Very low-depth sequencing has been proposed as a cost-effective approach to capture low-frequency and rare variation in complex trait association studies. However, a full characterization of the genotype quality and association power for very low-depth sequencing designs is still lacking.We perform cohort-wide whole-genome sequencing (WGS) at low depth in 1239 individuals (990 at 1× depth and 249 at 4× depth) from an isolated population, and establish a robust pipeline for calling and imputing very low-depth WGS genotypes from standard bioinformatics tools. Using genotyping chip, whole-exome sequencing (75× depth) and high-depth (22×) WGS data in the same samples, we examine in detail the sensitivity of this approach, and show that imputed 1× WGS recapitulates 95.2% of variants found by imputed GWAS with an average minor allele concordance of 97% for common and low-frequency variants. In our study, 1× further allowed the discovery of 140 844 true low-frequency variants with 73% genotype concordance when compared to high-depth WGS data. Finally, using association results for 57 quantitative traits, we show that very low-depth WGS is an efficient alternative to imputed GWAS chip designs, allowing the discovery of up to twice as many true association signals than the classical imputed GWAS design.https://www.ebi.ac.uk/ega/home): EGAD00010000518; EGAD00010000522; EGAD00010000610; EGAD00001001636, EGAD00001001637. The peakplotter software is available at https://github.com/wtsi-team144/peakplotter, the transformPhenotype app can be downloaded at https://github.com/wtsi-team144/transformPhenotype.The HELIC genotype and WGS datasets have been deposited to the European Genome-phenome Archive (Bioinformatics online. The contribution of low-frequency and rare variants to the allelic architecture of complex traits remains largely uncharted. Power to detect association is central to genetic studies examining sequence variants across the full allele frequency spectrum. Whole-genome sequencing (WGS)-based association studies hold the promise of probing a larger proportion of sequence variation compared to imputed genome-wide genotyping arrays. However, although large-scale high-depth WGS efforts are now underway , low-depet al., 2015) and UK10K  projects (As part of the Hellenic Isolated Cohorts (HELIC) study, we whole-genome sequenced 990 individuals from the Minoan Isolates (HELIC-MANOLIS) cohort at 1× depth, on the Illumina HiSeq2000 platform. In addition, 249 samples from the MANOLIS cohort were sequenced at 4× depth  and common (MAF > 5%) SNVs, respectively, when compared to variants present on the Illumina OmniExpress and HumanExome chips genotyped in the same samples. In 10 control samples from the Platinum Genomes dataset (n = 1127), although this positive predictive value varied across the MAF spectrum, from 8.9% for singletons to 95.1% for common variants . We used a reference panel containing haplotypes from 4873 cosmopolitan samples from the 1000 Genomes and UK10K projects, as well as the phased haplotypes from 249 MANOLIS samples sequenced at 4× depth. The best-performing pipeline, described in variants . GenotypThe genotype refinement and imputation step yielded 30 483 136 non-monomorphic SNVs in 1239 MANOLIS individuals. The number of variants discovered using 1× WGS is nearly twice as high as that from array-based approaches. In a subset of 982 MANOLIS individuals with both 1× WGS, OmniExpress and ExomeChip data, we called 25 673 116 non-monomorphic SNVs using 1× WGS data, compared to 13 078 518 non-monomorphic SNVs in the same samples with chip data imputed up to the same panel : 13, 671A crucial question is the proportion of true positives among these additional SNVs not found by GWAS and imputation. By comparing their positions and alleles with high-depth WGS in the same samples, we find that the positive predictive value (PPV) profile for these variants is much lower compared to when all variants are examined . As expeP < 5×10−7 (n = 1457) study based on 22× WGS on the same traits in the same cohort (P < 5 ×10−5). According to this metric, 52 of 182 independent signals (28.5%) were true in the imputed GWAS, in contrast to 108 of 462 (23.4%) in the 1× study  are not seen associated below that threshold in the imputed GWAS. As expected, there are significantly fewer  true associations in the imputed GWAS not recapitulated by the 1× study.1× WGS calls a larger number of variants and is noisier than imputed GWAS in the same samples. To evaluate how this difference affects association study power, we performed genome-wide association of 57 quantitative traits in 1225 overlapping samples with both imputed OmniExome and 1× WGS using both sources of genotype data. We then compared independent suggestively associated signals at < 5×10−7 . These s1× study . With anIn this work, we empirically demonstrate the relative merits of very low-depth WGS both in terms of variant discovery and association study power for complex quantitative traits compared to GWAS approaches. However, the advantages of 1× WGS have to be weighed against compute and financial cost considerations. As of summer 2018, 1× WGS on the HiSeq 4000 platform was approximately half of the cost of a dense GWAS array (e.g. Illumina Infinium Omni 2.5Exome-8 array), the same cost as a sparser chip such as the Illumina HumanCoreExome array, and half of the cost of WES at 50× depth. By comparison, 30× WGS was 23 or 15 times more costly depending on the sequencing platform . The number of variants called by 1× WGS is lower than high-depth WGS, but is in the same order of magnitude, suggesting comparable disk storage requirements for variant calls. However, storage of the reads required an average 650 Mb per sample for CRAMs, and 1.3 Gb per sample for BAMs.Genome-wide refinement and imputation of very low-depth WGS generates close to 50 times more variants than a GWAS chip. The complexity of the imputation and phasing algorithms used in this study is linear in the number of markers, linear in the number of target samples and quadratic in the number of reference samples , which rP-values in a discovery setting, we considered only suggestively associated signals and did not seek replication in a larger cohorts for the discovered signals. In our study of 57 quantitative traits, we show that an 1×-based design allows the discovery of twice as many of the signals suggestively associated in the more accurate 22× WGS study, compared to the imputed GWAS design. Almost 10% of the suggestive signals arising in the 1× data are not discoverable in the imputed GWAS, but the great majority (96%) of imputed GWAS signals is found using the 1×.As a proof of principle, we used imputed GWAS, 1 and 22× WGS in overlapping samples from an isolated population to assess how genotyping quality influences power in association studies. As we only wanted to study the implications of varying genotype qualities afforded by different designs on association P-value in the more accurate 22× based study  and of isolated populations  and imputation . There are ongoing efforts to leverage the specificities of both low-depth sequencing  MANOLIS (Minoan Isolates) collection focuses on Anogia and surrounding Mylopotamos villages on the Greek island of Crete. All individuals were required to have at least one parent from the Mylopotamos area to enter the study. Recruitment was primarily carried out at the village medical centers. The study includes biological sample collection for DNA extraction and lab-based blood