample. For qualitative GxE, if different genes are associated with the phenotype in different environments, selective ascertainment strategies could paradoxically reduce the predictive validity of PRS and obscure genetic associations. Accordingly, data collection targeting individuals with moderators that amplify heritability should be conducted with caution, as ascertainment can alter the genetic architecture of a trait in subtle ways that may be difficult to anticipate (Cai et al., 2025), potentially hindering the generalizability of findings.

Data Availability:

This project used individual-level genetic and phenotypic data from the UK Biobank (www.ukbiobank.ac.uk): Application Number 57923. Access to individual-level data from the UK Biobank is not publicly available but may be obtained through an application to access the data (https://www.ukbiobank.ac.uk/register-apply/).

Software Availability:

The following software tools are required: R v4.3.1 (R Core Team, 2014; https://www.R-project.org/); OpenMx v2(Boker et al., 2011; Neale et al., 2016; https://CRAN.R-project.org/package=OpenMx); GW-SEM v2 (Pritikin et al., 2021); https://github.com/jpritikin/gwsem); and LDSC v1.0.1 (Bulik-Sullivan et al., 2015; https://github.com/bulik/ldsc).