Joint genetic models for multiple traits have helped to enhance association analyses. Most existing multi-trait models have been designed to increase power for detecting associations, whereas the analysis of interactions has received considerably less attention. Here, we propose iSet, a method based on linear mixed models to test for interactions between sets of variants and environmental states or other contexts. Our model generalizes previous interaction tests and in particular provides a test for local differences in the genetic architecture between contexts. We first use simulations to validate iSet before applying the model to the analysis of genotype-environment interactions in an eQTL study. Our model retrieves a larger number of interactions than alternative methods and reveals that up to 20% of cases show context-specific configurations of causal variants. Finally, we apply iSet to test for sub-group specific genetic effects in human lipid levels in a large human cohort, where we identify a gene-sex interaction for C-reactive protein that is missed by alternative methods. Genetic effects on phenotypes can depend on external contexts, including environment. Statistical tests for identifying such interactions are important to understand how individual genetic variants may act in different contexts. Interaction effects can either be studied using measurements of a given phenotype in different contexts, under the same genetic backgrounds, or by stratifying a population into subgroups. Here, we derive a method based on linear mixed models that can be applied to both of these designs. iSet enables testing for interactions between context and sets of variants, and accounts for polygenic effects. We validate our model using simulations, before applying it to the genetic analysis of gene expression studies and genome-wide association studies of human blood lipid levels. We find that modeling interactions with variant sets offers increased power, thereby uncovering interactions that cannot be detected by alternative methods. Understanding genetic interactions with external context (GxC), including environment, is a major challenge in quantitative genetics. Linear mixed models (LMMs) have emerged as the framework of choice for many genetic analyses, mainly because the random effect component in this class of models provides robust control for population structure , 2 and oClassical single-variant models for GxC use fixed effects to test for differential effect sizes of individual variants between contexts, either using an ANOVA –16 or LMFig 1A, middle). We denote this class of interactions rescaling-GxC. More generally, however, there may also be differences in the configuration of causal variants between contexts , such that not all genetic variants show the same fold-difference between contexts, as some become more prominent in particular contexts and others less so. We denote these complex interactions heterogeneity-GxC. These two classes of interactions have different functional implications–the former suggest no difference in causal variants between contexts, and the latter suggest otherwise. Distinguishing between them is only possible using multi-variant models such as set tests, and is important for identifying different potential causal variants in different contexts.One way to generalize single-variant interaction tests to variant sets is using a model that assumes that context differences cause the same fold-differences in effect size across all genetic variants, such that all genetic effects in one context are proportional to the effects in a second context; a criterion that has also been considered to assess co-localization of multiple traits   and to distinguish between rescaling-GxC and heterogeneity-GxC. We find that iSet yields increased power for identifying interactions and uniquely is able to robustly differentiate between rescaling-GxC and heterogeneity-GxC. We first validate iSet using simulations before applying the model to test for gene-by-sex interactions in blood lipid levels  (2 × 1). Finally, in the most general case, the configuration of causal variants is independent between contexts, corresponding to W = AT, with genetic signals γ1 (S × 1) and γ2 (S × 1) and general scaling factor matrix A =  (2 × 2).To simplify the notation, we consider the case of two contexts, however the same derivation holds for larger numbers of contexts. Different genetic architectures between contexts are cast as specific assumptions on the γ, γ1 and γ2, assuming independent unit-variance normal prior distributions, results in a marginal likelihood of the formMarginalizing over the genetic signal Cs is the C × C trait-context covariance for the set component and Rs is the local realized relatedness matrix , where for persistent effects Cs is a block covariance (Cs = a012×2), rescaling-GxC correspond to Cs with a rank-one structure (Cs = aaT) and the most case of independent architectures in both contexts can be captured by a full rank covariance (Cs = AAT). In order to test for associations, we additionally consider a null model without a set component (Cs = 0). The trait-context covariances Cs, Cn and the fixed effect weights B are estimated using (restricted) maximum likelihood with constraints to obey the alternative structures of Cs. Model parameters are optimized based on the restricted log marginal likelihood as objective, using a low-memory Broyden-Fletcher-Goldfarb-Shanno optizmier (L-BFGS) [fmin_l_bfgs_b optimisation method of the SciPy python library. Specific tests are implemented as pairwise likelihood ratio (LLR) tests, considering LMMs with different trait-context covariances :Here, vec denotes the stack-column operation, ⊗ the Kronecker product, (L-BFGS) , implememtSet: full-rank versus nulliSet: full-rank versus block covarianceiSet-het: full-rank versus rank-oneiSet support these tests in fully observed designs and in stratified populations. Care is required when applying the heterogeneity-GxC test to stratified designs. Differences in allele frequencies between strata and other confounding factors could potentially result in false positive heterogeneity-GxC signals.Empirical P values are estimated from the distribution of LLRs under the null. As permutation procedures are not well defined for interaction tests, for both iSet and iSet-het we generate test statistics from an empirical null distribution using a parametric bootstrap procedure . Brieflyfit the no-association model (null), the block covariance model (block), the rank-one covariance model (rank-one) and the full-rank covariance model (full) and estimate LLRs for mtSet (full vs null), iSet (full vs block) and iSet-het (full vs rank-one);for each genomic region, sample J LLRs from the null for each of the three tests ;for each of the these tests, pool the JT null LLRs across regions to obtain an empirical null and compute empirical P values.Note that the number of parametric bootstraps/permutations will determine the minimum P value that can be obtained. For example, for T tests and J = 30 bootstraps the minimum P value that can be estimated is 1/(JT), which correspond to a FWER of 1/J ≈ 0.03. While 30–100 bootstraps will be sufficient to reach typical thresholds in genome-wide studies, more stringent thresholds on significance levels (FWER< = 1%) require a larger number of parametric bootstraps . For mtSet we use the same procedure but with permutations .N denotes the number of individuals, R denotes the number of variants in the region, NPC is the number of PCs and t corresponds to the number of function evaluations of the optimizer (See [S1 Text). For cohorts with related individuals and fully observed designs, iSet can also be applied with an additional random effect term in the model. In this case, we again re-use efficient inference schemes for multi-trait set tests [O(N3 + N2R + tNR2). Notably, the computational scaling as stated assumes that the number of variants in the testing set R is constant, and in particular does not increase as a function of the sample size N. In practice, however, there may be an implicit dependency between N and the number of variants R, for example when low frequency and rare variants are included in the analysis. Consequently, the empirical compute cost as a function of N could be larger than linear. A second setting are small cohorts, where the number of variants R could exceed N. In order to retain efficiency in this setting, iSet detects when the low rank assumption of the region set is violated and reverts to an eigen decomposition in the space of individuals, and hence the computational complexity is bounded by O would szer (See  for detaet tests . This moC is O(tNC2 + tC3).Moreover, although iSet is primarily designed for the analysis of pairs of contexts, the model can also be applied to multiple contexts. The computational cost of iSet as a function of the number of contexts S1 Fig were reported for different designs, using synthetic cohorts generated using data from the 1000 Genomes Project . We report the average per-region compute time measured on 100 regions with size 30 kb, considering a single core of an Intel Xeon CPU E5-2670 2.60-GHz to fit iSet. The runtime for all three considered tests, mtSet, iSet and iSet-het (including bootstraps) on the eQTL analysis took on average 28.7s per gene, resulting ~100h of compute time for a genome-wide analysis using a single core. Similarly, the runtime for all three tests for the NFBC data was on average 107s per testing region, resulting in ~1,500h of compute time for a genome-wide analysis.Empirical runtime estimates in Finally, we note that while the linear scaling allows for application to larger datasets, the computational cost of iSet is typically larger than for a conventional single-variant model. Consequently, for genome-wide analyses it will be required to parallelize the computational operations across multiple compute cores. The software implementation of iSet supports the parallelization across multiple compute nodes and cores.S1 Text).The LMMs in iSet can also be used to estimate the phenotypic variance explained by the variant set for their persistent, rescaling-GxC and heterogeneity-GxC effects . For the simulation experiments shown in Fig 2, we considered sets with 30kb, which captures local LD in the data . For the analysis of the stimulus eQTL study, we have considered gene-based sets using a 100kb cis genetic region centered on the TSS, which is in line with other cis eQTL analyses [As for any set test, the size of the region set is an important parameter in iSet. The specific choice will depend on the biological application, LD and marker density. We have previously explored trade-offs between the computational efficiency and power of association tests for different choices of the window size . We hereanalyses . Finallyanalyses  and consS1 Text). We considered single-nucleotide polymorphism with a minor allele frequency of at least 2% . In all simulations, we simulated two contexts, modeled as the sum of a genetic contribution from a 30kb causal region, effects due to population structure, hidden covariates and identically distributed Gaussian noise. Effects due to population structure and hidden confounders were simulated with partial correlations across contexts, explaining variable proportions of the total phenotypic variance in each context .Simulations were carried out using a synthetic cohort of 1,000 individuals derived from genotypes of European populations in the 1000 Genomes project  . For each simulation we randomly selected a 30kb region and generated phenotypes simulating persistent effects from four causal variants and tested for GxC interaction in the region. To estimate P values, we used 30 parametric bootstraps for each test, resulting in a total of 3,000,000 null LLRs to estimate P values. Analogously, we assessed the calibration of iSet-het, where exclusively rescaling-GxC effects were simulated . Again, we considered 30 parametric bootstraps for each test and pooled LLRs to estimate P values. We also