measurements, and interview-based questionnaire filling. The phenotypes collected include anthropometric and biometric measurements, clinical evaluation data, biochemical and hematological profiles, self-reported medical history, demographic, socioeconomic and lifestyle information. The study was approved by the Harokopio University Bioethics Committee and informed consent was obtained from every participant.The HELIC  ratio of 1.8 for high-quality tranches . We therFor quality control purposes, reads from 17 of the well-characterized Platinum Genomes sequenced by Illumina at 50× depth . Due to The authors of SHAPEIT (https://github.com/VertebrateResequencing/vr-runner). Pipelines involving Beagle with the use of a reference panel ranked consistently better  for phasing sites, however, the program took more than 2 days to run on 5000 SNV chunks and was abandoned.Halving the number of SNVs per refinement chunk (including 500 flanking positions) from the 4000 recommended by the vr pipelines resulted in only a modest loss of genotype quality in the rare part of the allelic spectrum , while an = 1092), 249 MANOLIS samples sequenced at 4× depth, as well as the two cohorts included in the UK10K study  [TwinsUK ]. Alleles in the reference panel were matched to the reference allele in the called dataset. Positions where the alleles differed between the called and reference datasets were removed from both sources. Indels were filtered out due to poor calling quality.The pipeline with best minor allele concordance across the board used Beagle v.4  to perfoBeagle provides two position level imputation metrics, allelic R-squared and dosage R-squared. Both measures are highly correlated . Values Due to the sparseness of the 1× datasets, sample-level QC was performed after imputation. A total of five samples were excluded from the MANOLIS 1× cohort following PCA-based ethnicity checks.A set of high-confidence genotypes was generated for the five exomes in MANOLIS using filters for variant quality (QUAL > 200), call rate  and depth (250×). These filters were derived from the respective distributions of quality metrics .When compared to five whole-exome sequences from the MANOLIS cohort, imputed 1× calls recapitulated 77.2% of non-monomorphic, high-quality exome sequencing calls. Concordance was high, with only 3.5% of the overlapping positions exhibiting some form of allelic mismatch. When restricting the analysis to singletons, 9105 (58%) of the 15 626 high-quality singletons in the exomes were captured, with 21% of the captured positions exhibiting false-positive genotypes (AC > 1). To assess false-positive call rate, we extracted 1× variants falling within the 71 627 regions targeted by the Agilent design file for WES in overlapping samples, and compared them to those present in the unfiltered WES dataset. A total of 103 717 variants were called in these regions from WES sequences, compared to 58 666 non-monomorphic positions in the 1× calls and 1419 (2.4%) of these positions were unique to the 1× dataset, indicating a low false-positive rate in exonic regions post-imputation.−3. Filtering for MAF increased the inferred kinship coefficients. Generally, the more a variant set was sparse and enriched in common variants, the higher the coefficients were. However, these differences only had a marginal impact on association statistics, as evidenced by a lambda median statistic difference of 0.02 between the two most extreme estimates of relatedness when used for a genome-wide association of triglycerides  R script was used to apply a standardized preparation for all phenotypes. Association analysis was performed using the linear mixed model implemented in GEMMA  were used due to residual inflation being present in some traits. Signals were extracted using the peakplotter software (https://github.com/wtsi-team144/peakplotter) using a window size of 1 Mb.For association, 57 phenotypes were prepared, with full details of the trait transformation, filters and exclusions described in in GEMMA  on all vbty1032_Supplementary_DataClick here for additional data file.


Recent and classical work has revealed biologically and medically significant subtypes in complex diseases and traits. However, relevant subtypes are often unknown, unmeasured, or actively debated, making automated statistical approaches to subtype definition valuable. We propose reverse GWAS (RGWAS) to identify and validate subtypes using genetics and multiple traits: while GWAS seeks the genetic basis of a given trait, RGWAS seeks to define trait subtypes with distinct genetic bases. Unlike existing approaches relying on off-the-shelf clustering methods, RGWAS uses a novel decomposition, MFMR, to model covariates, binary traits, and population structure. We use extensive simulations to show that modelling these features can be crucial for power and calibration. We validate RGWAS in practice by recovering a recently discovered stress subtype in major depression. We then show the utility of RGWAS by identifying three novel subtypes of metabolic traits. We biologically validate these metabolic subtypes with SNP-level tests and a novel polygenic test: the former recover known metabolic GxE SNPs; the latter suggests subtypes may explain substantial missing heritability. Crucially, statins, which are widely prescribed and theorized to increase diabetes risk, have opposing effects on blood glucose across metabolic subtypes, suggesting the subtypes have potential translational value. a priori. We introduce Reverse GWAS to simultaneously split samples into homogeneous subtypes and to learn differences in genetic or treatment effects between subtypes. Unlike existing approaches to computational subtype identification from high-dimensional trait data, RGWAS accounts for covariates, binary disease traits and, especially, population structure–important features of real genetic datasets. We validate RGWAS by recovering known genetic subtypes of major depression. We demonstrate RGWAS can uncover useful novel subtypes in a metabolic dataset, finding three novel subtypes with both SNP- and polygenic-level heterogeneity. Importantly, we show that RGWAS can uncover subtypes with differential treatment response: we show that statin, a common drug and potential type 2 diabetes risk factor, may have opposing subtype-specific effects on blood glucose.Complex diseases depend on interactions between many known and unknown genetic and environmental factors. However, most studies aggregate these strata and test for associations on average across samples, though biological factors and medical interventions can have dramatically different effects on different people. Further, more-sophisticated models are often infeasible because relevant sources of heterogeneity are not generally known  Distinguishing subtypes can be essential for treatment, prognosis, and learning basic disease biology. For example, breast cancer has subtypes distinguished by tumor hormone receptor status that have different genetic risk variants, population structure, comorbidities, treatment responses, and prognoses , 2. Manybiological distinction between the inferred subtypes by testing for genetic effect heterogeneity across subtypes, both at the SNP- and polygenic-levels. We also test effect heterogeneity for non-genetic covariates, like medical interventions, to assess pragmatic distinctions between subtypes [In this work we describe a novel method to learn and validate genetic subtypes in a two-step approach that we call reverse GWAS (RGWAS). In the first