A complex disease has, by definition, multiple genetic causes. In theory, these causes could be identified individually, but their identification will likely benefit from informed use of anticipated interactions between causes. In addition, characterizing and understanding interactions must be considered key to revealing the etiology of any complex disease. Large-scale collaborative efforts are now paving the way for comprehensive studies of interaction. As a consequence, there is a need for methods with a computational efficiency sufficient for modern data sets as well as for improvements of statistical accuracy and power. Another issue is that, currently, the relation between different methods for interaction inference is in many cases not transparent, complicating the comparison and interpretation of results between different interaction studies. In this paper we present computationally efficient tests of interaction for the complete family of generalized linear models (GLMs). The tests can be applied for inference of single or multiple interaction parameters, but we show, by simulation, that jointly testing the full set of interaction parameters yields superior power and control of false positive rate. Based on these tests we also describe how to combine results from multiple independent studies of interaction in a meta-analysis. We investigate the impact of several assumptions commonly made when modeling interactions. We also show that, across the important class of models with a full set of interaction parameters, jointly testing the interaction parameters yields identical results. Further, we apply our method to genetic data for cardiovascular disease. This allowed us to identify a putative interaction involved in Lp (a) plasma levels between two ‘tag’ variants in the LPA locus (p = 2. 42 ⋅ 10−09) as well as replicate the interaction (p = 6. 97 ⋅ 10−07). Finally, our meta-analysis method is used in a small (N = 16,181) study of interactions in myocardial infarction.
Large data sets are vital to counter low statistical power due to low allele frequency, small effect sizes, and multiple testing. This has driven the GWAS field towards more collaborative efforts as well as meta-analyses. Fortunately, there exists a standardized statistical methodology that allows for reliability and comparability between different studies. In contrast, in association studies aiming at identifying interactions, or epistasis, there are multiple competing methodologies with unclear relationships. As a consequence, collaborative GWAS efforts have almost exclusively focused on single variant associations. Conceptually, interactions in association studies are generated when multiple genetic variants affect the dynamic, non-linear and inter-connected networks that underlie complex traits [1]. Candidate-based medical genetic studies established early that interaction may play an important role in complex diseases [2–7]. Consequently, over the last five years, substantial attention has been devoted to resolving the statistical and computational problems associated with large-scale studies of interactions [8–15]. However, the formulation of these methods is still not easily compared, and crucial differences between the underlying interaction models are not always transparent. This makes comparability between studies still a major concern and hampers opportunities for meta-analysis. There is, therefore, a need to harmonize interaction models and investigate the implications of their assumptions. From a statistical point of view it is non-trivial to define interaction [16] and it can often be unclear how different assumptions, e. g. , on the main effect of each variant, affect the definition of interaction. There has been some work aimed at a standardized description of interaction models [17,18]. However, these studies specifically targeted a class of models, in which any two models are related by a linear one-to-one transformation, which limits their applicability. In this paper, we present a more general framework that enables modeling and interpretation of genetic interactions in the context of any generalized linear model (GLM). This can be applied to, in principle, any type of outcome (e. g. , continuous, binary, factor or count phentoypes) or model of interaction. We show how this new formulation can be used to analyze the relation between various interaction models. Multiple tests for interaction have been proposed for case-control data. However, these tests typically depend on strong assumptions about the main effects, marginal effects or LD to reduce the computational complexity [15,19,20]. Recently Yu et al. introduced a closed-form Wald test restricted to a specific parameterization of the logistic regression model [21]. Here, we introduce a general class of computationally efficient Wald tests, that enables analysis of case-control traits, quantitative traits, and in fact any trait modeled by a member in the exponential family. More importantly, these tests allow for any combination of parameterization and link function to be used, that is, it can be applied to all the models considered here. Moreover, we show that our Wald test can be applied in large-scale meta-analyses. A major complication in interpreting interactions is that they are inferred relative to a link function. This function determines the parameter subspace that belongs to the null model and is, in practice, unknown. Consequently, mis-specification of the link function causes an inflated error rate that increases with sample size, which cannot be resolved by replication in a separate cohort. Here we address this issue by testing interactions using a family of link functions. Specifically we use two families of link functions that has been proposed previously [22,23]. We also show that the previously suggested goodness-of-link test [24] is not appropriate for joint testing of interaction parameters. We implement these new tests in a GLM-based analysis tool for both case-control and quantitative data. We investigate the impact of different parameterizations on both the false positive rate and the statistical power. We finally apply our Wald tests in two genome-wide interaction analyses. Firstly, we study a continuous phenotype, Lp (a), in the PROCARDIS cohort. Secondly, we perform a meta-analysis of myocardial infarction by combining results from the PROCARDIS cohort and the Myocardial Infarction Genetics Consortium cohort.