assessed the calibration of iSet and iSet-het when simulating violations from the model assumptions . These included (i) deviations from the polygenic model, using causal regions with a single causal variant, (ii) effects due to epistatic interactions of pairs of variants in the region and (iii) deviation from the Gaussian residual assumption by introducing outlying samples. Outliers were simulated by adding a heavy tailed noise contribution to 0.1% of the samples . For simulated outliers, we compared iSet on un-normalized phenotype data as well when quantile normalizing the phenotypes to a normal distribution. Quantile normalization has previously been shown to be effective in combination with multi-trait LMMs [To assess the calibration of P values obtained from the interaction test (iSet) and the test for heterogeneity-GxC (iSet-het), we considered 100,000 datasets with two contexts where only persistent genetic effects (no interactions) were simulated .For comparison, we considered single-variant interaction tests as in   the number of causal variants in the region , and ii) the proportionality factor of the effect sizes between contexts . When simulating general effects that include heterogeneity-GxC, we randomly selected an equal number of context-specific causal variants and monitored the correlation of the total simulated genetic effects across contexts, thereby controlling the extent of heterogeneity-GxC. Again, local genetic effects were simulated to explain 2% of the total phenotypic variance in each context. We varied (see S2 Table) i) the extent of simulated heterogeneity-GxC  and ii) the total number of causal variants across contexts . For each parameter setting, we considered 1,000 repeat experiments. To obtain P values for set tests we considered 30 parametric bootstraps for each test and computed empirical P values from 30,000 null LLRs in each simulated scenarios. We used the Benjamini-Hochberg procedure to adjust for multiple testing across repeat experiments and assessed all methods in terms of power at a fixed FDR<5%.To assess power of iSet for alternative genetic architectures, we simulated interaction effects from a 30kb region either considering rescaling-GxC effects or more general effects that include heterogeneity-GxC, using the simulation settings in Fig 1C and 1D), we used a simulation procedure analogous to the strategy described above. Phenotypes were simulated as the sum of genetic effects from three distinct causal regions (30kb) within a 5Mb region on chromosome 13, harboring respectively persistent, rescaling-GxC and heterogeneity-GxC effects. The effects from individual regions was simulated to explain 5% of the total phenotype variance.For the simulated example region to illustrate iSet and alternative genetic architectures  and precision-recall curves  to assess the ability of discriminating between these types of genetic effects. For comparison, we also considered a univariate baseline approach, scoring regions with significant associations using the squared Pearson correlation between the lead variants in both contexts (low squared Pearson correspond to high rank). We considered alternative significance thresholds on region-based P values obtained using eigenMT .As an additional assessment of the accuracy of iSet-het to detect heterogeneity-GxC effects, we tested how well the model discriminates between regions with and without simulated heterogeneity-GxC. We considered the identical 10,000 regions in S10A Fig) or more general effects . For both sets of simulations, we considered the default simulation parameter values (S2 Table).To study the effect of alternative sizes of the testing region on the power of iSet and iSet-het under different simulated scenarios, we considered sliding-window experiments using simulated data, when varying both the size of the simulated causal and of the testing region. Phenotypes were simulated across two contexts using the approach as described above, considering a causal region with variable size . For each of these simulation settings, we carried out a sliding window analysis in the surrounding 1 Mb region with testing windows of 30kb, 100kb, 300kb and 1Mb . We used Bonferroni to adjust for multiple testing across regions. For comparison, we also considered the single-variant test for interactions (mtLMM-int), applied to the same variants in the 1Mb region, and used eigenMT to adjust for multiple testing across variants while accounting for LD. For each scenario, we considered 200 repeat experiments and assessed power at FDR = 5%. We considered either simulated pure rescaling-GxC (−3), we were left with 5,729,118 genome-wide variants .The dataset consists of gene expression levels from primary monocytes, both in a naïve state and three different stimulus contexts, profiled in 432 genotyped individuals of European ancestry. Gene expression levels in the naïve state, after exposure to IFN-γ, after 24-hour LPS and after 2-hour LPS were available for 414, 367, 322 and 261 individuals respectively. Normalization, correction for batch and probe filtering were done as in . FollowiS4 Fig). All tests were applied considering a pair-wise approach, jointly testing for eQTLs in the naive state and one of the stimulated states, considering set tests for association (mtSet), interaction (iSet) and heterogeneity-GxC (iSet-het). For comparison we also applied a single-variant tests using the same variants, testing for association (mtLMM) and stimulus interaction (mtLMM-int). For single-variant tests, we estimated gene-level significance using the P value of the lead cis variant .Association and interaction tests were carried out considering 100 kb regions centered on the transcription start site of genes corresponding to individual probes . EmpirieigenMT, . Resultscis regions, however independently modeling each cellular context. At FDR<5% this analysis revealed 4,187, 4,786, 4,240 and 4,620 probes with an eQTL respectively in the naive, IFN-gamma, LPS2h and LPS24h states (S4 Table). To estimate the cis-genetic contribution to gene-expression in each context we calculated the Best Linear Unbiased Predictor (BLUP) from the model as μ is the estimated mean, Kcis is the cis realized relatedness matrix, V−1 is the inverse of the total estimated covariance and y is the gene-expression vector in the corresponding context.To illustrate the properties of the heterogeneity-GxC QTLs detected by iSet-het, we additionally considered univariate set tests in the same S4 Table).We used a single-variant forward selection LMM  to charar2 < 0.20) as a shift in lead variants between the two contexts. Probes with a shared lead eQTL  were annotated using secondary effects. Among the 4,186 probes with shared main effects, this analysis revealed context-specific secondary QTLs were identified for 999 genes. Context-specific secondary effects were defined when either i) the secondary effect was significant in only one of the two contexts or ii) the secondary effects lead variants were in low LD (r2 < 0.20) .Results from step-wise analysis were used to annotate probes with significant heterogeneity-GxC. We denoted the 1,449 probes that have significant marginal associations in both contexts and independent lead variants .We classified the 4,186 eQTLs with shared lead eQTL into directionally consistent and opposite-effect eQTLs. Briefly, opposite effects were defined by three criteria, i) marginal significance in both contexts, ii) LD between contexts (r2>0.8) and iii) negative correlation of genetic effects. These criteria resulted in 67 opposite-direction QTLs. Directionally consistent eQTLs correspond to criteria i) and ii) but positive correlated genetic effects, resulting in 4,119 co-located QTLs. Statistical significance of the enrichment for significant heterogeneity-GxC effects in opposite-direction eQTLs rather than same-direction eQTLs was assessed using a one-sided Fisher’s exact test . Statistical calibration and power simulations were performed analogously to the approach used for fully observed designs. Population structure was accounted for using the first ten principal components of the realized relatedness matrix as fixed effect covariates. We did not consider tests for heterogeneity-GxC, as differential tagging of causal variants could potentially result in spurious heterogeneity-GxC signals, and hence additional controls would be required. However, in principle the test applies to stratified populations.S1 Text). As an additional comparison, we extended the single-variant interaction test in [S1 Text). GESAT was run using the function GESAT of the package iSKAT version 1.2. Both iSet and GESAT were applied on identically processed standardized variants.We compared iSet to the single-variant interaction tests as in  , triglycerides (TRIGL), LDL and HDL cholesterol levels) measured in 5,256 unrelated individuals from the NFBC1966 cohort  . For comparison we considered the single-variant interaction test [S6 Table.We applied mtSet and iSet to 318,653 genome-wide variants with an allele frequency of at least 1% using a sliding-window approach Click here for additional data file.S1 Tablea, parameters in S2 Table but without simulating rescaling) and the same setting while considering a single causal variant (b), outlying samples (c) and epistatic interactions between randomly selected pairs of variants (d). These results show that iSet and iSet-het yield P values with controlled type-I error rates under different types of model misspecification. The corresponding QQ plots are shown in Fig 2A (main) and S4 Fig.Shown are empirical type-1 error estimates for increasingly stringent significance level thresholds. Persistent genetic effects (No GxC effects) were simulated using the standard simulation parameters ((PDF)Click here for additional data file.S2 TableK = 10 unmeasured hidden confounding factors, and iid observation noise (S1 Text). We fixed the variance explained by the region (vr = 2%), and the fraction of shared background signal (α = 0.6), the fraction of residual variance that is explained by the hidden factors (β = 0.5). We considered variable numbers of causal variants and altered the extent of rescaling-GxC and heterogeneity-GxC. (a) The contributions to phenotypic variance of all simulated effects.  Parameter values considered for simulations of rescaling-GxC and heterogeneity-GxC effects, respectively. Individual parameters in  were varied one by one, while keeping others at their default values (bold face). When simulating general-GxC, and when varying the number of causal variants, the region effect correlation r was constrained to 0.2 < r < 0.8. See S1 Text for full details.Simulated phenotype data were generated as sum of effects from variants in a designated causal region, effects from a relatedness component/population structure, effects from (PDF)Click here for additional data file.S3 Tableη, x-axis in Fig 2C), the corresponding fold change of the effect sizes, the fold change of the variance explained by the region and the relative directionality of the genetic signals. For −1 < η < 1, the absolute fold change increases for small absolute values of η and tends to infinity in the limit η → 0. The range η < 0 corresponds to genetic effect with opposite effects. The setting η = 0 corresponds to a local genetic effect that is specific to one context (no effect in the second context). Finally, η = 1 corresponds to no interaction (the fold change is zero and the direction of genetic effects is the same).Shown is the relationship between the proportionality factor of the effect sizes in the two simulated contexts ((PDF)Click here for additional data file.S4 TableThe result table is provided as separate supplementary information file.(XLSX)Click here for additional data file.S5 Tablea), naive/LPS2 (b) and naive/LPS24 (c).Breakdown of probe/stimulus pairs with shared lead variants, stratified by concordance of the effect direction (opposite-direction versus same-direction eQTLs) and significance of the heterogeneity-GxC test (heter vs No heter) in naive/IFN ((PDF)Click here for additional data file.S6 TableThe result table is provided as separate supplementary information file.