step, RGWAS infers subtypes by clustering multiple traits with a finite mixture of regressions method we designed specifically for large, multi-trait GWAS datasets (MFMR). The core assumption of MFMR is that the subtypes differ in distribution for many traits, which creates a subtype structure that can be detected with computational algorithms. In the second step, RGWAS assesses the causal subtypes –21.p-values, polygenic subtype validation tests, and covariate adjustment. These advances substantially reduce both type I and type II errors in computational subtyping.RGWAS offers several important advances in both the identification and testing steps over other recent computational approaches to uncover subtypes –31. FirsWe first evaluate RGWAS through extensive simulations over a wide range of parameter settings and generative models, including several scenarios that violate our assumed model. In comparison to previous methods, as well as several novel methods and obvious extensions, we find that RGWAS is substantially more powerful, more robust, and better calibrated. We then validate RGWAS in real data by recovering a recently discovered stress subtype in MD, as well as known subtype-specific SNP effects.a priori and find strong statistical support for genetic effect heterogeneity across dozens of complex phenotypes: first, a subtype-aware mixed model substantially increases heritability, from 20.7% to 30.2% on average across traits; second, we identify dozens of subtype-specific SNP effects that could not be discovered in standard analyses, including three SNPs with previously identified metabolic interactions; third, subtype-aware GWAS increases the number of hits, from 60 to 70 across traits. Crucially, we find that statin, a widely prescribed drug that may increase diabetes risk [Finally, we apply RGWAS to a metabolic cohort where subtypes are unknown tes risk –34, has K = 2 model and a K = 1 model where no subtypes of any sort are present. For each parameter setting we aggregate results from roughly 300 simulated datasets. We explore several variations to this baseline later.We examine the relative behaviors of RGWAS and other recent methods for computational subtype identification through application to simulated datasets. We simulated from a range of generative models and parameter settings meant to reflect many of the complexities of real data (z. We focus on our Multi-trait Finite Mixture of Regressions (MFMR) approach .For each simulated dataset we first cluster individuals (RGWAS step 1) into a subtype vector approach  and threapproach . First,  interaction test in . Intuitively, subtypes are clusters of covariate-adjusted traits. Variables that confound subtype structure, then, should included as covariates, while variables that may have different distributions between subtypes should be included as traits. Nonetheless, this distinction can be murky in practice, and so we perform simulations where we treat a covariate like a trait or vice versa. MFMR remained calibrated when swapping traits and covariates, unlike GMM, suggesting exact covariate/trait specification is not essential for valid inference with MFMR. However, MFMR did lose power, emphasizing the statistical utility derived from properly modelling the distinction between covariates and traits .Fourth, we studied the impact of SNP-subtype correlation (Supplementary Section 3.2), which is a known source of bias in genetic interaction tests . We founFST = .1 (Supplementary Section 3.3). We repeated our simulations using 12 randomly selected SNPs  and adding population main effects of varying strength. For MFMR and the oracle, we condition on three genetic PCs and their interactions with z in the step 2 tests. MFMR remains calibrated and powerful while CCA and GMM suffer substantial FPR inflation, even for completely null SNPs  of deeply phenotyped (31 binary and 10 quantitative traits) Han Chinese women with recurrent MD and matched controls. Cases were carefully ascertained to minimize environmental heterogeneity, comparatively amplifying signals for biological heterogeneity. RGWAS analysis is also compelling because only a small number of genetic associations with MD have been found to date–consistent with the existence of genetically distinct disease subtypes–and there are few known genetic interactions in complex human traits.The results over the simulated datasets showed that RGWAS was powerful and calibrated across a wide range of parameter settings. To see if RGWAS could perform well in a real setting, we used CONVERGE , a majorK = 2 are summarized in The inferred MFMR subtypes (RGWAS step 1) with p = .038); we also found suggestive polygenic score interaction and suggestive differences in the respective heritability estimates per subtype [We tested five SNPs for effect heterogeneity (RGWAS step 2) across subtypes  with our subtype , furtherK = 2 using prior knowledge that MD can be split by (binary) “Stress”. We assessed this empirically by evaluating the MFMR likelihood on held-out data, which supported K > 1 subtypes  to metabolic traits measured in METSIM . By combK = 3 to compromise between parsimony and the cross-validated log-likelihood, which broadly supports 1 < K ≤ 7 . We first compared the large blue group to the combined orange and green groups, which suggested the blue group had less-esterified cholesterol in small HDL and higher histidine and relative amounts of omega-3 fatty acid. Next, comparing orange to green indicated orange had more free but less esterified overall cholesterol, especially in large LDL, and that orange has higher polyunsaturated fats and phenylalanine.The three inferred metabolic subtypes are summarized in . However, with ∼6,000 samples, we do not have power for genome-wide heterogeneity tests. Instead, we test known metabolic GWAS SNPs–68 from T2D and 13 from CHD . We foun= .05/81 . The oraSUGP1 that affects downstream splicing in the gene targeted by statins, HMGCR [These SNPs have several known metabolic interactions, providing additional evidence that the subtypes are meaningful. rs10401969 is a splice variant for s, HMGCR ; it alsos, HMGCR . rs71388s, HMGCR  and feats, HMGCR . rs78009s, HMGCR , suggests, HMGCR , and is s, HMGCR .p = .05/81  were comparatively modest, with a maximum of 1.33 . Despite this modest inflation for some traits–which can be readily detected, and avoided, by evaluating λGC–RGWAS results are usable, unlike the covariate-unaware results that mimic existing approaches. Similar conclusions hold for the global, K df test.To mimic prior approaches, we performed a covariate-unaware GxE GWAS by using GMM subtypes (in step 1) and by excluding covariates from the heterogeneity tests (in step 2). Genome-wide tests were highly inflated for nite λGC . NotablyIdentification of heterogeneous effects at individual SNPs provides both evidence of differential causal effects between subtypes and the specific loci that distinguish them. To complement these results, we additionally employed a polygenic linear mixed model (LMM) test for genetic subtype heterogeneity (RGWAS step 2). Such polygenic approaches have greater power to detect genetic signal than SNP-level tests, but they do not identify individual causal loci.Briefly, the polygenic model estimates both the ordinary heritability and the subtype-specific heritability, the latter aggregating all subtype-specific SNP effects. For example, when subtypes are two sexes, the subtype-specific heritability aggregates all SNP effects that are active only in one sex . For non-sex linked traits, the resulting sex-specific heritability is zero, hence nonzero subtype-specific heritability demonstrates the existence of subtype-specific genetic effects.We fit the subtype specific LMM with IID GxEMM , which iWe report estimates for binary traits on the observed scale as there is no commonly accepted approach for rescaling heterogeneous heritability estimates to a liability scale. Similarly, we do not report p-values for these binary traits, to be maximally conservative. Nonetheless, we do observe that IID GxEMM increases the estimated heritability for preT2D and CHD, which is consistent with the existence of subtype-specific heritability.p < .01/13, We found significant subtype-specific heritability for 4 of the 13 quantitative traits used in the MFMR decomposition (p < .01/228), dramatically increasing the number of significantly polygenically heterogeneous traits. On average, the subtype-aware heritability is 31.8% and the standard heritability is 17.5% . There is no obvious FPR inflation as statin has no other significantly heterogeneous effects across other traits (p = 2.95 × 10−6). Statin effect heterogeneity is further supported by tests with K = 4 (T2D-adjusted heterogeneity p = 9.67 × 10−7); for K = 2, however, the test is insignificant (p = .39), providing further evidence that K = 2 gives insufficient subtype resolution.While identifying genetic heterogeneity is important for showing that subtypes have differential causal biology, identifying nongenetic sources of subtype heterogeneity can be pragmatically important. We tested for statin effect heterogeneity to assess the ability of the metabolic subtypes to differentiate medical intervention effects. Using our standard interaction test with K = 3 , only glr traits . These rWe next estimated the subtype-specific statin effects with our primary metabolic subtypes (derived treating statin as heterogeneous inside MFMR). We performed a heterogeneous linear regression on glucose conditional on our standard covariates and T2D, which indicated that statin increases blood sugar in most people–consistent with , 33–but p = 0.0036), with orange and green converting less often than blue, which remained after conditioning on our standard covariates (p = .031). This also demonstrates the subtypes persist at least partially over time, in contrast to prior, directly age-dependent T2D subtypes [Since METSIM measured two time points, we next tested the pragmatic ability of our baseline subtypes to predict conversion from preT2D to T2D. We fit logistic regression on time 2 T2D status for the 1,924 baseline prediabetics with time 2 data. Subtypes significantly predicted T2D conversion  Finally, MFMR could be applied only within diseased individuals to directly define subtypes of disease, though this requires fundamentally different step 2 tests .Our polygenic approach to subtype validation with GxEMM provides a much needed power advantage over SNP-level heterogeneity tests at the cost of resolution; conceptually, polygenic risk score tests lie between , 62. ButK = 2 disease-only subtypes [K, on the other hand, StructLMM is a natural complement to GxEMM: the latter is more powerful because it uses genome-wide information and a richer GxE model, but the former has SNP-level resolution and scales to dramatically larger N and K. Similarly, large-K subtypes could be post-processed with hierarchical clustering and tested with TreeWAS [Although we focused on fixed- and random-effect interaction tests to establish heterogeneity between subtypes in RGWAS step 2, it may also be useful to apply recent, complementary heterogeneity tests. For example, Subtest could be used to assess differences between subtypes . For lar TreeWAS . BroadlyIn the future, we will actively encourage MFMR to prioritize tissue-specific subtypes by incorporating tissue-specific genetic risk scores as heterogeneous covariates. This is particularly interesting for traits, including metabolic diseases , 66, thaFinally, as MFMR seeks clusters that are unaffected by confounders like population structure, age, or sex, it may be useful for clustering in settings where protecting certain information is important for privacy or fairness . In thisCONVERGE: The study protocol was approved centrally by the Ethical Review Board of Oxford University  and the ethics committees of all participating hospitals in China. All participants provided written informed consent.METSIM: The Ethics Committee of the University of Eastern Finland and Kuopio University Hospital approved the METSIM study, and this study was conducted in accordance with the Helsinki Declaration. All participants provided written informed consent.Reverse GWAS first clusters samples into subtypes (step 1) and then tests for covariate effect heterogeneity between subtypes (step 2). RGWAS always uses MFMR in step 1. Step 2 always features interaction tests between subtype membership and focal covariates, which may be non-genetic, a SNP, or all SNPs in the genome.y, covariates X, discrete subtypes z, and a focal covariate g putatively interacting with z, our model is:Xi, is a vector of Q control covariates, like genetic PCs or sex, with homogeneous effect sizes α. zi ∈ {1, …, K} is a K-level factor specifying the subtype for individual i, and γk are the subtype main effects. β is the vector of subtype-specific g effect sizes. We say g is homogeneous if β1 = … = βK; otherwise, g is heterogeneous. We assume ϵ is i.i.d. Gaussian with mean zero.We derive a novel clustering algorithm, multitrait finite mixture of regressions (MFMR), beginning from the standard regression model for interaction. Assuming a single quantitative trait z) rather than assume they are known  by assuming zi are i.i.d. Categorical:Our full MFMR model generalizes  in severy to a matrix Y of Multiple traits (MFMR), which adds power for subtypes that affect the distribution of many traits. This power is important in practice because genetic interactions are often weak but phenotypic relationships are often strong. We also generalize the single heterogeneous covariate g to a matrix G of multiple covariates, which adds power when there are many heterogeneous effects.Second, we generalize the single trait k-means. For example, this issue led others to discard roughly half their data post hoc [B, which we address with a novel conditional independence assumption. This induces constraints in our optimization which we solve analytically with block matrix identities (Supplementary Section 2.3).Finally, we model binary traits with probit link functions to mitigate the spurious local modes that plague methods like post hoc . The fulz updates in E-steps and parameter updates (e.g. α and β) in (C)M steps.Computationally, we fit MFMR with an Expectation Conditional-Maximization (ECM) algorithm. Our ECM generalizes standard EM for Gaussian Mixture Models. Both iterate between X , or included in G (MFMR+). In Gaussian mixture models, covariates can only be ignored (GMM) or added as traits (GMM+) [When fitting MFMR in step 1, a covariate that will be tested for heterogeneity in step 2 can either be ignored (MFMRX), included in s (GMM+) . MFMR+ as (GMM+) . For simX are excluded and the covariates G are reduced to an intercept, MFMR becomes GMM. When P = 1 and z is known, MFMR becomes a standard gene-environment interaction (GxE) model with discrete environments/subtypes. Finally, if P = 1 and βk = β0 for all k, MFMR reduces to linear/probit regression.We note that MFMR generalizes several well-known