We will now introduce five different strategies for testing interactions. We will use these strategies together with the Wald test described in Section Fast estimation and testing of interaction in generalized linear models, above, in our investigations of statistical power and false positive rate. The first three are based on two different saturated parameterizations: G × G and AD × AD (Fig 1). The G × G test models is the standard approach for regression on discrete variables in the GLM literature, the AD × AD test model corresponds to the F∞ model discussed by [17] (the careful reader will notice that, compared to [17], we have changed the coding of the second column of AD from (0, −1,1) to (0,1, 2) to allow for easy comparison with G). A parameterization that features multiple interaction parameters can be tested either jointly or separately. Let the interaction parameter vector be denoted by δ = {δ11, δ21, δ12, δ22, }. A joint test evaluates the hypothesis that all interaction parameters are zero δ = 0 and a separate test evaluates the hypothesis that one or more interaction parameters are zero ∪h{δh = 0}; notice for the separate test that, while each test has a lower degree of freedom, the multiple testing burden will increase by a factor of 4. In our first two strategies, the interaction parameters are tested by a joint test and we will refer to these tests as G × G joint and AD × AD joint, respectively, while the third strategy is based on a separate test and the G × G parameterization, which we refer to as G × G separate. For the next strategy, we introduce a non-saturated parameterization that assumes that the genotypic effect is completely additive in the number of minor alleles, and thereby ignores possible deviations. This corresponds to the Kronecker product of two additive-encoded uni-variant GLMs (A ⊗ A). This model is often written g (μ a b) = α + a β 1 + b γ 1 + a b δ 11 This is a much more restricted test model and the interaction is now represented by a single parameter δ11 instead of four as in the previous models. We refer to this test as the A × A test. The last strategy, which has been applied in multiple studies of interactions [9,27], is to first encode the genotypes into binary variables according to dominance or recessiveness. These binary variables are then analyzed separately in an interaction test. This encoding corresponds to the (saturated) parameterizations D × D, R × D, D × R and R × R: g (μ a b) = α + I (a ≥ 1) β + I (b ≥ 1) γ + I (a ≥ 1) I (b ≥ 1) δ g (μ a b) = α + I (a ≥ 1) β + I (b = 2) γ + I (a ≥ 1) I (b = 2) δ g (μ a b) = α + I (a = 2) β + I (b ≥ 1) γ + I (a = 2) I (b ≥ 1) δ g (μ a b) = α + I (a = 2) β + I (b = 2) γ + I (a = 2) I (b = 2) δ In each GLM, interaction is measured by a single parameter δ. In analogy with separate testing, we evaluate if δ in any of the test models deviate from zero, and the multiple testing burden will increase by a factor of 4. We refer to this family of tests as R/D × R/D. We performed two experiments to investigate how potential model misspecifications affect the false positive rate (FPR) for the interaction tests described previously. In the first experiment we generated synthetic data from (null) models with no interaction, but where some assumptions of the evaluated tests fail (specifically, the presence of recessive and dominance for the A × A test and the presence of an additive component for the R/D × R/D test); we also tested whether linkage disequilibrium (LD) affects the FPR. The results in Fig 2 show that all tests controlled the error rate when there is no LD for a Normal dispersion distribution. However, for the Binomial dispersion distribution, the error rate of the A × A and R/D × R/D tests was inflated by the presence of dominant and recessive inheritance patterns. A second source of errors is LD, and the A × A test had a strongly inflated error rate when data was generated from R × A and R × D under both dispersion distributions. Moreover, also the R/D × R/D parameterization had an inflated error rate for all generative models under LD. The error rate was highest for data generated from R × A and a Binomial dispersion distribution. In general, the G × G joint, G × G separate and AD × AD joint tests are safe to use in the presence of LD, whereas the other tests must be treated with caution. In the second experiment, we investigated the impact of link function misspecification on the FPR. As discussed in Section Relating different GLMs, it is known that interactions inferred with one link function might be absent when another link function [25,26] is used, and vice versa. Here we test how often link misspecification introduce “false interactions”. We generated data from the A × A model with the log link function, whereas we performed each test using the identity link function. We measured the FPR as a function of the main effect of the second variant, while the main effect of the first variant is constant. Notice that when the second main effect is zero, the phenotype depends only on the first variant and, thus, no inflation of “false interactions” is expected. The results (Fig 3) show that all tests were affected similarly by link misspecification, and the false positive rate quickly increased with the second main effect’s deviation from zero; this effect was slightly more pronounced for the A × A test. The effect became more pronounced as the sample size increased. This demonstrates, as previously predicted [25], that use of an erroneous link function may lead to false inference of interaction. Designing a convincing and realistic experiment for measuring the expected statistical power is challenging. Ideally, we want to generate data from a biologically relevant model. However, neither the effect sizes, nor the structure of such a model are known. In addition, there is a large number of possible interaction models, even for single pair interactions where we have nine model parameters and two allele frequencies. In an attempt to balance efficiency and exhaustive exploration of this parameter space, we, perform two separate experiments. The first considers data generated from a small set of common interaction models, and a second, that considers data generated from a larger set of models, to give an overall picture of how different tests perform in relation to each other. In the first experiment, we estimated the statistical power on data generated from the A × A, A × Afailed, D × D, D × Dfailed, H × H, and R × D GLMs. The A × Afailed and D × Dfailed are AD × AD generative models that were designed to violate the assumptions of the A × A and D × D models, respectively (see further Section Materials and methods). The results in Fig 4 shows that, for data generated from these specific models, there is no universal winning test strategy. However, both the G × G joint and AD × AD joint are generally among the best test under each generative model. Surprisingly, and in contrast to a single variant association test [28], the A × A test generally has a large loss of power when the generative model underlying the data is not A × A. Separate testing of parameters in the G × G separate or R/D × R/D tests sometimes incur a small loss of power compared to the joint tests. This can be expected because the application of Bonferroni correction in the separate test implicitly assumes independence of the individual interaction parameters, while the joint test accounts for any correlation among them; this will, overall, result in a small power advantage for the joint test. In the second experiment, we investigate the statistical power for the same tests over a large number of randomly sampled generative models. The results in Fig 5 reinforce those from the first power experiment. The joint tests, G × G joint and AD × AD joint, consistently perform best, followed by the G × G separate and R/D × R/D tests (where each parameter is tested separately), whereas the A × A test has on average 20% lower power. There is, as expected, no difference between the G × G joint and AD × AD joint tests. Despite the increased number of parameters of the joint tests these results holds also for lower sample sizes, see S3 Fig. However, we can not exclude that when many additional covariates are required, the tests with fewer parameter might gain statistical power. We applied the G × G joint test in two different association analyses aimed at the quantitative outcome blood concentration of Lp (a) lipo-protein particles [29] and a case-control outcome for myocardial infarction (MI), respectively. Our results, above, show that joint tests using saturated models has the best power. Moreover, joint tests using the G × G and AD × AD parameterizations are equivalent; our choice to use the G × G is arbitrary, but perhaps it is more easy to directly relate to the actual genotypes. The link family test was used to test for link function invariance. As this test is quite computer-intensive, we apply the following strategy for each association analysis: We first perform the large-scale discovery analysis using the canonical link function in the respective analysis. The significant interactions from the discovery are then re-analyzed including the full test for link function invariance. The full link function invariance test is also applied in the replication analysis.
Our major contribution is computationally efficient tests for the complete family of GLMs, applicable to any combination of parameterization and link function. Our tests can be used for case-control studies as well as studies of quantitative traits. We have also shown how this methodology facilitates computationally efficient meta-analyses. The G × G and AD × AD tests clearly had the overall best power to detect interaction. However, for certain data sets generated from specific interaction models, the G × G separate test and the R/D × R/D test performed comparable to the joint saturated full GLMs. Interestingly, the commonly used additive test had the overall worst power performance among the evaluated tests. It had strikingly poor power in several experiments. Nevertheless, for low minor allele frequencies the difference in power was less pronounced. In conclusion, our empirical results together with the equivalence of full saturated models, suggest that tests using saturated full GLMs are superior for interaction studies in general. It is important that high power does not come at the expense of control of the false positive rate (FPR). Our simulations show that, under certain circumstances, incorrect parameterization in unsaturated or constrained GLMs may lead to incorrect interaction inferences. Specifically, when the data includes LD, tests using the constrained R/D × RD or the unsaturated A × A GLMs failed to control FPR, while it was still controlled by the tests using saturated full GLMs. Moreover, also for data generated without LD, but then only when generated with a Binomial dispersion distribution, incorrect parameterization caused the A × A test to display substantial FPR inflation. We demonstrate the applicability of our method on experimental data aimed at Lp (a). Levels of circulating Lp (a) have an inverse association with coronary heart and are predominantly regulated by genetic variation in proximity of the LPA gene [29]. Although the mechanism remains elusive, it has been hypothesized that the association between Lp (a) levels and CVD is indeed causal, in part because a number of genetic polymorphisms have effects on both LP (a) levels and risk of cardiovascular events. We discovered a new genome-wide significant interaction between two variants in proximity to the LPA locus. This could potentially reflect an interaction between AGPAT2, involved in triglyceride metabolism, and SLC22A2, a transmembrane transporter, which previously has been indicated as involved in lipoprotein metabolism [46]. However, as often is the case, we cannot exclude the possibility that the discovered interaction may be a proxy for an unsampled, either single variant or another interaction, association. We demonstrated the applicability of our meta-analysis method by analyzing two MI case-control cohorts. The genetic architecture of MI is known to be complex and single variant analysis has so far explained only a small fraction of its heritability. This motivated us to investigate the potential impact of interactions on MI. Our meta-analysis did not result in any interactions of genome-wide significance, which require passing the severe Bonferroni-corrected threshold p ≤ 5. 72 ⋅ 10−12. The link function, a. k. a. the “scale”, is a crucial component of GLMs that relates the linear predictors to the phenotype mean. Because the link function used in a test can have a considerable impact on the resulting FPR, it is of interest whether the interaction is invariant of the scale, and a few tests for this have been devised. We applied the link family test, which evaluates invariance across a family of link functions [22,23]. This test is unfortunately computationally demanding and can be restrictive for large-scale analyses. To circumvent this, we used the so-called canonical link function in the large-scale discovery phase, and performed the test for link function invariance only on those interactions that passed the discovery phase. Alvarez-Castro and Carlborg [18] derived conditions for when the parameterization is orthogonal in a linear model. Parameterization orthogonality facilitates decomposition of the variance into components corresponding to main and interaction effects. For a GLM the concept of orthogonality is complicated by the link function and the genotypic dependence of the variance (see S1 Text). Nevertheless, our tests are applicable to both orthogonal and non-orthogonal parameterizations alike. Our test is currently limited to discrete genotype data, i. e. , directly genotyped or hard-called imputed data [47]. Meta-analysis of cohorts with non-overlapping genotype data requires imputed data. However, with currently available algorithms for incorporating imputation uncertainty in interaction tests, this would substantially compromise computational efficiency. Similarly, in the present implementation it is not obvious how to include covariates in the analysis. A solution that we are currently working on could be to use the GLM weights to model both imputation uncertainty and covariate adjustment. Ideally, this will retain the computational efficiency at the cost of statistical efficiency. However, for the time being, the simplest solution to the covariate issue is a two-step strategy, which, a posteriori to the initial analysis, fits a standard GLM including relevant covariates for the identified interactions and checks that the results still hold (this approach was used in the Lp (a) replication analysis). If this causes too many pairs to be identified, an alternative that is commonly applied in univariate meta-analyses is to first regress out relevant covariates and then model the residuals. Because residuals are expected to be normally distributed for all GLM models this can be generally applied; however, some interpretation is lost.
We first considered two scenarios that may produce false positives: submodels in which one or more assumptions fails to hold, and tests where the incorrect link function is used. In both experiments, we evaluated the following five different tests: the G × G joint, AD × AD joint, G × G separate, A × A and R/D × R/D tests. In the first experiment we investigated FPR where the data violates two model assumptions, erroneous parameterization and presence of linkage disequilibrium (LD). LD was measured by Lewontin’s D (LD) and we used two cases, high linkage (LD = 0) and low linkage (LD = 0. 8). The genotypes were generated according to the following probability distribution p 00 2 2 p 00 p 01 p 01 2 2 p 00 p 10 2 p 00 p 11 + 2 p 01 p 10 2 p 01 p 11 p 10 2 2 p 10 p 11 p 11 2 (4) where p 00 p 01 p 10 p 11 = (1 - r) (1 - q) + D r (1 - q) - D (1 - r) q - D r q + D (5) and q is the minor allele frequency of the first variant, and r is the minor allele frequency of the second variant. The parameter D = LD ⋅ Dmin where Dmin = min (1 − qr, 1 − (1 − q) (1 − r), q (1 − r), (1 − q) r). The minor allele frequency was set to q = r = 0. 3 and the sample size to 4,000. For each genotype, continuous phenotypes were then generated using six different null GLMs, obtained as the combinations of (i) a linear predictor from either of the A × A, R × A and R × D parameterizations, (ii) the identity link function, and (iii) either a Binomial and normal dispersal distribution (parameters can be seen in S5 Table). For each parameter combination we generated 1,000 data sets and estimated the false positive rate as the average number of incorrectly identified pairs. In the second experiment, we investigated the false positive rate when the link function is misspecified. Specifically, we generated the phenotype using a log link function, while the tests were performed using the identity link function. The genotypes were generated under Hardy-Weinberg equilibrium with a minor allele frequency of 0. 3 for both variants. We used a GLM with the A × A linear predictor, the identity link function and a normal dispersion distribution to generate phenotype from each genotype; we varied the size of the second variant’s main effect between −0. 4 and 0. 4 and the sample size over 2000,3000 and 4000 (parameters are shown in S6 Table). For each parameter combination, we generated 1000 data sets and estimated the false positive rate as described above. We performed two different simulation experiments for statistical power; one smaller comprising a set of specific GLMs and one larger comprising a wide range of interaction models. In both experiments we generated a continuous phenotype from a Normal dispersion distribution using the identity link function. We evaluated the same tests as in the FPR experiments, described above. In the small power experiment we generated data from six different parameterizations, A × A, A × Afailed, R × D, D × D, H × H, and D × Dfailed. The A × Afailed and D × Dfailed are AD × AD models designed to violate the assumptions of the A × A and D × D, respectively, that is, in the A × Afailed, the values of the interaction parameters are switched with respect to those in a A × A, while in the D × Dfailed, half of the interaction parameters are in the opposite and wrong direction compared to D × D. We generated genotypes for 4000 individuals under Hardy-Weinberg equilibrium with a minor allele frequency 0. 3 for both variants. For each genotype, we then generated phenotypes from each GLM. We varied the effect size between -1. 0 to 1. 0; depending on the selected parameterization, this required modification either of a single or multiple parameters in β (parameters can be seen in S7 Table). For each effect size, we generated 1000 data sets and estimated the statistical power of each test as the average number of correctly identified pairs under Bonferroni correction assuming 1012 variant pairs. In the large power experiment, we focused on different G × G parameterizations from a range of parameter combinations that can be found in S8 Table—notice that, since this is a saturated full parameterization, this approach will cover a large number of other, unsaturated and saturated, constrained and full, parameterizations. The genotypes were generated under Hardy-Weinberg equilibrium and we varied the allele frequency between 0. 2,0. 3 and 0. 4 (for both variants), and the sample size between 2000,3000 and 4000. For each genotype a phenotype was generated using a linear predictor from the G × G parameterizations described above, the identity link function and the Normal dispersion distribution. For each parameter combination we generated 1000 data sets and estimated the statistical power as described above (using a Bonferroni correction assuming 1012 variant pairs). The PROCARDIS study was approved by the Regional Ethics Review Board at Karolinska Institutet, Stockholm in Sweden (approval number 98-482 and 03-491) and by the Institutional Review Boards of the Mario Negri Institute, Milano in Italy, the University of Munster, Munster, in Germany, and the University of Oxford, Oxford, United Kingdom. The PROCARDIS study was supplemented with controls from the WTCCC study, UK. The WTCCC was approved by the relevant research ethics committees. All study participants provided their written informed consent to participate in the study, which was conducted in accordance with the Helsinki Declaration. The SCARF and SHEEP studies were approved by the local ethics committees at Karolinska Hospital, Stockholm (approval number 95-397 and 02-091), and Karolinska Insitutet, Stockholm (approval number 01-097), respectively. The MIGEN data was accessed from dbGAP (access. nr. phs000294. v1. p1) [45]; all participants in the MIGEN studies gave written informed consent in accordance with the guidelines of local ethical committees. The PROCARDIS multicentre study was designed to investigate early onset CAD. Cases with a documented CAD event before the age of 66 years were collected from Sweden, the UK, Germany and Italy. The full PROCARDIS cohort comprise 8410 cases and 5188 matched controls free from CAD. Here we have used a subset of the PROCARDIS cohort previously genotyped with the Illumina Human1M Quad chip and the Illumina Human610K chip [33]. We only included unrelated individuals genotyped on these chips. The intersection of these chips contain 566,865 variants. The subset of PROCARDIS used in this study depends on the phenotype. Lp (a) plasma levels were measured in 3,741 individuals. For the meta-analysis aimed at MI, the PROCARDIS cohort was extended with 5,667 control samples from the Wellcome Trust Consortium (WTCCC) [44]. This resulted in a total of 2,809 cases and 7,330 controls for the MI disease phenotype. The SCARF [35] and SHEEP [34] cohorts included unrelated MI cases from the Stockholm region of Sweden, with age and sex-matched controls collected from the general population of the same region. The comparable design and demographics of the 2 cohorts means that they can be combined as one cohort. The SCARF-SHEEP cohort was previously genotyped with the CardioMetabo chip, a custom Illumina iSelect genotyping array that targets genetic variants likely to be involved in metabolic and cardiovascular disorders [48]. The chip contains 116,540 variants. Lp (a) levels were measured in 2,345 individuals. The MIGEN cohort is a case-control study aimed at investigating the genetic basis of MI [45]. The cohort contains samples from 6 collection sites: Boston, MA (Masschusetts General Hospital Premature Coronary Artery Disease Study), Seattle, WA (Heart Attack Risk in Puget Sound), Helsinki, Finland (FINRISK), Malmö, Sweden (Malmö Diet and Cancer Study), Barcelona, Spain (REGICOR), and Milan, Italy (Italian Atherosclerosis Thrombosis and Vascular Biology Working Group). The MIGEN cohort was previously genotyped with the Affymetrix Genome-wide Human SNP Array 6. 0. The data was approved by and downloaded from dbGAP with accession phs000294. v1. p1. This data contains 3,068 cases and 2,957 controls. We performed multiple experiments on biological data: exhaustive interaction scan in Lp (a), vGWAS scan in Lp (a), and a exhaustive meta analysis of CAD. All interaction tests was performed using the G × G joint test. In the discovery phase, a canonical link function and the appropriate dispersion distribution was used in an initial analysis. The significant pairs was then tested using the same G × G joint test, but using the link family test to step through a family of link functions; the latter approach was also used in the replication phase. For the continuous Lp (a) phenotype, the Normal dispersion distribution was used first with the canonical identity link and then with the symmetric Box-Cox family. For the binary MI case-control data, the Binomial dispersion distribution was used first with the logit link and then with Pregibon link family. The Lp (a) phenotype was log-transformed and outliers were removed in both PROCARDIS and SCARF-SHEEP. For the exhaustive interaction scan we applied the following QC filters: minor allele frequency 0. 05, genotyping rate 0. 1. We then removed variants in linkage disequilibrium using the Plink “–indep-pairwise” option, using a window size of 100, step size of 5 and a LD threshold of r2 < 0. 5 (we additionally check D′ for any significant interaction discoveries). This resulted in 180,947 variants, and we tested for interaction between all possible pairs of these. For the vGWAS scan, we applied the same QC filters as above; the resulting set of SNPs is called L. We then tested for variance heterogeneity of each variant with a Brown-Forsythe test [37]. The set of variants with p-value less than 10−4 is called V. We constructed all pairs of one variant from V and one variant from L (1. 26 ⋅ 107 pairs) and tested these for interaction. In the meta-analysis, we only considered variants that was genotyped both in MIGEN and PROCARDIS, this resulted in 132,181 variants. We applied the following QC filters: minor allele frequency > 0. 05, genotyping rate > 0. 1. We only tested pairs for which all genotypes had at least five samples in both cases and controls (8. 17 ⋅ 109 pairs). For each analyzed cohort and phenotype combination we checked for potential p-value inflation (genomic inflation) due to population stratification. In all cases genomic inflation was low (see S4, S5, S6 and S7 Figs). The QQ-plots for the two Lp (a) cohorts (S6 and S7 Figs) showed a surprisingly large deviatin from the diagonal line, expected under the null mode. After investigation, it became clear that this inflation was due to substantial LD at the previously identified strongly associated Lp (a) locus on chromosome 6 [29]. A c++ implementation of the software can be accessed at https: //github. com/mfranberg/besiq.
Interaction between organic molecules forms the basis of all biological systems. The availability of high-throughput genotyping and sequencing platforms enables us to cost-effectively genotype a large number of individuals. For sufficiently large datasets it is possible to reconstruct the genetic dependencies that underlie complex traits and diseases. However, there is a need for efficient statistical methodologies that can tackle the large sample size and computational resources required to study interaction. In this work we provide theory that reduces the required computational resources, and enable multiple research groups to effectively combine their results.
genome-wide association studies
medicine and health sciences
variant genotypes
random variables
covariance
alleles
genetic mapping
probability distribution
mathematics
coronary heart disease
statistics (mathematics)
genome analysis
cardiology
research and analysis methods
genomics
mathematical and statistical techniques
genetic loci
probability theory
myocardial infarction
normal distribution
heredity
meta-analysis
genetics
biology and life sciences
physical sciences
computational biology
vascular medicine
statistical methods
human genetics
Fast and general tests of genetic interaction for genome-wide association studies