(XLSX)Click here for additional data file.S7 TableR lower than the number of individuals N or not). Here, N denotes the number of individuals, R is the number of variants in the region, NPC is the number of principal components and t is the number of function evaluations of the optimizer. Operations that do not include t as factor refer to the cost of upfront computations that do not need to be evaluated in every step of the optimization. Operations that entail substantial computational burden (cubic operation in N) are highlighted in red.Shown is the computational complexity of iSet for alternative designs (either complete or stratified data designs), strategies to adjust for confounding  and analysis settings (number of variants (PDF)Click here for additional data file.S1 FigMethods). Reported CPU times are empirical averages to perform the tests for association (mtSet), interaction (iSet) and heterogeneity-GxC (iSet-het) and include the cost of 30 parametric boostraps for each test to estimate P values (Methods). CPU times are averaged across 100 genomic regions . Note that the reported costs do not include the up-front cost of the principal components and the up-front eigenvalue decomposition of the global relatedness matrix for the random-effect method. Runtime estimates were obtained using a single core of an Intel Xeon CPU E5-2670 2.60 GHz processor.Shown is the average CPU time (in seconds) for one in interaction set test using a 30kb region for alternative designs and cohort sizes. By default iSet uses principal components to adjust for confounding (PC). Alternatively, iSet can be combined with an additional random-effect (RE) to adjust for structure such as relatedness (see (PDF)Click here for additional data file.S2 Figa), and stratified cohorts .iSet supports efficient interaction set tests both for complete designs Click here for additional data file.S3 FigMethods) for three different scenarios: (a) positive proportionality factor of effect sizes (0.3) and 1 causal variant, (b) positive proportionality factor of effect sizes (0.3) and 4 causal variants, (c) negative proportionality factor of effect sizes (-0.5) and 4 causal variants.Shown is the QQ plot for the P values obtained from the heterogeneity-GxC interaction test (iSet-het) when simulating rescaling-GxC Click here for additional data file.S4 Figa, b) Simulated phenotypes that violate the assumption of the infinitesimal model iSet is based on : i) a single causal variant in the region (a) and ii) epistatic interactions .  Violations of Gaussian distributed residuals, by simulating outlying samples (Methods). Notably, while iSet and iSet-het using unnormalised phenotypes yield inflated P values (c), both methods are well calibrated after quantile normalization to a unit variance Gaussian distribution . All the results from the real data applications are based on quantile normalized phenotypes (Methods).Shown are the QQ plots for the P values obtained from the interaction set test (iSet) and the heterogeneity-GxC set test (iSet-het) when simulating different types of model mismatch. ((PDF)Click here for additional data file.S5 Figr2) across all variants in each region; scatter plot of the number of effective tests versus the number of variants. Shown in color is the within-region average correlation (r2) across all pairs of variants. From top to bottom: (a) 10,000 30kb regions from the simulated data based on 1000 Genomes individuals, (b) 100kb cis regions (centered on the TSS) considered in the cis stimulus eQTL analysis  and (c) 100kb regions considered in genotype-sex interaction analysis in the NFBC1966 cohort .From left to right: distribution of the number of variants across the analyzed regions; distribution of the number of effective tests as estimated by eigenMT; distribution of the average pairwise squared Pearson correlation ((PDF)Click here for additional data file.S6 FigFig 2B, however, considering synthetic genotypes without LD.Shown are results analogous to those presented in (PDF)Click here for additional data file.S7 Figa) and precision-recall curve (b) for alternative approaches to classify heterogeneity-GxC. Considered was the iSet test to score the extent of heterogeneity (iSet-het) and a baseline approach based on single-trait single-variant LMMs. Briefly, for the baseline model the considered score is defined as 1 − r2 (where r is the Pearson correlation coefficient between lead variants identified in each context) for regions with significant associations in both contexts . Regions that were not marginally significant in either of the two contexts were assigned a score of zero (Methods).Receiver operating curve ((PDF)Click here for additional data file.S8 Figr < 0.8, where r is the correlation of the simulated genetic effects in the two contexts).Shown is the power of iSet, iSet-het and a single-variant interaction test (mtLMM-int) for detecting GxC effects when simulating heterogeneity-GxC for increasing numbers of causal variants (0 < (PDF)Click here for additional data file.S9 FigShown are the estimates of the genetic variance explained by the set component across all simulated settings when considering different covariance models. The variance component estimates are from a model that either considers a full-rank covariance , a rank-one covariance  and a block covariance matrix . Both designs with fully observed cohorts  and stratified samples  are considered. While the full-rank iSet model yields calibrated variance components, other methods yield biased estimates in some settings. In particular, we considered scenarios with either rescaling-GxC effects  or heterogeneity-GxC (where we vary the number of SNPs). For each simulated scenario, we considered 1,000 simulated regions and altered the variance explained by the region . Grey horizontal lines denote the true simulated local genetic variance.(PDF)Click here for additional data file.S10 Figa,b) Power of iSet when simulating causal regions of different sizes (x-axis) and for sliding window analyses with increasing window sizes (y-axis) (see Methods). Considered were both settings with pure rescaling (a) and settings with heterogeneity-GxC effects (b). For comparison, we also considered the power of mtLMM-int . (c) Power of iSet in the same settings as considered in (b) when simulating heterogeneity-GxC effects and for the iSet-het test. Note that mtLMM-int is not specific to detecting heterogeneity-GxC; hence the reported power was set to zero.((PDF)Click here for additional data file.S11 FigShown is the number of probe/stimulus pairs with significant effects, detected by alternative single-variant and set-based tests, varying the genome-wide false discovery rate (FDR) threshold. Considered were the single-variant tests for associations (mtLMM-any) and interactions (mtLMM-int), as well as set tests for associations (mtSet), interactions (iSet) and heterogeneity-GxC effects (iSet-het). Venn diagrams on top show the overlap of significant probe/stimulus pairs for the three interaction tests at selected FDR thresholds .(PDF)Click here for additional data file.S12 Figlog10P values from single-variant LMMs and set tests for association tests (mtLMM-any vs mtSet) and interaction tests (mtLMM-int vs iSet) for different stimulus contexts . P values for single-variant models correspond to the minimum P-value across variants in the considered testing region, adjusted for the effective number of tests . Venn diagrams on the top of individual panels show the overlap of probes with significant associations or interactions identified using alternative methods (5% FDR).Shown are the scatter plots of the −(PDF)Click here for additional data file.S13 Figa) Scatter plot of the maximum Q values across pairs of jointly analyzed contexts , comparing set tests and single-variant tests for different stimulus pairs. Heterogeneity-GxC cases without clear single-variant interpretation (highlighted in red) tend to be associated with increased significance when using set tests, suggesting that differences in power may impair the single-variant annotation of these eQTLs. (b) Histogram of the squared Pearson correlation (r2) of cis genetic effects predicted using the best linear unbiased predictor, BLUP (Methods) across pairs of contexts. Shown is the distributions either for all probes with significant effects in the naive context and probes with significant heterogeneity-GxC. As expected, genes with heterogeneity-GxC tend to be associated with lower correlation of genetic effects across contexts. (c) Scatter plot of the correlation coefficient of genetic effects (as in b) versus the maximum q-value across the naive and the stimulated state. Heterogeneity eQTLs (highlighted in red) tend to have stronger marginal associations and lowly-correlated cis genetic signals.((PDF)Click here for additional data file.S14 Figcis variants for OAS (a), LMNA (b) and PTK2B (c) across the contexts showing opposite effects. While the three examples are identified as opposite effects when using single-variant methods , iSet identified significant heterogeneity-GxC, suggesting changes in the configuration of causal variants. Lead variants in individual contexts are annotated using triangles and are in high LD (r2 > 0.8).Shown is the z-score statistics for (PDF)Click here for additional data file.S15 Figa) QQ plot for the P values obtained when applying iSet and GESAT to synthetic datasets where only persistent genetic effects (No GxC) were simulated. (b) Power comparison of alternative interaction tests and variance decomposition results from iSet when simulating rescaling-GxC effects, for different factors of proportionality of the variant effect sizes across contexts. Considered were iSet, GESAT and a single-variant interaction test (mtLMM-int). (c) Analogous results as in b when simulating general GxC effects . Results are stratified by the correlation between the simulated genetic effects between the two contexts.((PDF)Click here for additional data file.S16 Figcrp3dec, a), LDL cholesterol , HDL cholesterol , and triglycerides  obtained from univariate set tests ignoring sex-specific differences (stSet), an association test that accounts for differences in genetic effect across strata (mtSet), iSet, GESAT and single-variant interaction test (mtLMM-int). Red arrows indicate the interaction effects that are discussed in the main text. Blue arrows indicated associations that can only be detected when modeling differences in effect sizes across strata (mtSet vs stSet).Shown are Manhattan plots for C-reactive protein ((PDF)Click here for additional data file.S17 Figcrp3dec), LDL cholesterol (FS_KOL_L), HDL cholesterol (FS_KOL_H), and triglycerides (FS_TRIG) obtained using a univariate association set tests ignoring sex , an association test modeling sex-specific genetic effects , iSet (c), GESAT (d) and single-variant interaction test (mtLMM-int).Shown are the QQ plots for C-reactive protein ((PDF)Click here for additional data file.S18 FigShown is the Manhattan plot for C-reactive protein using single-variant interaction tests applied to common variants (MAF>0.5%) on imputed data. Vertical grey lines indicate the 100kb region with significant genotype-sex interaction (FWER = 10%) detected using iSet. Non-imputed typed variants are highlighted in red, showing that for this locus imputation strategies did not increase the power of single-variant methods.(PDF)Click here for additional data file.S19 Figa) Local Manhattan plot (1Mb around significant region) for single-variant association tests, either considering males (black) or females (pink). For comparison, shown are also the P values from the iSet (red), mtSet (green) and stSet (grey). (b) C-Reactive protein level stratified by different alleles of rs4660378 . Rho in the caption correspond to the Spearman rank correlation coefficient and the corresponding P value, both for male and female individuals.((PDF)Click here for additional data file.


In humans, prosocial behaviour is essential for social functioning. Twin studies suggest this distinct human trait to be partly hardwired. In the last decade research on the genetics of prosocial behaviour focused on neurotransmitters and neuropeptides, such as oxytocin, dopamine, and their respective pathways. Recent trends towards large scale medical studies targeting the genetic basis of complex diseases such as Alzheimer’s disease and schizophrenia pave the way for new directions also in behavioural genetics.Based on data from 10,713 participants of the American Health and Retirement Study we estimated heritability of helping behaviour–its total variance explained by 1.2 million single nucleotide polymorphisms–to be 11%. Both, fixed models and mixed linear models identified rs11697300, an intergene variant on chromosome 20, as a candidate variant moderating this particular helping behaviour. We assume that this so far undescribed area is worth further investigation in association with human prosocial behaviour. Prosocial behaviour–voluntary behaviour intended to benefit others –is essenEver since Hamilton  the evolOn the individual level, however, we are only just beginning to understand the genetic influences on human (pro)social behaviour. Research on the regulatory effects of neuropeptides such as oxytocin and vasopressin on social cognition and behaviour ,11 and tThe predominant method in investigating the genetic basis of prosocial behaviour and decision-making is the application of incentivized laboratory-based experiments derived from the field of experimental economics. These complement behavioural genetics approaches –22. All Nonetheless, there are several drawbacks to this approach, varying in their severity with the field of application. The sample size of laboratory-based experiments is often small, limiting the generalizability of the results . The traToday, the increasing number of predominantly medical studies provides a vast collection of genetic data of large study samples. Their aim is to reveal genetic influences on complex diseases such as Alzheimer’s disease, breast cancer, and schizophrenia using genome-wide association approaches –28.http://www.ssc.wisc.edu/wlsresearch/, Health and Retirement Study http://hrsonline.isr.umich.edu/index.php, Avon Longitudinal Study of Parents and Children http://www.bristol.ac.uk/alspac/). The study teams also collect comprehensive phenotypic data beyond basic demographic information and medical condition. Therefore, these data sets provide an excellent opportunity to investigate genetic influences on 'every day' prosocial behaviour beyond strictly controlled laboratory-based experiments and on a much larger sample base. Simultaneously, recent progress in estimating heritability from whole genome sequence data [These studies are often designed as longitudinal studies to keep track of their participants over a longer period of time  have not been used very frequently to identify the genomic basis of behavioural traits, besides the GWAS used in mental diseases research. Although GWAS have historically only explained a small proportion of the variance in a variety of complex traits being studied, they are well suited to detect unknown causal variants associated with a trait as in contrast to candidate gene tests GWAS are hypothesis free. They therefore offer the opportunity to gain completely new insights into the genetic basis of behaviour. In addition, large study data sets of unrelated individuals allow for an estimation of genome-wide variance explained which due to the availability of common causal variants usually present underestimates. A typical problem of GWAS is their limited potential to describe biological mechanisms on basis of GWAS results. Gene set analysis addresses this issue and uses GWAS results which describe a limited number of significantly associated SNP’s with a trait to estimate associations between the trait and entire gene sets known for their specific biological functions . GWAS reAll these points taken together, large study data sets provide a promising basis to explore new directions in behavioural genetics.The goal of this study is to demonstrate new ways of exploring and investigating the genetic basis of (pro)social behaviour and decision making using established methods from medical/complex disease research. Not unlike complex diseases the genetic basis of a certain human behaviour is complex and heavily interdependent on various influence factors. However, unlike at least some complex diseases human prosocial behaviour is much more difficult to measure, quantify and describe compared to diseases and conditions with specified measurable symptoms.This leads to the probably single most important limitation of the study presented here: the phenotypic representation of human prosocial behaviour by self-reported helping behaviour. The amount of time a person spends in order to help out his/her family, friends and neighbours without getting paid covers by no means the entire spectrum of prosocial behaviour. However, we feel that it constitutes a valid real-life approximation of a well-defined characteristic of prosocial behaviour. Observations on real-life human helping behaviour with friends and family basically approximates the degree of helpfulness a person exhibits in its everyday life. Unlike in standardized laboratory experiments we can only speculate on the reasons for these observations based on the information we have at hand (the questionnaire). Generally, helping behaviour towards friends and family may be accounted for by Hamilton’s rule of kin selection (family) or the basic principle of direct reciprocity . The latUsing the data from the Health and Retirement Study we are able to go beyond this question. We can actually assess a degree of helpfulness in real-life. This comes of course with the cost of not being able to reproduce the motivations underlying these decisions.The study at hand is limited to investigate a very narrow spectrum of human prosocial behaviour–namely individual differences in helping behaviour towards family and friends. And although it is not able to give answers similar to standardized (laboratory) studies, its exploratory approach might very well show new directions in investigating human prosocial behaviour.Based on the University of Michigan's Health and Retirement Study (HRS), an on-going longitudinal panel study that collects survey data, anthropometric measurements, and physical performance tests, where more than 10,000 Americans have been genotyped, we used self-reported helping behaviour (SHB) to run a genetic association analysis on 1.2 Million SNPs.P = 6.96 × 10−10). P-values.One locus–rs11697300 –exceeded genome-wide significance in association with self-reported helping behaviour. Rs11697300 is an intergenic variant located between solute carrier family 52 (riboflavin transporter), member 3 (SLC52A3), and scratch family zinc finger 2 (SCRT2) on chromosome 20  = 30.7%, https://genome-euro.ucsc.edu/cgi-bin/hgTracks?db=hg38&lastVirtModeType=default&lastVirtModeExtraState=&virtModeType=default&virtMode=0&nonVirtPosition=&position=chr20%3A737890%2D737906&hgsid=225572689_6w3VFIL5xieR9y6lUUDNVbK6pABP). The Chimpanzee, the Orang—there is no data available for the Gorilla—as well as the phylogenetically closely related Gibbon show no differences in the region of interest.Rs11697300 is located in a conserved region in the Hominidae. This is based on data from the UCSC Genome Browser  is located in a narrow region, spanning 215,932 base pairs, on chromosome 4 covering transmembrane protein 33 (TMEM33), DDB1 and CUL4 associated factor 4-like 1 (DCAF4L1), solute carrier family 30 (zinc transporter), member 9 (SLC30A9), ATPase, Na+/K+ transporting, beta 1 polypeptide pseudogene 1 (ATP1B1P1), and BEN domain containing 4 (BEND4)  . S1 TablP = 2.52 × 10−9). And again, the area around SCL30A9 was revealed to be heavily populated with SNPs approaching genome-wide significance. gc = 1.0466).We confirmed the robustness of the results of the genetic association analysis with a linear model including six covariates from principal component analysis . Again, only one locus exceeded genome-wide significance in association with SHB . Using New genetic loci implicated in fasting glucose homeostasis and their impact on type 2 diabetes risk, by Dupuis et al. 2010 [P = 0.0435); b) Genome-wide study for circulating metabolites identifies 62 loci and reveals novel systemic effects of LPA, by Kettunen et al. 2016 [P = 0.0055); and c) Genome-wide Association Studies Identify Genetic Loci Associated With Albuminuria in Diabetes, by Teumer et al. 2016 [P = 0.0313; P = 0.0434). Studies a) and b) are flagged as “Caution” by LDHub because “using this data may yield results outside bounds due to relative low Z score”. However, there seems to be a genetic correlation between the presented GWAS on SHB and GWAS on metabolism and diabetes (for a summary of the genetic correlations see Applying LDHub we found significant genetic correlations to the following GWAS: a) al. 2010   with SNPs from the present SHB GWAS , dendritic (P = 0.00004) and neuron spine (P = 0.00004) and hormone receptor activity (P = 0.0007). Interestingly, some genes previously highlighted to influence prosocial behaviour are part of gene-sets significantly associated at P < 10−4: OXTR , AVPR1a , DRD4 . Other interesting associated gene sets are negative regulation of behaviour , learning , associative learning , and regulation of behaviour .Gene set analysis revealed a total of 343 gene sets significantly associated , member 3 (SLC52A3), and scratch family zinc finger 2 (SCRT2) on chromosome 20, is associated with self-reported helping behaviour. To date, no literature is available on the function of this variant or variants in strong linkage disequilibrium (LD) with rs11697300  as the Interestingly, albeit intuitively there no association between urinary albumin-to-creatinine ratio  would be expected, the genetic correlation between SHB and Albuminuria may make sense as Albuminuria is known of being associated with lower cognitive functioning particularly in elderly individuals ,43. If cAlso the gene set analysis did find significant associations of the results to some gene sets that make biological sense including the dopamine receptor genes (DRD1 to DRD5), OXTR, and AVPR1a, all well known in the research of social behaviour. Especially associations with (associative) learning and (negative) regulation of behaviour appear intuitive and supportive of the results of the GWAS. However, as a “correlative approach” a GWAS is not able to transfer the vague concept of “genetic influence” in causality and determination. Accordingly, the relevance of the gene sets found to be associated with the results of the present GWAS may not be over-interpreted, but may provide a starting point for future analysis and deliver ideas where to start looking for causality and determination.Based on our results we suggest that i) the potential function of rs11697300 and its surrounding area, as well as the other nearly genome-wide significant SNPs on and around SLC30A9, should be investigated in more detail; ii) rs11697300 and the other nearly genome-wide significant SNPs should be investigated in candidate-gene approaches, particularly in studies involving both laboratory-based experimental studies and studies on “every day” prosocial behaviour; iii) on the phenotypic level the accordance between lab and field data  should be investigated in more detail because this issue is still under debate ,45; and In conclusion, this study points towards new possible directions for research in behavioural genetics. We present results suggesting an association between yet undescribed genetic variants and human prosocial behaviour.We encourage other studies to replicate and expand upon our findings. This would be an important step forward in clarifying the biological functioning of loci detected and supporting the notion that these areas are associated with prosocial behaviour.The University of Michigan Health and Retirement Study (HRS) is an on-going longitudinal panel study designed to monitor changes in labour force participation and health transition of individuals toward the end of work life and beyond. The current sample population consists of 22,037 Americans over age 50. The sampling mechanism is based on a national probability sample to represent the entire American population. HRS collects survey data , anthropometric measurements, and physical performance tests , as well as blood and saliva samples.http://hrsonline.isr.umich.edu/index.php. This individual research project was approved by the Ethics Committee of the University of Vienna (Reference number 00077), data use was approved by the National Center for Biotechnology Information Genotypes and Phenotypes Database (NCBI dbGaP) Data Access Request system at the National Institutes of Health (Project ID 6192).The Health and Retirement Study (Project #6192) genetic data is sponsored by the National Institute on Aging  and was conducted by the University of Michigan . Collect−5. Further quality control steps were taken by teams at the University of Washington (UWGCC), the Health and Retirement Study investigator's team, and dbGaP. In total, 2,443,179 SNPs were genotyped. After several steps of stringent quality control measures, 1,244,134 SNPs were left for each participant Quality control steps included dropping dublicate SNPs and SNPs with a missing call rate > = 2%, Hardy-Weinberg-Equilibrium (HWE) P-value < 10−4 in either European or African samples, and a MAF < 0.05. http://hrsonline.isr.umich.edu/sitedocs/genetics/HRS_QC_REPORT_MAR2012.pdf).Based on voluntary participation, genotyping was performed on saliva samples. In total, 12,507 individuals have been genotyped since 2006. Genotyping was performed at the Center of Inherited Disease Research (CIDR) using the Illumina HumanOmni2.5-4v1 array and using the calling algorithm GenomeStudio version 2011.2, Genotyping Module 1.9.4 and GenTrain version 1.9. The medium call rate is 99.7% and the error estimated from 336 pairs of the study sample duplicates is 6 × 10After removing 172 related individuals  of the initial pool of 12,507 study participants 12,235 individuals were left in the subject pool. Families were defined as individuals being connected by a kinship coefficient (KC) > 0.1. The threshold corresponds to the expected KC of half-siblings minus two standard deviations.Self-reported helping behaviour (SHB) is coded in four questions  in section G  of the Core questionnaire catalogue , public use dataset). The questions read as follows:MG198, Have you spent any time in the past 12 months helping friends, neighbors, or relatives who did not live with you and did not pay you for the help? MG199, Altogether, would you say the time amounted to less than 100 hours, more than 100 hours, or what? MG200, Would it be less than 200 hours, more than 200 hours, or what? MG201, Would it be less than 50 hours, more than 50 hours, or what? Based on these questions, we merged the eight possible combinations of answers into five categories of hours spent helping others: 0, 1 to 50, 51 to 100, 101 to 200, and 200+. 10,713 individuals with non-missing answers to SHB were matched to 1,244,134 SNPs. Genetic association analysis was carried out with i) a linear mixed model with a genetic relatedness matrix (GRM) and the effects of SNPs treated as random and ii) a standard linear regression approach with six principal component analysis eigenvectors as covariates.GCTA (version 1.25) provides options to perform mixed linear model (MLM)-based association analyses . The MLMWe implemented the GCTA-LOCO approach, which evaluates markers on a given chromosome using a GRM calculated from the remaining chromosomes. This 'leaving-one-chromosome-out' (LOCO) method avoids double-fitting the candidate marker and increases power of the analysis compared to regular MLM approaches as well as linear regression , 49.http://hrsonline.isr.umich.edu/sitedocs/genetics/HRS_QC_REPORT_MAR2012.pdf.In PLINK (version 1.07) we used the implemented standard linear regression for quantitative trait data  to find https://github.com/bulik/ldsc/wiki/Heritability-and-Genetic-Correlation). LDSC calculates genomic inflation as the proportion of the inflation in the mean χ2 that the LD Score regression attributes to causes other than polygenic heritability [gc, not biased by sample size in the presence of polygenicity [We used the python tool LDSC to estimate genomic inflation (tability . Using tgenicity .We estimated genetic variance based on GCTA's GREML-LDMS method  using whhttp://ldsc.broadinstitute.org) to estimate potential genetic correlations among SHB and 177 diseases and traits gathered from publicly available resources and consortia. Estimation is done on the basis of the summary level results of the present GWAS on SHB and the summary results of those 177 GWAS [We used the online tool LDHub  on the present GWAS results . GSA assigns SNPs to a gene that encompasses the SNP with some padding. Genes are clustered in gene sets of known function. As gene set we used the set “Gene Ontology” (default) with a padding size of +/- 20,000 and k-th best P value (default 2). P values are corrected according to Benjamini and Hochberg [We applied the gene set analysis (GSA) approach developed by Nam et al  implemenHochberg . The GSAHochberg . DetailsHochberg  and Kim Hochberg .S1 Fig(a) Manhattan plot of genome-wide association for self-reported helping behaviour. (b) Quantile-quantile plot of GWAS for self-reported helping behaviour.(EPS)Click here for additional data file.S1 FileP < 0.05) with SNPs. Information includes set name, gene count, set size, z-score, p-value, corrected p-value, FDR, and gene symbols.Gene set analysis revealed a total of 343 gene sets significantly associated ((CSV)Click here for additional data file.S1 Tableb), standard error (se), and p-value (p). PLINK: linear regression implemented with PLINK association analysis and PCA eigenvectors as covariates with regression coefficient (b), t-statistic (stat), and p-value (p). 17 SNPs located within 215,932 base pairs on chromosome 4 are highlighted in bold.SNP: Single nucleotide polymorphism, Chr: chromosome, Pos: base pair position, ID: SNP name, Ref: reference allele, Alt: alternative allele, Freq: reference allele frequency. GCTA-LOCO: mixed-linear model implemented with GCTA's leaving-one-chromosome-out method with regression coefficient ((DOCX)Click here for additional data file.S2 TableEst) and standard errors (s.e.); for details see main text.GREML-LDMS : Linkage disequilibrium and minor allele frequency stratified GREML analysis with estimates ((DOCX)Click here for additional data file.S3 TableSHB: self-reported helping behaviour. The list of gene sets is grouped in to the top results (Top) and interesting results (Misc).(DOCX)Click here for additional data file.S4 Tablehttp://www.ensembl.org/Homo_sapiens/Variation/Explore?db=core;r=20:737398-738398;v=rs11697300;vdb=variation;vf=107862839. ASW: African Ancestry in Southwest US, CEU: Utah residents with Northern and Western European Ancestry, MXL: Mexican Ancestry in Los Angeles, California. Populations were chosen to represent the Health and Retirement Study sample population.SNP: single nucleotide polymorphism. Data available on (DOCX)Click here for additional data file.S5 Table2: LD score with rs11697300. GCTA-LOCO: mixed-linear model implemented with GCTA's leaving-one-chromosome-out method with regression coefficient (b), standard error (s.e.), and p-value (p).SNP: single nucleotide polymorphism, Chr: chromosome, Pos: genomic position, ID: SNP name, Ref: reference allele, Alt: alternative allele, Freq: reference allele frequency, r(DOCX)Click here for additional data file.


Genotype-phenotype association has been one of the long-standing problems in bioinformatics. Identifying both the marginal and epistatic effects among genetic markers, such as Single Nucleotide Polymorphisms (SNPs), has been extensively integrated in Genome-Wide Association Studies (GWAS) to help derive “causal” genetic risk factors and their interactions, which play critical roles in life and disease systems. Identifying “synergistic” interactions with respect to the outcome of interest can help accurate phenotypic prediction and understand the underlying mechanism of system behavior. Many statistical measures for estimating synergistic interactions have been proposed in the literature for such a purpose. However, except for empirical performance, there is still no theoretical analysis on the power and limitation of these synergistic interaction measures.In this paper, it is shown that the existing information-theoretic multivariate synergy depends on a small subset of the interaction parameters in the model, sometimes on only one interaction parameter. In addition, an adjusted version of multivariate synergy is proposed as a new measure to estimate the interactive effects, with experiments conducted over both simulated data sets and a real-world GWAS data set to show the effectiveness.We provide rigorous theoretical analysis and empirical evidence on why the information-theoretic multivariate synergy helps with identifying genetic risk factors via synergistic interactions. We further establish the rigorous sample complexity analysis on detecting interactive effects, confirmed by both simulated and real-world data sets.The online version of this article (10.1186/s12864-018-4552-x) contains supplementary material, which is available to authorized users. HoweverTo find important features considering interactive effects, one possible solution is to derive a full Logistic Regression model that incorporates the interactive effects as feature multiplication terms . HoweverDue to the prohibitive sample complexity and computational cost when considering the full model with different orders of interactions, most of the existing biomarker identification approaches take a two-step procedure: 1) First, some heuristic measures based on correlation, mutual information, or simplified regression models, are adopted to estimate the statistical association among pairs of features and the outcome , 23, 24;The primary goal of this work is to establish rigorous mathematical theories for feature screening and selection approaches with the consideration of interactive effects under a specific system model based on logistic regression , 10, whid independent binary profiled variables X1,X2,…,Xd and a binary outcome variable Y. The profiled variables are assumed to have the probability distribution Pr(Xi=+1)=pi and Pr(Xi=−1)=qi with pi,qi>0, pi+qi=1 for 1≤i≤d, and the conditional probability of the outcome variable Y is assumed to take the following form: Consider σ(x):=1/(1+ex−) is the sigmoid function and {βS:S⊆{1,2,…,d}} is a family of real parameters. For any subset S of {1,2,…,d}, parameter βS measures the amount of the cooperative interaction among the variables Xi’s (i∈S). We call this model as the “full” model. Assume that all parameters βS are bounded, i.e., |βS|<C for all S⊆{1,2,…,d}. It is a highly generic model based on the classical logistic regression model, since it incorporates the cooperative interaction of any subset of profiled variables X1,X2,…,Xd. We can estimate the cooperative interactions among candidates Xi’s and Y via the help of multivariate information measures, which are suggested to quantify the correlation among two or more random variables. Such measures include multivariate mutual information ormation .RSN|N−1” was used) and recently proposed for interaction and association studies in bioinformatics by Anastassiou  quantifying the effect size for each of these features and pairs. For each logistic regression model, we generate random training sets of 500, 1000, 1500, 2000, 2500, and 3000 samples. Each training sample consists of an observation of each covariate  quantify synergy , 37, 2)  synergy , 4) Igna synergy , 5) Wata synergy , and 6) p-value is less than 10−5, which shows the statistically significant difference between the detection accuracies. The adjusted multivariate synergy is directly related to the interaction parameter according to Corollary 3. Thus it can well capture the interactive effect via the normalization. It needs a little more samples to get a relatively accurate estimate (both its numerator and denominator need to be estimated) compared to multivariate synergy. We conjecture that the combination of these two measures probably could serve as a more useful tool for interaction detection. The other methods based on Schneidman’s normalized synergy, Watabene’s total correlation, or Han’s dual total correlation have inferior performance when identifying the interactions, since these measures have no clear relationships with the interaction parameters in logistic regression models.For each measure, we consider the three pairs with the largest estimated values of this measure as the interacting pairs, and evaluate the detection correctness. Figure We further study the relationship between the sample number and the detection accuracy of the interactive effects. It is observed that the curves obtained by both the multivariate synergy and the adjusted multivariate synergy fit very well with a logarithmic relation: i.e., Here we remark that the same conclusion can be drawn with different settings on the number of features in the model. We also generate 1000 logistic regression models, each of which contains 20 features. In each model, we randomly generate 200, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, and 2000 samples by the same way as in the previous simulation. Figure K from a uniform distribution on {1,−1}, and then each covariate Xi is drawn from a conditional probability Pr(Xi=1|K)=pi+μK and Pr(Xi=−1|K)=qi−μK, for 1≤i≤50, with μ controlling the dependency among covariates. Here, pi is randomly chosen from a uniform distribution over {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9} as in the previous simulations. The output Y is generated by the conditional probability . The trends are clear that these measures still can help accurately identify the interactions even when all pairs of features are weakly dependent, especially when μ is small (< 0.1). The relationship between the sample complexity and detection error rate still follows the derived logarithmic relationship with weakly dependent features.Although we derive the theoretical results with the assumption that the features are independent of each other, the multivariate synergy and the adjusted multivariate synergy can still serve as good measures of interactions for the cases when the features are weakly dependent in practice. We further simulate such weakly dependent cases to empirically evaluate their interaction detection performance. In each simulated full logistic regression model, we first randomly choose Type 1 Diabetes (T1D), previously known as Insulin-Dependent Diabetes Mellitus (IDDM), is an autoimmune disease resulting from the deficiency of insulin. This disease is conjectured to be caused by both genetic and environmental factors and has attracted tremendous research interests, especially in detecting pairwise or high-order genome-wide interactions for T1D –46. Herep-value 1.467×10−18. Also, the pairs with the 681 largest estimates selected by our algorithm cover all 91 pairs in Table W, giving a p-value 1.044×10−27. Further, Table W contains 17 pairs  with the lowest PLINK p-value [p-value) 1.100×10−16, and they are included in the set of the pairs with the 76  largest estimates selected by our algorithm. The corresponding p-values in Fisher’s exact tests are 1.678×10−22 and 2.268×10−15, respectively. These significant p-values shows the highly significant overlap between the interacting SNP pairs found by our algorithm and those by BOOST and PLINK.Fisher’s exact test  has been p-value   with p-values less than 0.05. Note that in the T1D literature [We further check the biological interpretation of the top ranked SNPs with significant interactions. The associated genes with the SNPs in the top 15 interacting pairs are listed in Table terature –55, MHC Genome mapping of SNPs is further illustrated in Fig. As discussed in previous simulation and real-world data experiments, multivariate synergy measures are effective in identifying interactions among candidate risk factors for genotype-phenotype association studies. This is expected from our derived theoretical connection between synergy and interaction parameters in logistic regression modeling of genotype-phenotype relationships. On the other hand, accurate and reproducible identification of interactive factors requires that the number of samples grows exponentially with the order of interactions to detect. Based on the given number of samples, identified interactions should be thoroughly validated with caution.In this paper, we study why the multivariate synergy can serve as a measure to quantify the interaction among multiple factors for feature selection with interactions. We further have established the theoretical analysis on sample complexity, which is general for feature selection when considering interactions. For risk factor identification in GWAS, when the genotype-phenotype association is modeled by logistic regression, we show that the multivariate synergies have a close relationship with the corresponding multiplication parameters capturing the interactive effects among features. Based on such derived relationships, we have proposed an “adjusted multivariate synergy” as a new interaction measure. The experiments showed the adjusted multivariate synergy achieves an excellent empirical performance in risk factor identification with interactions over both simulated and real-world T1D GWAS data.Additional file 1Some useful lemmas and the proofs of Theorems 1 and 5. (PDF 89 kb)


Brassica napus. Understanding the genetic regulation of lodging resistance is therefore of key interest to breeders worldwide. Current strategies aimed at minimising lodging risk involve the incorporation of dwarfing genes or the application of plant growth regulators. However, despite these efforts, lodging continues to be a persistent problem and it is therefore of high interest that novel, complimentary strategies for lodging control are implemented. One approach would be to focus on understanding the genetic properties underlying stem mechanical strength. With this in mind, we screened a training genetic diversity panel of B. napus accession for variation in stem mechanical strength and related traits. Using Associative Transcriptomics, we identified molecular markers for a suite of valuable traits. Using an independent test genetic diversity panel, we show that the methods employed are robust for identification of predictive markers. Furthermore, based on conserved synteny with Arabidopsis thaliana, we are able to provide a biological context to the marker associations detected and provide evidence for a role in pectin methylesterification in contributing to stem mechanical strength in Brassicaceae.Lodging continues to be a major cause of yield loss in important crop species such as The online version of this article (10.1007/s11032-018-0781-6) contains supplementary material, which is available to authorized users. Brassica napus L.) is a crop of growing economic importance and is second only to soybean as a commercial oilseed crop. In 2014, oilseed rape contributed 71 million metric tonnes to global oilseed production (FAOSTAT). In addition to providing a high-value oil for human consumption, oilseed rape also provides a source of protein meal which is used widely as animal feed. Oilseed rape is widely used in rotation with cereal crops, aiding the management of weeds, pests and diseases. Finally, more recently, oilseed rape has been recognised as a potential feedstock for biofuel production  background resulted in a significant height reduction and decreased lodging risk. However, additional pleotropic effects of this allele, including a delay in flowering time, may make the incorporation of this allele undesirable under certain growth conditions . The two commonly used PGRs for lodging control of oilseed rape are metconazole and tebuconazole . We show that Associative Transcriptomics is a powerful approach for identification of molecular markers that have potential for use in marker-assisted breeding for increased stem strength. Furthermore, by making use of the genomic resources available in the closely related species, Arabidopsis thaliana, we provide evidence of an important role of cell wall pectin demethylesterification in contributing to variation in stem mechanical strength in Brassicaceae.Our aim was to explore the available variation for lodging-related traits across a panel of B. napus accessions grown at KWS UK Ltd., Thriplow (with field location varying between years). At both sites, accessions were grown in single, randomised, 1-m2, open-field plots. Forty-six B. napus accessions were grown in 2010  and 44 accessions in 2011 . In each year, where possible, ten plants were harvested from each accession. In 2012, a larger trial consisting of 79 B. napus accessions was conducted under field conditions at The John Innes Centre . This trial consisted of plants grown in three randomised blocks. For each accession, one plant was grown in each block. These plants were grown within a net-enclosed cage to reduce damage caused by pigeons, which were a problem on this site. Supplementary Data File Plant material was harvested across 3 years. The first 2 years (grown in 2010 and 2011) consisted of small panels of Prior to harvest, all plants grown in trial J2012 were screened for lodging risk. This was carried out at maturity just prior to harvest using a digital force gauge mounted on a pulley system  and collection of other morphological data), the transverse of the most basal stem end was photographed. All images were later analysed using the digital analysis software Sigma Scan , allowing the following cross-sectional measurements to be determined: whole stem area , stem hollow area  , the area of stem parenchyma (or pith) and the thickness of the stem outer cortex.To allow for an in-depth analysis of the relationship between plant morphology and stem strength/lodging risk, plant height and stem weight were scored across all genotypes. Following this, a 20-cm section was taken from the base of the stem. To obtain stem cross-sectional measurements . For this reason, prior to mechanical testing, all stem samples were dried for a minimum of 6 days at 55% RH and 23 °C in the silica chamber.B. napus, a three-point bend test method was used. This screen was carried out on the lower portion of the stems of each plant (following the equilibration of stem moisture content with the previously described silica chamber) and was performed using the Texture Analyser (TA)  with a three-point bend test setup   Fig. b. These UK) Fig. . The TA ned Fig. c. These ned Fig. , were usned Fig.  and the It is a measure of the geometrical properties of, in this case, a hollow beam which summarises the distribution of material around the central axis.D2diameter of whole stem calculated from stem cross-sectional areaD1adiameter of stem hollow calculated from stem hollow areawhereThe modulus of rupture describes the resistance of the stem material to breakage under load.MOE describes the flexural stiffness of the stem material under load.Lthe length of the stem sample between the two supportsaL/2where10 transformation and the ANOVA repeated using transformed data values. To determine whether the accessions varied significantly for the various traits, a minimum significance level of P < 0.05 was implemented. Following the calculation of trait means, Genstat was used to undertake regression analysis.To assess trait normality and trait variance components, analysis of variance (ANOVA) was performed using Genstat 15th edition. Where non-normality was observed, the raw trait values were transformed using a logB. rapa , were tr et al.  identifir et al.  and Harpr et al. .B. napus accessions between the two detected populations, was constructed.To assess the level of broad-scale population ancestry, STRUCTURE analysis was used  analysis in TASSEL . Supplementary Data File http://signal.salk.edu/tdnaprimers.2.html.To allow for extraction of genomic DNA, leaf samples were ground to a fine powder and 200 μl of TE extraction buffer added . Following further mixing, samples were centrifuged at maximum speed for 3 min. The resulting supernatant was then added to 150 μl of 100% isopropanol. Following thorough mixing by inverting, samples were centrifuged for 7 min at maximum speed. The resulting pellet was then washed with 70% ethanol and resuspended in 40-μl distilled water. Genotyping of T-DNA mutants was carried out using primers listed in Supplementary Data File Following the identification of homozygotes, mutants were grown for a second generation for phenotyping. After 4 weeks of growth, plants were transferred to ARRASYSTEM trays with supportive tubing (Beta Tech bvba), allowing for the plants to bolt and mature with no need for staking.B. napus. To accommodate the much smaller stem samples, a support stand was made using two razor blades (blade covered and pointing downwards) held within a clamp. These supports were set to be 1.2 cm apart. The Texture Analyser was fitted with a 1-kg load cell. The probe was set at a start height of 4 cm from the test bed and the probe was set to descend a further 0.5 cm following contact with the stem sample.At maturity, plants were harvested and prepared for mechanical testing as follows. A 2.5-cm section was taken from the base of the stem using a scalpel. This was carried out, where possible, from six plants per genotype. This section was then stored in a small breathable bag and stored within a silica humidity tank at 55% RH and 23 °C for 2 days. The cross-section of the lower stem transverse was photographed to allow for stem morphological measurements to be collected as described previously for −1 using a dynamic alignment FTIR spectrophotometer . Prior to analysis of stem tissue, a spectrum of air was generated which acted as the reference medium. Each stem sample was macerated using a pestle and mortar and the ground tissue clamped against a diamond element. To reduce error, the crystal was completely covered with stem tissue. Between samples, the crystal and clamp were cleaned with 100% ethanol. This was left to dry thoroughly before analysing the next sample. For each sample, two spectra were taken, with three plant replicates per genotype analysed. All spectra obtained were truncated to the fingerprint region (800–1800 cm−1). The baseline was anchored to 1800 cm−1 and the curve areas normalised. Average spectra were then obtained for the mutant line and the values subtracted from those of WT Arabidopsis. This allowed for any difference in cell wall composition of the T-DNA insertion line relative to WT to be detected.Fourier transform infrared (FTIR) spectra were collected across the spectral range of 800–4000 cmB. napus (genotype unknown) were harvested from a test panel grown at Nottingham University. These genotypes formed a subset of the Renewable Industrial Products from Rapeseed (RIPR) diversity panel . Following confirmation that the designed primers were able to effectively screen for the target variation, they were further used to genotype the 86-accession test panel. All genotyping was performed using AMPLITAQ Gold polymerase ). Prior to sequencing, PCR reactions were purified using the ExoSAP protocol (Etchevers t test (Genstat 15th edition), the trait data and genotype data obtained were assessed for any significant marker-trait segregation patterns. Supplementary Data File Eighty-six accessions of P = < 0.01), as might be expected for an association panel of diverse accessions.We first assessed lodging resistance across the diversity panel. Using ANOVA, the trait data obtained from the largest trial (J2012) was assessed for normality. In all cases, approximately normally distributed residuals were observed. Through ANOVA, for all traits included in the study, we observed high levels of phenotypic variation between genotypes (P = 0.618), significant differences were seen between accessions for all of the traits assessed (P < 0.05). Data summaries and all ANOVA outputs for each of these field trials can be found in Supplementary Data File Two smaller trials were conducted, K2010 and K2011. In the majority of cases, accessions were replicated across both trials. However, a number of accessions were represented in just a single year (see Supplementary Data File F/V correlate positively with an R2 of 0.88 (P < 0.001). This suggests that these traits are highly related, i.e. stems resistant to bending are also more resistant to breaking. These absolute strength traits also correlate significantly with a number of stem morphological and structural traits. The structural strength trait, second moment of area, correlates positively with both Fmax and F/V with respective R2 values of 0.63 (P < 0.001) and 0.66 (P < 0.001). Stem diameter was found to explain a greater level of variation than second moment of area in these absolute strength traits with an R2 value of 0.70 (P < 0.001) for both Fmax and F/V. Stem outer cortex thickness and parenchyma area were also found to correlate positively with Fmax and F/V. These traits are also highly related to stem diameter and second moment of area. It is therefore difficult to assess whether these tissues themselves are contributing to strength, or whether the correlations are due to their relationship with a more general increase in stem thickness.To assess the importance of stem morphological and structural traits on stem mechanical strength, regression analysis was conducted. A summary of the results obtained from this analysis is given in Table R2 values of 0.62 (P < 0.001) and 0.64 (P < 0.001) were seen for Fmax and F/V respectively. When assessing the relationships between this biomass trait and the stem structural and morphological traits, we observed that the correlations with Fmax and F/V were stronger. Similar correlations were also identified between stem weight and stem diameter/second moment of area. It is therefore likely that the relationship observed between stem weight and absolute strength reflects this, but may also suggest that stem density is important. Stem absolute strength was also seen to correlate positively with plant height, with R2 values of 0.35 and 0.39 obtained for F/V and Fmax respectively.A strong relationship was detected between the absolute strength traits and stem weight where F/V are used in calculating MOR and MOE respectively. Based on the equations used, it would be expected that an increase in these absolute strength traits would result in increased material strength values. However, when looking more closely at the raw data, it can be seen that in the majority of cases, high absolute strength values can be largely explained by high structural strength (large diameter/high second moment of area) and in these accessions, low material strength is often observed. While this suggests that stem structural strength is the most important contributor to absolute stem strength in these accessions, it does not mean that the material strength properties do not have a role to play. When looking at the mean trait values, there are several examples which illustrate a certain importance of material strength in determining overall stem strength. An example can be seen when comparing the trait data for the varieties, Dwarf Essex and Taisetu. The absolute strength values for Dwarf Essex are 24.2 and 19.9 kg/s for Fmax and F/V respectively. The mean values for stem diameter and second moment of area for this accession are 9.2 and 362 mm4 respectively. When comparing these data to those obtained for Taisetu, it can be seen that despite displaying very similar values for stem diameter and second moment of area (9.288 and 365.8 mm4 respectively), Taisetu shows much lower absolute strength values of 6.259 and 8.79 kg/s for Fmax and F/V respectively. This may be explained by the much lower material strength seen for this accession in comparison to that of Dwarf Essex. For Taisetu, MOR and MOE values of 2.52 and 173.4 N/mm−2 were seen. In comparison, Dwarf Essex was found to have a mean MOR of 11.03 N/mm−2 and a mean MOE of 39.1 N/mm−2. This clearly shows the effect that material strength can have in the absence of variation in stem structural strength. As seen in the regression analysis results carried out for the 2012 JIC trial material, in the analyses carried out on material from the 2010 and 2011 KWS trials, the absolute strength traits Fmax and F/V were found to correlate most significantly with stem diameter, second moment of area and stem weight (data not shown).Interestingly, when looking at the relationship between stem absolute and stem material strength traits, a negative relationship can be seen. Stem material strength describes the mechanical strength contributed by the material composition of the stem and is considered independent of stem structural strength which is described by the stem geometry. Of course, these strength components are not truly mutually exclusive, but it is useful, when aiming to understand the underlying genetics, to break down complex traits into component traits—in this case material strength, structural strength and absolute strength . Fmax and P = 0.043). Through a further regression analysis, the relationship between this field-based scoring and the mechanical strength traits obtained from the three-point bend test were explored. Some of the key relationships observed are plotted in Fig. R2 values of 0.42 (P < 0.001) and 0.33 (P < 0.001) for Fmax  for MOR and 0.21 (P < 0.001) for MOE  and 0.37 (P = 0.001) for stem diameter  and 0.26 (P < 0.001). However, given that these correlations are weaker than those seen between SLR and the absolute strength measures, it seems that the variation in these morphological phenotypes has not had a confounding effect on the SLR measurements. The detection of these significant correlations is very promising and suggests that the mechanical strength measures obtained through the three-point bend test are capturing important variation relevant to stem mechanical strength under field conditions. Furthermore, as with the results obtained from the three-point bend test assay, the results presented here suggest that understanding the genetic control of stem absolute strength and stem structural strength  would be of great interest for the improvement of lodging resistance in B. napus. The work presented also confirms the widely accepted link between plant height and lodging susceptibility.Using the apparatus illustrated in Fig. In order to identify molecular markers that might be useful to support breeding for traits of relevance for lodging resistance, we conducted association analysis by Associative Transcriptomics.F/V. Given the very high correlation observed between Fmax and F/V, we limited the consequent Associative Transcriptomics analysis to Fmax only. We also explored marker-trait associations for several stem structural traits that showed high correlation with absolute stem strength. The structural traits included were stem diameter, stem second moment of area, stem parenchyma area, outer cortex thickness and stem weight. Finally, given the known role that height has to play in contributing to lodging risk (which was confirmed in the present study), variation in plant height was also included in the Associative Transcriptomics analysis. An overview of the most highly significant SNP and GEM marker associations identified for each trait is provided in Supplementary Data File Through regression analysis, we showed that stem lodging risk is most highly related to the absolute stem strength traits Fmax and P = 4.45E-05, trait effect = 30.9% (a percentage based on trait range across accessions)). This region was also detected (more weakly) in the SNP association analysis carried out for the K2010 material for absolute stem strength . No clear candidate genes were identified in close proximity to this marker.The second SNP association for this absolute strength trait was detected on chromosome A3/C3 . Weak signals in this region were also identified for stem weight in the Associative Transcriptomics analysis carried out on both the J2012 and K2011 trials  (AT4G27710). The expression of this gene correlates negatively with Fmax  . The second association, seen on A9/C8, had a much stronger signal, with the most significant marker being JCVI_7691:268 (P = 5.44E-07). In close proximity to the SNP association on A6/C6, there are several genes involved in cell wall biosynthesis. The first of these is an orthologue of Arabidopsis INCURVATA 4 (AT1G52150). This gene is predicted to act as a transcriptional regulator, important for vascular development and cell differentiation. In addition, two genes known to contribute to cell wall mannan structure were identified. Firstly, there is a gene orthologous to Arabidopsis MANNAN SYNTHESIS-RELATED 2 (AT1G51630). Cell wall mannans are thought to bind to cellulose, providing structural integrity to the cell wall  Fig. . To furtP value of 0.00023. No clear candidate genes for this trait were identified in this region.As seen for Fmax, a clear SNP association can be seen on chromosome A2/C2 for stem diameter with the most significant marker being JCVI_31359:1651   SUPPRESSOR 1 (AT4G30610). BRI1 is known to be involved in regulation of plant height in Arabidopsis. Mutants defective for this gene have a very clear dwarfing phenotype . No clear candidate genes were detected in this region.One clear association signal was detected for plant height. This was seen on chromosome A3/C7 where the most significant marker is JCVI_26003:352 (MICROTUBULE ORGANISATION 1 (MOR1) (AT2G35630). Arabidopsis mutants for this gene exhibit a temperature-dependant reduction in organ size. This was found to be due to the inability of the mutant plant to correctly organise cortical microtubules ), were mapped as a trait against the SNP data. In doing this, a very clear SNP association on chromosome A2/C2 was uncovered. The most highly associated marker within this peak was JCVI_20133:246. No clear candidate genes were identified within this region. However, the results presented here suggest that there may be a gene, which resides at this A2/C2 locus, which either directly or indirectly regulates the expression of MOR1, contributing to the observed variation in plant height in B. napus.The GEM analysis identified two highly associating GEM markers for plant height on chromosome A5, both of which correspond to a gene orthologous to Arabidopsis B. napus accessions of previously unknown genotype was screened for marker variation detected through Associative Transcriptomics. Following mechanical testing, it was then possible to assess whether the allelic variation segregates with the target trait as would be expected based on the Associative Transcriptomics results.The discovery of marker-trait associations through methods such as Associative Transcriptomics has the potential to contribute greatly to crop improvement through marker-assisted breeding. The durability of such markers can however depend on many factors including environmental interactions. It is also possible that the associations may be the result of false-positive errors due for example to any unaccounted-for population structure or relatedness between individuals used within the study. It is therefore important that the efficacy of detected markers in selecting for the trait of interest is validated. To do this, a test panel  of P value 9.32E-05; trait effect of 26.5%). Following the assessment of the efficacy of the developed genome-specific marker assay in screening for the target variation (using a subset of Associative Transcriptomics accessions of known genotype), the marker assay was used to explore variation across the 86-accession test panel. High levels of variation in Fmax are seen across accessions and there is no discernible relationship with the sub-populations identified through STRUCTURE analysis, as shown in Fig. P < 0.05) increase in Fmax associated with the increasing (A/G) allele, as shown in Fig. Given the importance of the absolute strength trait, Fmax, in contributing to stem lodging resistance, we focused this marker validation study on the marker association detected on chromosome A2/C2 with the most significant marker being JCVI_31359:1657. This marker was not only found to show association with Fmax across multiple field trials (across different years and environments), but was also seen for additional traits which were found to be related to stem absolute strength through regression analysis. Due to low sequence read-depth in the region surrounding the most significant marker in this association peak, an alternative, tightly linked marker, JCVI_31359:1723, was used 1 to pme(i)3.Given that two independent SNP associations identified for Fmax  were in close proximity to genes implicated in pectin methylesterification, we decided to explore the potential causality of these genes through screening Arabidopsis T-DNA mutants. Although no T-DNA insertion mutants were available for the PME/PMEI candidate genes identified on chromosome A2/C2, several mutants were available for the PME/PMEI candidate identified on chromosomes A5/C5. These mutants each carried an insertion affecting the coding regions of Arabidopsis gene model AT3G12880. Supplementary Data File Following genotyping, homozygous mutants were grown alongside wild-type (WT) control plants and the mature stem tissue mechanically tested using a three-point bend test method. Figure pme(i)-2 homozygous and WT plants. Figure −1. The second is a peak at between 1600 and 1630 cm−1, which would show a higher or lower absorbance relative to the 1740 cm−1 in low methylesterified and high methylesterified pectins respectively (Szymanska-Chargot and Zdunek −1 and depletion at 1624 cm−1. This suggests that this pectin-related gene is functioning as a pectin methylesterase and that a lack of demethylesterification in these mutant lines is contributing negatively to absolute stem strength.Based on information available from the Arabidopsis Information Resource (TAIR), it is not clear whether AT3G12880 has a role in promoting or inhibiting pectin methylesterification. Studies have reported high similarities in the sequence of proteins performing these seemingly antagonistic functions  of pectins as having an important role in determining stem strength in B. napus genotypes included in the panel has enabled the detection of high levels of genetic variation and high mapping resolution from a relatively small Associative Transcriptomics panel. We have also illustrated the importance of carrying out marker validation experiments to reveal the potential of markers identified through association analysis for the selection of traits of agronomic importance. Such confirmation provides greater confidence in the usefulness of the markers identified, strengthening the translation of scientific discovery from the lab into the field.The work presented here illustrates the power of Associative Transcriptomics for understanding the genetic regulation of complex traits in a polypoid crop species of global economic importance. The careful selection of the ESM 1(XLS 41 kb).ESM 2(XLS 23241 kb).ESM 3(XLS 326 kb).ESM 4(TXT 46.4 mb).ESM 5(TXT 62.3 mb).ESM 6(TXT 62.7 mb).ESM 7(XLS 15 kb).ESM 8(PDF 2556 kb).