models. If binary traits and K = 2 subtypes and just one interacting covariate, g. These assumptions mean the output from step 1 is just a vector z, where zi is the subtype 1 probability for sample i, and that the interaction model takes a simple form:K > 2 and a matrix G instead of a single covariate g.For simplicity, we assume there are tructure , 29, 31.g: the homogeneity test for δ ≠ 0 given β = 0; the heterogeneity test for β ≠ 0 with free δ; and the global test for δ, β ≠ 0 [K − 1 df, and the global test has K df. We focus on the heterogeneity test, which establishes that g has differential effects across subtypes and thus that the subtypes differ in causal biology (if g is genetic) or pragmatically (e.g. if g is a treatment). We also investigate the global test’s ability to increase power over the typical homogeneous test in the GxE GWAS analysis in ϵ is i.i.d and test with linear or logistic regression.We consider three tests for δ, β ≠ 0 . The homδ and β across all SNPs with random effects. When there are no interaction effects, i.e. β = 0, this gives exactly the standard GREML approach to estimate heritability from genome-wide similarity across unrelated samples [δ, as a small Gaussian variable, and then aggregating the size of each SNP’s δ across the genome to estimate the total genetic contribution to phenotypic variability, i.e. heritability.We also developed a polygenic version of the interaction test in  to joint samples . GREML aβ, a random effect distribution. As the homogeneous mixed model used in GREML aggregates the δ estimates across the genome to estimate homogeneoues heritability, the interaction mixed model aggregates the β estimates along the genome to estimate the heterogeneous heritability explained by subtype-/environment-specific genetic effects. The latter provides an estimate of subtype-specific heritability, which can be combined with the homogeneous heritability estimates to partition broad-sense heritability into shared- and subtype-specific components. This approach was pioneered for unrelated humans in [Polygenic interaction models go further by also giving the interaction effect, umans in , but thiumans in , which aPolygenic interaction tests are essential for genetic subtyping at modest sample sizes because the test for nonzero subtype-specific heritability is much more powerful than testing individual SNPs in complex traits. Polygenic interactions can demonstrate that subtypes have partially distinct genetic bases even when power is too low to discover individual subtype-specific SNP effects.G and the joint binary and quantitative phenotype matrix (Yb: Y), each column-wise centered and scaled, and taking z to be the top phenotypic CC. CCA (and phenotypic PCA) defines z as a linear trait combination. We prove that this causes the interaction tests to have inflated FPR when a mixture of heterogeneous and homogeneous traits are studied (Supplementary Section 4), which is likely in practice. Nonetheless, sparse estimators can resolve this problem in some theoretical settings, and CCA is computationally efficient . We used the 65 CHD SNPs in r2 < .1). We filtered the original 10,070 person dataset so all pairwise kinships were below 0.05, as in [We selected metabolically relevant SNPs by taking published GWAS SNPs for T2D or CHD. We used the 153 T2D SNPs in 5, as in .Yb to {0, 1} in order to retain the downstream logistic regression framework. By contrast, discarding samples with any missing data reduces sample size by roughly half and the known positive SNP interactions were no longer recovered.We imputed missing data before running MFMR in CONVERGE. We jointly imputed covariates and traits with a sample-wise i.i.d. Gaussian model (MVN-impute from ). We thrWe imputed METSIM similarly, including all 228 NMR traits at the imputation step. We used softImpute to accommodate the wide matrix .We note that complete-data analyses performed in similar contexts substantially reduce sample size, e.g.  discard rgwas R package, available with a vignette at https://github.com/andywdahl/rgwas.RGWAS is implemented in the simple, free https://github.com/andywdahl/rgwas-scripts.All summary data and code necessary to reproduce the main and supplementary figures and tables are available at: S1 TextFull description of MFMR model and EM algorithm. Also includes full descriptions of simulations and proofs about PCA/CCA subtype estimators.(PDF)Click here for additional data file.S1 Fig5-fold when N = 100, 000. Accuracies are estimated for roughly 300 simulations per point in the plot. MFMR+ is shown for simplicity because MFMR gives different clusters per tested SNP.Left: Average running times in main (TIF)Click here for additional data file.S2 FigK is the number of true, simulated subtypes and B is the number of binary traits. ρGE is the gene-subtype correlation term, with ρGE = 0 giving non-heritable subtype statuses and ρGE = 1 giving perfectly heritable subtypes. h2 term not governed by the x-axis. In (a), all methods fit K = 2 subtypes; there is no true heterogeneity for K = 1, where the oracle is not defined, and for K > 1 and the oracle picks a true cluster at random. Generally, increasing the heterogeneous factors (Tests for truly heterogeneous SNPs are shown in the top 6 panels (a-f), and the corresponding tests for SNPs with only homogeneoues effects are shown in the below 6 panels (g-l). (TIF)Click here for additional data file.S3 FigY PC) are not trait-specific.Left: the tested trait has no genetic heterogeneity or main subtype effect. Center: the tested trait has only a main subtype effect but no heterogeneity. Right: the full heterogeneity simulation. Linear subtype estimators (CCA and (TIF)Click here for additional data file.S4 FigN ≥ 10, 000 . The other methods, with low power, define subtypes as the top PC of Y or G, optionally thresholded to be binary (“G PC+disc”).MFMR+ varies MFMR by treating the tested SNP as heterogeneous. GMM+ varies GMM by including the SNPs as traits when clustering. As expected, MFMR+ and GMM+ are miscalibrated. GMM+ often fails to converge, especially for (TIF)Click here for additional data file.S5 Figϵ, has marginal t5 distributions. ϵ is simulated by drawing i.i.d. t5-distributed random variables, arranging into an N × P matrix, and then right-multiplying with Σ1/2, where Σ is the noise covariance matrix and is drawn as in the main simulations in main Purely homogeneoues simulations, without subtypes, where the noise, (TIF)Click here for additional data file.S6 Fig2) have zero correlation. In (e-h), the true SNPs are exponentiated before inclusion in MFMR, so the true SNP effect is log-linear. Results are partitioned by whether the tested traits are quantitative or binary, as well as by whether the true SNP effect is null or homogeneous.Purely homogeneoues simulations, without subtypes, where SNPs truly have a non-linear effect. In (a-d), the SNPs are squared before use in MFMR, so that the true SNP and the utilized covariate (i.e. SNP(TIF)Click here for additional data file.S7 Figz is chosen to be Gaussian. Top: Effect sizes are chosen so that power roughly matches main z simulations. Bottom: All heterogeneoues effect sizes are doubled relative to top panels.(TIF)Click here for additional data file.S8 FigTop: SNP effect heterogeneity tests are applied to binary traits, not quantitative traits as in main (TIF)Click here for additional data file.S9 FigK = 2. (e-h) Tests are shown for the misplaced trait/covariate in ; for the truly homogeneous SNPs in ; and for the truly heterogeneous SNPs in .Simulation modification where decompositions falsely treat a trait as a SNP/covariate  or vice versa . (a-d) No genetic or main subtype heterogeneity is simulated, so that the positive heterogeneity associations are unambiguously false. We test both the variable that we misplace  and the correctly place SNP/covariates and traits . (e-j) Simulations are drawn as in main text (TIF)Click here for additional data file.S10 FigK = 1; this is analogous to using likelihood ratio statistics to compare a general K to the null with K = 1. The average across folds are shown in red, and the maximizer of K is highlighted in green.Samples are split into 5 folds; parameters are fit holding one fold out; the parameters’s likelihood is evaluated on the held out fold; and the process is repeated for each fold. The log-likelihoods are shown relative to the baseline likelihood of each fold at (TIF)Click here for additional data file.S11 Figp = .05/81. We test all 228 NMR-based metabolomic traits here rather than using their top PCs as in main SNP-phenotype pairs where the test for effect heterogeneity across subtypes is significant at (TIF)Click here for additional data file.S12 Figp = 5 × 10−8, the conventional GWAS threshold. Each point is a SNP, and colors indicate which analyses were significant for the SNP. T2D, CHD, WHR and insulin are omitted because they have no genome-wide significant hits in either analysis; preT2D is omitted because the only hit is shared between both analyses. NMR PC 5 is omitted because it is badly inflated in GxE GWAS (λGC = 1.83); this trait has one hit in GWAS.Guide lines are drawn at (TIF)Click here for additional data file.S13 FigLeft: total IID GxEMM heritability (which adds the homogeneous and heterogeneous estimates) compared to the ordinary heritability estimated with GREML. Right: histogram of per-trait heritability increases from replacing GREML with IID GxEMM.(TIF)Click here for additional data file.S1 TableGC is very large. We do not perform SNP tests on NMR PC 5.GWAS means the standard regression approach conditioning on known covariates and genetic PCs. RGWAS is our approach, which uses covariate-aware clusters (MFMR) and tests for genetic variant effect heterogeneity (Het) or globally for any genetic effect . “Previous Subtyping” is like RGWAS, except using covariate-unaware clustering (GMM) and heterogeneity tests. “Inf” means our calculations suffered numerical problems, meaning that λ(CSV)Click here for additional data file.S2 Tabledroptest” columns test using the large-effect covariate test implemented in the rgwas R package and described in the Methods in the main text. The “+Condition on T2D” columns run the same linear model as in droptest, except that T2D status is additionally included as a covariate; this analysis is performed to add confidence that the heterogeneous statin effect on glucose is not merely driven by simple confounding from T2D status.The “MFMR (CSV)Click here for additional data file.


We replicate an association between locomotor activity and Csmd1, which we identified in an earlier generation of this AIL, and show that Csmd1 mutant mice recapitulate the locomotor phenotype. Our results demonstrate the utility of the LG x SM AIL as a mapping population, identify numerous novel associations, and shed light on the genetic architecture of mammalian behavior.The LG/J x SM/J advanced intercross line of mice (LG x SM AIL) is a multigenerational outbred population. High minor allele frequencies, a simple genetic background, and the fully sequenced LG and SM genomes make it a powerful population for genome-wide association studies. Here we use 1,063 AIL mice to identify 126 significant associations for 50 traits relevant to human health and disease. We also identify thousands of  Multigenerational outbred populations from an advanced intercross line (AIL) of mice represent useful resources for genome wide association analysis. Here, the authors analyze 1,063 LG x SM AIL mice to identify significant associations for 50 traits relevant to human health and disease. Because the majority of associated loci are in noncoding regions, expression quantitative trait loci (eQTLs) are useful for elucidating underlying molecular mechanisms4. However, it remains challenging to obtain large, high-quality samples of human tissue, particularly from the brain. In contrast, tissue for gene expression studies can be collected from model organisms under optimal conditions. Finally, the genomes of model organisms can be edited to assess the functional consequences of specific mutations.Genome-wide association studies (GWAS) have revolutionized psychiatric genetics; however, they have also presented numerous challenges. Some of these challenges can be addressed by using model organisms. For example, human GWAS are confounded by environmental variables, such as childhood trauma, which can reduce power to detect genetic associations. In model organisms, environmental variables can be carefully controlled. Furthermore, it has become clear that phenotypic variation in humans is due to numerous common and rare variants of small effect. In model organisms, genetic diversity can be controlled such that all variants are common. In addition, allelic effect sizes in model organisms are dramatically larger than in humans2 crosses, outbred animals provide improved mapping resolution for GWAS. This is because outbred populations have higher levels of recombination, meaning only markers very close to the causal allele will be associated with the phenotype. However, there is a necessary tradeoff between mapping resolution and statistical power, and this is further aggravated when the causal allele is rare, which is sometimes the case in commercially outbred mice6. In an effort to combine the resolution afforded by an outbred population with the power of an F2 cross, we performed GWAS using an advanced intercross line (AIL) of mice. Originally proposed by Darvasi and Soller7, AILs are the simplest possible outbred population; they are produced by intercrossing two inbred strains beyond the F2 generation. Because each inbred strain contributes equally to an AIL, all variants are common. This avoids the loss of power that results from rare alleles and simplifies phasing and imputation. Each successive generation of intercrossing further degrades linkage disequilibrium (LD) between adjacent markers, which improves mapping resolution relative to classical inbred crosses such as F2s.Unlike classical F8. In 2006, we established an independent AIL colony with mice from generation 33 at the University of Chicago . Because the LG and SM founder strains were independently selected for large and small body size, this AIL has frequently been used to study body weight, musculoskeletal, and other metabolic traits12. LG and SM also exhibit a variety of behavioral differences, and previous studies from our lab have used the LG  × SM AIL to fine-map behavioral associations identified in F2 crosses between LG and SM15.An AIL derived from the LG/J (LG) and SM/J (SM) inbred strains was initiated by Dr. James Cheverud at Washington University in St. Louis5. In this paper, we apply a similar approach to AIL mice, taking advantage of the more favorable allele frequencies and the ability to impute founder haplotypes. Using mice from AIL generations 50–56 , we investigate novel behavioral traits, including conditioned place preference for methamphetamine (CPP), as well as other biomedically important traits including locomotor activity following vehicle and methamphetamine administration, which have been extensively studied for their relevance to drug abuse16, and prepulse inhibition (PPI), which has been studied for several decades as an endophenotype for schizophrenia17. We also examine body weight and a constellation of musculoskeletal phenotypes relevant to exercise physiology, which are known to be heritable in this AIL18. In total, we report 52 associations for 33 behavioral traits and 74 associations for 17 physiological traits. We use RNA-sequencing (RNA-seq) to measure gene expression in three brain regions of ~200 mice and identify thousands of cis- and trans-eQTLs, which we use to identify quantitative trait genes (QTGs). Finally, we use a mutant mouse line to validate one of our strongest candidate QTGs. Our work demonstrates that the LG × SM AIL is a powerful tool for genetic analysis and provides a methodological framework for GWAS in multigenerational outbred populations. The numerous associations described in this work benefit the complex trait community by shedding light on the genetic architecture of complex traits in a system with only two founders. Although we present only a subset of our findings in detail, AIL genotype, phenotype and gene expression data, complete with a suite of tools for performing GWAS and related analyses, are publicly available on GeneNetwork.org19.We recently used commercially available outbred CFW mice to perform a GWAS20 (GigaMUGA), which yielded only 24,934 polymorphic markers  to genotype 1063 of the 1123 mice that were phenotyped  to address this issue23. To demonstrate the appropriateness of a LOCO-LMM, we performed a GWAS for albinism, which is known to be a recessive Mendelian trait caused by the Tyrc allele, using all three approaches: a simple linear model, an LMM, and a LOCO-LMM  are now commonly used to perform GWAS in populations that include close relatives because they can incorporate a genetic relationship matrix (GRM) that models the covariance of genotypes and phenotypes in samples of related individualsele Fig. . As expeted Fig. . This inted Fig. . The LOCted Fig. . Was theted Fig. . Even inted Fig. . The stated Fig. . The LOCect Fig. . These rp = 8.06 × 10−6 at α = 0.05 . We did not use a Bonferroni-corrected significance threshold because we have previously shown that permutation adequately controls type I and II error rates25. We identified 52 loci associated with 33 behavioral traits and 74 loci associated with 17 physiological traits  estimates may differ depending on whether pedigree data or genetic data is used to calculate the GRM26. Pedigree data captures shared environmental effects and epigenetic effects in addition to additive genetic effects. We used GEMMA27 to estimate h2 using a GRM calculated from the AIL pedigree 2 estimates obtained using a GRM calculated from 523,028 SNPs 2 for physiological traits than for behavioral traits , mean 2 than 2 estimates and their standard errors are reported in Supplementary Data In populations of related individuals, heritability h estimateWe expected many of the traits measured in this study to be correlated with one another (as explained in the Methods and n = 208), prefrontal cortex  and striatum  , 626 PFC trans-eQTLs (p < 1.04 × 10−5) and 653 STR trans-eQTLs (p < 8.68 × 10−6) at a genome-wide significance threshold of α = 0.05 after testing 49,642 genes across the three tissues , which we refer to as trans-eQTLs that regulate the expression of many genes30; we refer to these as master eQTLs . We identified several master eQTLs, including one on chromosome 12 (70.19–73.72 Mb) that was associated with the expression of 97 genes distributed throughout the genome . Other genome-wide significant associations included total D1 side changes (0–30 min), distance traveled on D1 (0–5 min), and distance traveled after an injection of methamphetamine on D2 , which had a significant cis-eQTL in HIP. Although Crim1 may appear to be the best candidate to explain the associations with locomotor behavior, two nearby genes, Qpct (glutaminyl-peptide cyclotransferase) and Vit (vitrin), though physically located outside of the locus, also had cis-eQTLs within the region associated with locomotor behavior  across all traits that we measured in this study , which revealed that this region pleiotropically affected multiple other traits, including locomotor activity following saline and methamphetamine administration  is not clear, and we are not certain whether the pleiotropic effects are due to one or several genes in this interval. In this case, the mapping resolution we obtained was insufficient to address these questions.One of the most significant loci we identified was associated with startle response  spanned 2.31 Mb and completely encompassed the narrowest locus, which spanned 0.74 Mb ; therefore, we focused on the smaller region. Oprd1 (opioid receptor delta 1) had a cis-eQTL in all three brain regions; the SM allele conferred an increase in locomotor activity and was associated with decreased expression of Oprd1. Oprd1 knockout mice have been reported to display increased activity relative to wild-type mice33, suggesting that differential expression of Oprd1 could explain the locomotor effect at this locus. However, we note that there are several other genes and eQTLs within this locus that could also contribute to its behavioral effects.We identified seven overlapping loci for locomotor activity on chromosome 4  was most strongly associated with rs108610974, and may be the QTG  in PFC. Mice lacking Nr4a2 in midbrain dopamine neurons exhibit a 40% reduction in body weight34. Consistent with this, the LG allele was associated with decreased expression of Nr4a2. Taken together, these data strongly implicate Nr4a2 as the QTG for this locus.Because LG and SM were created by selective breeding for large and small body size, this AIL is expected to segregate numerous body size alleles35. Therefore, we measured the weight of five hindlimb muscles, tibia length, and tail length (Methods). We identified 22 associations for muscle weight and five for bone length (Supplementary Data Tpp1 (tripeptidyl peptidase 1), which was a cis-eQTL gene in all tissues and a trans-eQTL gene targeted by the master HIP eQTL on chromosome 12 37. Dysfunction of Tub (tubby bipartite transcription factor) causes late-onset obesity in mice, perhaps due to Tub’s role in insulin signaling38. In addition, several trans-eQTL genes map to this interval, including Gnb1 (G protein subunit beta 1), which forms a complex with Tub39. Another trans-eQTL gene associated with this interval, Crebbp (CREB binding protein), has been associated with juvenile obesity in human GWAS34.LG and SM also exhibit differences in fat and muscle weight, bone density, and other morphometric traitsp = 2.03 × 10−13) on chromosome 13 , is differentially expressed in LG and SM soleus muscle18, with LG exhibiting greater expression. Nln has been shown to play a role in mouse skeletal muscle40.The strongest association we identified in this study was for EDL weight  and plantaris  were located in an intron of Astn2 (astrotactin 2), which is differentially expressed in LG and SM soleus12. SM, which exhibits lower expression of Astn2 in soleus relative to LG12, has a 16 bp insertion in an enhancer region 6.6 kb upstream of Astn2 (ENSMUSR00000192783)21. Two other genes in this region have been associated with muscle or bone phenotypes traits in the mouse: Tlr4 (toll-like receptor 4), which harbors one synonymous coding mutation on the SM background (rs13489095) and Trim32 (tripartite motif-containing 32), which contains no coding polymorphisms between the LG and SM strains.Finally, we identified an association with EDL, plantaris, and TA weight at another locus on chromosome 4 , a smaller sample size, and a higher proportion of first-degree relatives18. In the present study, we addressed these limitations. The number of SNPs used in our prior studies was increased by several orders of magnitude by using GBS42 followed by imputation from the sequenced founders21. Unlike our previous studies using the G34 AIL, in this study we did not use AIL mice to fine map loci identified in an F2 cross, but instead used them as both a discovery and fine mapping population.Our previous work with this AIL used mice from LG × SM G34 to fine-map loci identified in an F6. Although our approach to the AIL was similar to the CFW studies, certain practical considerations, along with the AIL’s simpler genetic background, affected the design of this study and its outcomes in important ways. For example, non-sibling CFW mice can be obtained from a commercial vendor, which avoids the expenses of maintaining an AIL colony and reduces the complicating effects that can occur when close relatives are used in GWAS. However, haplotype data from the CFW founders are not available, and many CFW alleles exist at low frequencies, limiting power and introducing genetic noise6. In contrast, the LG and SM founder strains have been fully sequenced21 and AIL SNPs have high MAFs, which simplifies imputation of SNPs and founder haplotypes and enhances statistical power22. Parker et al. demonstrated that GBS is a cost-effective strategy for genotyping CFW mice5; however, in the present study, we took advantage of the fact that in the LG × SM AIL, all alleles that are identical by state are also identical by descent. This allowed us to use imputation to obtain 4.3 million SNPs despite using only about half the sequencing depth that was necessary for CFW mice 46. We conclude that the low heritability of CPP likely reflects a lack of relevant genetic variation in this specific population. It is possible that even lower doses of methamphetamine, which might fall on the ascending portion of the dose-response function, would have resulted in higher heritability. Responses to other drugs or different CPP methodology may also have resulted in higher heritability. It is also possible that testing for non-additive genetic effects would have increased the heritability of CPP.The primary goal of this study was to identify the genes that are responsible for the loci implicated in behavioral and physiological traits. We were particularly interested in uncovering genetic factors that influence CPP, which is a well-validated measure of the reinforcing effects of drugsero Fig. . This wa49. This was expected because environmental effects can be shared within but not between inbred strains, which will produce higher estimates of heritability26. In addition, epistatic interactions can contribute to heritability in inbred panels, which leads to higher estimates of heritability50. These observations share many similarities with the higher heritabilities obtained using twins compared to chip heritabilities in human studies26.In general, trait heritability in the LG × SM AIL Fig.  was lowe5. Despite this, we identified many more genome-wide significant loci in the current study (126 associations in the AIL compared to 17 in CFW at α = 0.05). The greater number of significant associations generated in this study may be due to enhanced power that comes with the more favorable distribution of MAFs in the AIL . Heritability estimates were very similar between those two populations4. Therefore, we used RNA-seq to quantify mRNA abundance in three brain regions that are strongly implicated in the behavioral traits that we measured: HIP, PFC, and STR. We used these data to identify 7081 cis-eQTLs (FDR < 0.05) and 2002 trans-eQTLs (α = 0.05 for a single gene test) , genetic diversity in an AIL is limited to variants segregating in the founder strains. Still, the proportion of eQTLs that were significant in all three brain regions was smaller for trans-eQTLs versus cis-eQTLs  versus testing only nearby SNPs (cis-eQTLs). Thus, in addition to integrating QTLs with eQTLs, we incorporated data about gene expression in other tissues18, coding SNPs, mutant mice, and human genetic studies to parse among the implicated genes. Although that was broadly similar to the approach we applied to CFW mice, the eQTL sample in this study was more than two times greater. Indeed, to our knowledge, this is the largest eQTL analysis that has been performed in outbred mice.Our ability to identify QTGs was critically dependent on mapping resolution in the LG × SM AIL Fig. . HoweverFDR < 0.0 and 200215, contextual and conditioned fear15, and startle response14 in previous studies of LG x SM G34, suggesting that the locus has a broad impact on many behavioral traits. These results support the idea that pleiotropy is a pervasive feature in model organisms and provides further evidence of the replicability of the loci identified by this and prior GWAS.We also used PheWAS to identify pleiotropic effects of several loci identified in this study. In many cases, pleiotropy involved highly correlated traits such as body weight on different days or behavior at different time points within a single day  were independently selected for high and low body weight at 60 days53. The LG × SM AIL was derived from an F1 intercross of SM females and LG males initiated by Dr. James Cheverud at Washington University in St. Louis8. Subsequent AIL generations were maintained using at least 65 breeder pairs selected by pseudo-random mating54. In 2006, we established an independent AIL colony using 140 G33 mice obtained from Dr. Cheverud . Since 2009, we have selected breeders using an R script that leverages pairwise kinship coefficients estimated from the AIL pedigree to select the most unrelated pairs while also attempting to minimize mean kinship among individuals in the incipient generation  to a four-week battery of tests over the course of two years. 1063 of these mice  had high-quality GBS data; therefore, we restricted our analysis to these individuals. This sample size was based on an analysis suggesting that 1000 mice would provide 80% power to detect associations explaining 3% of the phenotypic variance  on D1. On D2 and D4, mice were administered methamphetamine  and restricted to one visually and tactically distinct environment; on D3 and D5 mice were administered vehicle and restricted to the other, contrasting environment. No testing occurs on D6 and D7. On D8, mice were allowed to choose between the two environments after administration of vehicle; we measured CPP at this time. Other variables measured during the CPP test include the distance traveled (cm) on all testing days, the number of side changes on D1 and D8, and locomotor sensitization to methamphetamine (the increase in activity on D4 relative to D2). We measured CPP and locomotor traits across six five-minute intervals and summed them to generate a total phenotype for each day.CPP and locomotor behavior: CPP is an associative learning paradigm that has been used to measure the motivational properties of drugs in humans56. PPI and startle are measured across multiple trials that occur over four consecutive blocks of time14. The primary startle trait is the mean startle amplitude across all pulse-alone trials in blocks 1–4. