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A robust and adaptive framework for interaction testing in quantitative traits between multiple genetic loci and exposure variables [1]

['Julian Hecker', 'Channing Division Of Network Medicine', 'Department Of Medicine', 'Brigham', 'Women S Hospital', 'Boston', 'Massachusetts', 'United States Of America', 'Harvard Medical School', 'Dmitry Prokopenko']

Date: 2022-12 

Abstract The identification and understanding of gene-environment interactions can provide insights into the pathways and mechanisms underlying complex diseases. However, testing for gene-environment interaction remains a challenge since a.) statistical power is often limited and b.) modeling of environmental effects is nontrivial and such model misspecifications can lead to false positive interaction findings. To address the lack of statistical power, recent methods aim to identify interactions on an aggregated level using, for example, polygenic risk scores. While this strategy can increase the power to detect interactions, identifying contributing genes and pathways is difficult based on these relatively global results. Here, we propose RITSS (Robust Interaction Testing using Sample Splitting), a gene-environment interaction testing framework for quantitative traits that is based on sample splitting and robust test statistics. RITSS can incorporate sets of genetic variants and/or multiple environmental factors. Based on the user’s choice of statistical/machine learning approaches, a screening step selects and combines potential interactions into scores with improved interpretability. In the testing step, the application of robust statistics minimizes the susceptibility to main effect misspecifications. Using extensive simulation studies, we demonstrate that RITSS controls the type 1 error rate in a wide range of scenarios, and we show how the screening strategy influences statistical power. In an application to lung function phenotypes and human height in the UK Biobank, RITSS identified highly significant interactions based on subcomponents of genetic risk scores. While the contributing single variant interaction signals are weak, our results indicate interaction patterns that result in strong aggregated effects, providing potential insights into underlying gene-environment interaction mechanisms.

Author summary The understanding of gene-environment interactions provides potential insights into the pathways and mechanisms underlying complex diseases, but they are hard to detect since effect sizes are expected to be small. To facilitate the detection of such interactions in quantitative traits, we propose a robust and flexible approach called RITSS that can incorporate sets of genetic variants and multiple environmental factors. RITSS can utilize any suitable machine/statistical learning approach to screen for interactions and rigorously tests these aggregated signals using sample splitting and robust test statistics. We demonstrate the validity and power of our approach in extensive simulation studies. Furthermore, in an application to lung function and height data in the UK Biobank, RITSS discovers highly significant interactions based on subcomponents of genetic risk scores.

Citation: Hecker J, Prokopenko D, Moll M, Lee S, Kim W, Qiao D, et al. (2022) A robust and adaptive framework for interaction testing in quantitative traits between multiple genetic loci and exposure variables. PLoS Genet 18(11): e1010464. https://doi.org/10.1371/journal.pgen.1010464 Editor: Heather J. Cordell, Newcastle University, UNITED KINGDOM Received: January 12, 2022; Accepted: October 4, 2022; Published: November 16, 2022 Copyright: © 2022 Hecker et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: UK Biobank data can be accessed following successful application at https://www.ukbiobank.ac.uk/enable-your-research/apply-for-access. The R package RITSS as well as the code for the simulation studies is available at https://github.com/julianhecker/RITSS. Funding: This research was supported by the National Heart, Lung, and Blood Institute [U01HL089856, U01HL089897, P01HL120839, P01HL132825] and National Human Genome Research Institute [R01HG008976, 2U01HG008685]. MHC was supported by R01HL149861, R01HL135142, R01HL137927, and R01HL147148. MM was supported by T32HL007427 and K08HL159318. DQ was supported by K01HL129039. SML was supported by R01MH129337. BDH was supported by K08HL136928. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: I have read the journal’s policy and the authors of this manuscript have the following competing interests: MHC has received grant funding from GSK and Bayer, and speaking or consulting fees from AstraZeneca, Illumina, and Genentech. EKS has received grant support from GSK and Bayer. WK is a current employee of Eisai Inc. MM received grant support from Bayer. STW reports personal fees from UpToDate, outside the submitted work; STW is on the Scientific Board of Histolix.

Introduction Genome-wide association studies (GWAS) have identified thousands of genetic variants associated with complex diseases and traits [1]. However, the effect of a genetic variant on a complex phenotype or disease can be modified by environmental exposures [2,3]. The knowledge about interactions between environmental exposures (e.g., tobacco smoke or occupational exposure) and genetic variants can provide insights into the underlying pathways and biological disease mechanisms [4]. Many of the methodological approaches for gene-environment interaction testing have focused on scenarios in which a single genetic variant and a single environmental variable are tested for potential interaction. Methods have been developed for case-control data, case-only data, and quantitative trait studies, summarized in detail by Gauderman et al. [5]. Since most gene-environment interaction effect sizes are expected to be small and statistical power is consequently limited [6], power-increasing approaches were proposed. This includes so-called screening statistics that prioritize genetic variants to reduce the multiple testing burden [6–14]. Also, researchers proposed to aggregate genetic information in a genomic region in set-based tests [15–23]. Recent approaches utilize mixed models, reaction norm models, or incorporate multiple environmental variables to derive a combination of environmental factors that potentially modifies the genetic effect [24–28]. Current data analyses observed that genetic risk scores, (weighted) sums of genetic risk variants, interact with smoking exposure in Coronary heart disease [29] and lung function [30,31]. While detecting significant interactions on this aggregated level is encouraging, identifying individual genes and pathways underlying these effects is challenging based on such results. Besides the power limitation, another caveat of gene-environment interaction testing is that the misspecification of the marginal main effect models can lead to false positive findings in interaction testing [32,33]. This is especially problematic for continuous environmental factors where the implicit linearity assumption that is commonly used in the model might not be correct. One example is pack-years of smoking, where it is not straightforward to assume that every additional pack-year has the same, constant effect on the outcome. In this communication, we propose RITSS, a robust and flexible framework for gene-environment interaction testing in quantitative traits. The general idea is to derive an interaction score comprised of the (weighted) sum of individual genetic variant/environmental factor product pairs, such that the combination of these signals increases the power while maintaining biological interpretability. Our approach utilizes a sample splitting strategy and test statistics that are robust against misspecifications of the main effects. The general form of RITSS allows the incorporation of user-specified screening/learning approaches in constructing candidate interaction scores while providing valid statistical inference without restrictive assumptions. In extensive simulation studies, we demonstrate the robustness and power of RITSS in various scenarios. We also applied RITSS to lung function phenotypes and human height in the UK Biobank, incorporating sex information and smoking exposure.

Description of the method Ethics statement This research was conducted by using the UK Biobank resource under application number 20915. All participants provided written informed consent, and study protocols were approved by the North West Multi-centre Research Ethics Committee and ethical procedures were controlled by the UK Biobank Ethics Advisory Committee. RITSS In this section, we describe the RITSS framework and the corresponding algorithm. Set up For sample i, we denote the quantitative trait by Y i , the m-dimensional genotype information by X i , and the d-dimensional environmental information by E i . Also, we define an additional p-dimensional covariate vector by Z i that includes, for example, age, study indicators, and genetic ancestry principal components [34]. The genotype information X i represents a pre-selected set of m genetic variants. We assume the following model where E[ε i |X i , Z i , E i ] = 0. The unknown function μ describes the main effect of the environmental factors E i and other covariates Z i . The genetic contribution of variant j is modeled by π j (E i , Z i )X ij , where π j is an unknown function. The null hypothesis of no gene-environment interaction is described by , where π 0j is an unknown function depending on Z i only. Therefore, we implicitly assume the absence of gene-gene-interactions but, in general, allow for interactions between the genetic variants and the covariates Z i under the null hypothesis. The rationale of RITSS is to incorporate sets of genetic variants and/or multiple environmental factors to increase power and detect interactions based on aggregated signals. This step is motivated by the observation that single variant approaches are often underpowered. There are two critical aspects that RITSS aims to address: 1.) testing of aggregated interaction signals can increase power, but the results are more difficult to interpret since individual factor contributions are unobserved, and 2.) recent results demonstrated that the misspecification of main effects in interaction analyses can lead to false positive interaction findings [32,33]. Using sample splitting to screen for interaction signals Given the input set of genetic and environmental factors, RITSS derives interaction scores of the form U i = ∑ j ∑ l π jl X ij E il that combine signals adaptively while maintaining biological interpretability by keeping the number of involved factors of moderate size. RITSS uses sample splitting since screening and testing, in general, cannot be performed using the same data. Testing of interaction scores using a robust approach Standard interaction tests are based on a product between the candidate interaction score and the phenotype adjusted for main effects . Zhang et al. showed that, for a quantitative trait (identical link function), main effects misspecifications can lead to invalid interaction tests, especially if genetic variants and environmental factors are not independent [33]. To avoid main effect misspecifications, they proposed to apply generalized additive models (GAMs) to model the main effects flexibly. RITSS follows this idea and incorporates rich models for the main effects, including GAM-based solutions. However, while Zhang et al. considered scenarios of single variant and single environmental factors, the dimensions of X i , E i , and Z i , that is m, d, and p, in our setting are of substantial size. The potential high dimensionality of the models leads to slower convergence rates so that standard results might not apply here. Furthermore, considering RITSS as a general framework, we aim to derive an algorithm that can incorporate any other suitable future approach. Therefore, we utilize a concept based on recent theoretical results in causal inference [35]. Specifically, instead of considering the standard test statistic , RITSS is based on , where is an estimate of a transformation of U i that is orthogonal to (i.e., has covariance zero with) the environmental and genetic main effects. This orthogonality improves the robustness against main effect model misspecifications since it reduces the impact of residual main effects in on the product test statistic. Furthermore, we extend the sample splitting and estimate the main effect models as well as the transformation in non-overlapping subsamples of the data that also do not overlap with the subsamples used for the screening step and the subsamples used for evaluating the test statistic. This enables to apply any suitable approach for the screening step and the estimation of the main effect models/transformations, without the need to characterize the asymptotic behavior. It reduces the necessary assumptions to suitable convergence rates [36]. Algorithm We here describe the algorithm underlying RITSS. The algorithm is also visualized in Fig 1. PPT PowerPoint slide

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TIFF original image Download: Fig 1. Visualization of the RITSS algorithm. https://doi.org/10.1371/journal.pgen.1010464.g001 Input: data X i , E i , Z i , Y i , screening function Screen(S) including the parameter S (number of candidate scores), parameter K for the number of splits, splitting fractions c = (c 1 , c 2 , c 3 ). Algorithm: Split the data (Y i , X i , E i , Z i ) randomly into K non-overlapping subsamples I k , k = 1,…,K, of approximately equal size.

, X , E , Z ) randomly into K non-overlapping subsamples I , k = 1,…,K, of approximately equal size. For each subsample I k : Denote all samples not in I k by Perform interaction signal screening based on In is , s = 1,…,S, as predictors in a linear model while incorporating flexible models for the main effects i . Using the alternating conditional expectation (ACE) algorithm [37], estimate/derive the orthogonalized transformation i in S1 Appendix), denoted by Based on the predictions of the trained models, compute k .

: Compute the overall statistic

Properties and discussion Under the null hypothesis of no gene-environment interaction, the overall z-score z is asymptotically normal, that is z→N(0,1) [35,38] and an overall RITSS interaction p-value can be computed. This theoretical result applies if, besides standard regularity conditions, the main effects + and the transformations are estimated with sufficiently fast convergence rates (as a function of sample size). These assumptions are closely related to the corresponding dimensions of genetic information, environmental factors, and covariates (m, d, and p, respectively), since the increasing dimensions increase the complexity of estimating these models sufficiently precise. This is where the advantage of the robust approach is apparent over, for example, solely application of generalized additive models (GAMs). The robust approach reduces the required assumptions to the statement that the specific products of convergence rates (main effects and transformation) are fast enough instead of requiring fast convergence rates for the main effects directly. As a remark, if we assume the absence of interactions between genetic variants and the covariates Z i and an additive genetic model, that is π 0j (Z i ) = π 0j , the genetic main effect model becomes less complex, and faster convergence rates can be achieved. Nevertheless, since general models become complex with growing dimensions rapidly, we consider sets of m genetic variants with m in the magnitude of 102−103 and d≤10 and p≤10 as suitable input dimensions for RITSS. We emphasize that it is therefore only required to estimate the transformation of a given interaction score U i accurately, but the interaction score U i itself can be constructed using any suitable method in the screening step, including machine learning or other advanced statistical learning procedures. Under the null hypothesis, the test is ensured to be valid. On the other hand, the power of RITSS depends on the ability of the screening strategy to detect signals and capture them in the interaction score U i . Therefore, although the screening step can be implemented flexibly, it is crucial for statistical power. We discuss and investigate two different screening strategies in the next section. It is important to note that the K different interaction scores U i , in general, will include different interaction factors since the corresponding screening was performed in different parts of the data. Investigating the overlap between the interaction scores provides further fine mapping of the results. Specific implementation The screening strategy influences the statistical power of RITSS. There is no uniformly optimal strategy since the optimal screening depends on the interaction signal architecture. Specific implementations of RITSS differ in the realization of steps b.) and c.); steps a.), d.), and e.) are fixed throughout different implementations. Here, we describe two different implementations; we will refer to them as RITSS1 and RITSS2. These two implementations will be investigated in the simulation studies in the next section. Screening strategy 1: interaction between single environmental factor and subcomponent of genetic risk score The first screening strategy aims to identify subcomponents of genetic risk scores that interact with a single environmental factor E it , where t is the index of the selected factor. In step b.), we first fit the model in to obtain estimated genetic main effects . Then, we perform approximate best subset selection [39] to identify sets J s , s = 1,2, that capture interaction scores of the form , describing subcomponents of genetic risk scores multiplied with the environmental factor E it . The best subset selection is performed using a 2-sample split of and a step size of 5 between 10 and the maximum number of variants available after additional filtering. J 1 contains all variants that intersect between the best subsets, J 2 contains all variants that were included exactly once. More details are described in S1 Appendix. In step c.), we construct the overall interaction score U i = U i1 +c 2 U i2 , where c 2 is set to 1, if the corresponding regression coefficient p-value is below 0.05, otherwise c 2 = 0. We denote the set of genetic variants in the interaction score U i that is tested in I k by m(I k ). Screening strategy 2: aggregated single environmental factor interactions based on single variant testing In step b.), we fit the models and select variants together with the estimated into the at most two interaction scores based on False Discovery Rate (FDR) q-values to correct for multiple testing. Step c.) is implemented as in screening strategy 1. We refer to S1 Appendix for more details. As for the screening strategy 1, we denote the set of genetic variants in the interaction score U i that is tested in I k by m(I k ).

Results Table 2 contains the results of our UK Biobank main analysis using 180,000 samples based on RITSS1 and RITSS2. For each of the four traits, we observed a highly significant interaction between sex and subcomponents of the genetic risk score using RITSS1. For the lung function traits, we also observed similar interaction findings with height. We investigated this in more detail in the validation analysis below. PPT PowerPoint slide

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TIFF original image Download: Table 2. Results of the interaction testing using the two approaches RITSS1 and RITSS2 in the UK Biobank. The environmental factor tested for interaction is denoted by E it . |m| is the number of total SNPs in the analysis, |m 4 | and |m 3 | are the number of SNPs that are shared by all four and exactly three interaction scores, respectively. P-Y-S: pack-years of smoking, E-S: ever-smoking. https://doi.org/10.1371/journal.pgen.1010464.t002 Furthermore, in line with previous results in the literature, we observed a significant interaction with pack-years of smoking in FEV 1 /FVC [30,31] that maintains significant after Bonferroni correction for n = 32 tests at a significance level of α = 0.05. Based on the K = 4 different interaction scores, we investigated the overlap between the genetic variants included in the sets m(I k ), k = 1,…,4. We denote all genetic variants that were in all four scores by m 4 and all genetic variants included in exactly three scores by m 3 . For the validation analysis, we considered all Y i /E it pairs that had a Bonferroni corrected significant interaction p-value at a significance level of α = 0.05 and non-empty m 4 or m 3 sets; this included RITSS1 results only. The validation analysis was based on the remaining and independent 74,033 samples and a standard regression interaction test while fitting X i , E i , and Z i . We tested two interaction scores, the first based on the variants in the corresponding m 4 set and the second based on variants in m 3 . Here, the genetic main effects for the genetic risk scores were estimated based on the 180,000 samples in the main analysis. The regression interaction p-values were evaluated based on model-based standard errors as well as standard errors obtained from a robust sandwich estimator. The results of our validation analysis are described in Table 3. The results show that all findings replicate. We also note that the directions of effects were consistent between the main analysis and validation analysis. PPT PowerPoint slide

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TIFF original image Download: Table 3. Results of the validation analysis in the 74,033 remaining samples in the UK Biobank. The two sets m 4 and m 3 were identified in the corresponding main analysis (Table 2) and correspond to the results of the RITSS1 analysis. The interactions between the scores (corresponding to m 4 and m 3 respectively) and the environmental factor E it were tested using a linear regression, adjusting for all X i , E i and Z i . The robust p-values are based on standard errors obtained by a robust sandwich variance estimator. P-Y-S: pack-years of smoking. https://doi.org/10.1371/journal.pgen.1010464.t003 To disentangle these findings in more detail, we examined the single variant interaction p-values for the genetic variants in the combined set m 4 /m 3 with the respective environmental factor in the validation dataset. This analysis was based on a linear regression that fits X i , E i , and Z i , and the respective interaction term. Table 4 lists the number of genome-wide significant single variant interactions, i.e., p<5×10−8, as well as the number of significant single variant interactions using a Bonferroni correction adjusting for the number of variants in the corresponding combined set m 4 /m 3 at a significance level of α = 0.05. Both numbers are either 0 or very small, for all trait/environmental factor combinations. This suggests that the interaction effects at the single variant level are weak. However, we made an interesting observation by investigating the empirical distribution of directions of effect across the single variant interaction tests. Table 4 provides these empirical distributions of directions of effect for a.) the unweighted interaction terms X ij ×E it and b.) the weighted terms , where is the estimated genetic effect based on the main analysis dataset. In addition, we compared the interaction p-value distributions according to the directions of effect of the weighted interaction terms. As we can see by the enrichment of positive effects, the weighting by the genetic main effect, as it is applied in the computation of the genetic risk score, “aligns” the interaction effect directions across the genetic variants. Furthermore, the p-values for weighted interaction terms with positive effect show substantially higher inflation. This indicates that RITSS1 identified genetic variants whose interaction signals are weak at the single variant level but when summed up in the genetic risk score, they lead to a significant interaction. Since we used a coding of sex of male = 1 and female = 0, this suggests that the genetic effects in males and females across these variants have the same direction, but the magnitudes of effect in males are slightly larger. Related, we note that the genetic variants in m 4 /m 3 between the sex and height interactions in lung function highly overlap, which is in line with the observation that height and sex are strongly positively correlated. However, if we test the height interaction separately in males and females in the validation dataset, the signal vanished or was strongly diminished. Therefore, we hypothesize that the observed interactions are a result of sex-differential effects, but further analyses are required to disentangle the exact mechanisms. For example, investigating the impact of sitting or thoracic height, instead of standing height [40]. In this context, we note that, related to recent results regarding so-called participation biases and their impact on sex-related analyses [41], the genetic variants in m 4 /m 3 across the three lung function phenotypes and height were not significantly associated with sex, but replication of the interaction signal in an independent cohort is preferable. Detailed information about the selected variants in the gene-by-sex interactions for lung function and height can be found in S2 Appendix. PPT PowerPoint slide

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TIFF original image Download: Table 4. Results of the validation analysis in the 74,033 remaining samples in the UK Biobank II. The two sets m 4 and m 3 were identified in the corresponding main analysis based on the RITSS1 approach (Table 2). All variants in m 4 /m 3 were tested for interaction with the respective environmental factor E it . In this context, gws. interaction refers to a single variant interaction p-value p<5×10−8. Similar, bfs. interaction corresponds to p<0.05/(|m 4 |+|m 3 |) (Bonferroni correction). The last columns report the empirical distribution of effect directions in these tests and the corresponding 5% quantile of p-values (weighted analysis). In the weighted analysis, the genotype/env.factor product term was multiplied by the estimated genetic main effect (obtained from the independent main dataset). P-Y-S: pack-years of smoking. https://doi.org/10.1371/journal.pgen.1010464.t004 We also applied GESAT in the validation analysis. For each of the four traits, we applied GESAT to test for interaction with sex (using the remaining E i and Z i as covariates) based on 1.) all genetic variants in the analysis and 2.) all genetic variants in the corresponding combined m 4 /m 3 set. For all traits, these tests were at least nominal significant (p<0.05), and we obtained similar p-values in the two analyses 1.)/2.) (except for FVC): FEV 1 : p = 1.81×10−11/p = 8.92×10−7, FVC: p = 2.71×10−14/p = 2.29×10−4, FEV 1 /FVC: p = 2.61×10−2/p = 2.29×10−2, height: p = 1.08×10−3/p = 6.19×10−3. These results add further evidence that RITSS1 identified a fine mapped set of interacting genetic variants.

Discussion We propose RITSS, a robust and flexible interaction testing framework for quantitative traits. Our framework aims to identify aggregated interaction signals and tests these signals using sample splitting and robust test statistics. Since interactions at the single genetic variant level are hard to detect due to small effect sizes, we hypothesize that strategies that aggregate suitable signals across a limited number of factors/loci have higher statistical detection power while, at the same time, they still permit an interpretation and can reveal potential biological mechanisms. In extensive simulations, we investigated the performance of RITSS based on two distinct implementations and demonstrated that our framework controls the type 1 error rates across different scenarios. In contrast, alternative approaches showed inflated type 1 error rates in these scenarios. Additionally, we demonstrated the interplay between the screening strategy and RITSS’s power based on selected power simulations. We applied our approach to UK Biobank data and observed potential interactions between subcomponents of risk scores and sex in lung function and height. Since interactions for complex traits at the single genetic variant level were rarely detected, even in recent large-scale analyses, we hypothesize that our approach will be an important tool for the identification of genetic interactions and the underlying mechanisms. Fawcett et al. performed a genome-wide genotype-by-sex interaction analysis of lung function in the UK Biobank, using more than 300,000 samples [40]. Although they found a small number of five genome-wide significant interactions, only one interaction signal was replicated in an independent study. In a different study, Bernabeu et al. utilized the UK Biobank to analyze genotype by sex interactions for 530 complex traits. This analysis revealed several genome-wide significant interaction findings, but heritability and genetic correlation analyses suggest that substantial proportions of the sex-differential genetic architecture are yet to be discovered and identified [42]. Our results are in line with these findings since our interaction signals are not driven by strong effects at the single variant level but aggregations across sets of variants. Furthermore, recent work by Zhu et al. hypothesizes and demonstrates that so-called amplification is a common mode of genotype-by-sex interactions [43]. Amplification describes the setting of shared genetic variants, with the same direction of effect, but different magnitudes. This is exactly what we observed in our UK Biobank sex interaction analysis of lung function and human height. However, we note that analyses based on independent datasets are required to further replicate, validate, and disentangle these potential gene-by-sex interactions. Our approach has the following limitations. A significant interaction p-value of the aggregated score does not imply that all included pairs of genetic variants/environmental factors necessarily truly contribute to the interaction signal. Another limitation is that the set of genetic variants needs to be pre-selected and, as described in the Methods section, the size m is assumed to be of moderate magnitude in the range of 102−103 (assuming typical GWAS sample sizes). This restriction is related to the required assumption that the main effects and transformations can be estimated accurate enough given the sample size. The computational burden, depending on the specific implementation of the screening strategy, also increases with m. Therefore, an important question is how to pre-select the set of genetic variants for RITSS testing. It is possible to integrate an external screening approach. For a quantitative trait, one can screen for potential interaction signals based on marginal effects or variance-effects [10,14]. Based on the results of such a screening, one can choose sets of genetic variants for RITSS testing and introduce weighted hypotheses reflecting the results of this external screening approach. It is important to note that this screening should not be confused with the screening and testing steps within the RITSS framework. In the UK Biobank application, we restricted the input to previously reported GWAS findings for the respective trait, which can be interpreted as a special case of this prior external screening. However, interaction effects do not need to be accompanied by marginal effects and therefore our analysis provides only a limited insight. In general, the interpretation of significant RITSS p-values requires careful consideration. RITSS aims to identify deviations from the null hypothesis model , represented as significant interaction scores U i = ∑ j ∑ l π jl X ij E il . However, a significant contribution by U i does not imply interaction between E i and X i without assumptions regarding no unmeasured confounding/omitted variables and the sampling design [44,45]. Moreover, measurement of environmental factors is often subject to noise or even bias. This can result in false positive interaction findings since RITSS assumes that environmental factors are measured without error. Furthermore, due to the computational burden, our type 1 error simulations are based on 10,000 replicates only and, therefore, the validity at small significance levels cannot be empirically investigated. Nevertheless, we emphasize that the RITSS approach is supported by theoretical results describing the asymptotic normality of the test statistic. The simulation results demonstrated the concordance with these theoretical approximations. Finally, due to the application of sample splitting and several independent estimation steps, RITSS requires datasets with sample sizes that are in the order of recent GWAS publications (30,000 samples or more). Future directions include more detailed follow-up analyses of the observed gene-by-sex interactions, the incorporation of other screening techniques, such as, for example, LASSO [46] or RaSE [47], and the improvement of the computational efficiency. We plan to extend RITSS to incorporate dichotomous traits and implement these extensions in the R package RITSS. We note that RITSS can also be used to analyze gene-gene-interactions and interaction models can be based on more general, non-linear approaches. In addition, RITSS can be applied to other omics data layers, such as DNA methylation, transcriptomics, or metabolomics. The R package RITSS as well as the code for the simulation studies is available at https://github.com/julianhecker/RITSS.

Supporting information S1 Appendix. ACE algorithm and RITSS main effects, Screening strategies, Implementation of GAMsv, Simulation details, Additional simulation study results, and UK Biobank data. This appendix contains additional information regarding the ACE algorithm and RITSS main effects, the screening strategies, the implementation of GAMsv, simulation details, additional simulation study results, and the UK Biobank data. https://doi.org/10.1371/journal.pgen.1010464.s001 (PDF) S2 Appendix. Additional information regarding genetic variants tested in the UK Biobank analysis. These tables contain all genetic variants as well as the m 4 /m 3 information for the sex-interaction analysis in the UK Biobank. https://doi.org/10.1371/journal.pgen.1010464.s002 (XLSX) S1 Fig. Density plots for standardized residuals for all four traits in the analysis. FEV 1 : forced expiratory volume in 1 second, FVC: forced vital capacity. https://doi.org/10.1371/journal.pgen.1010464.s003 (TIF)

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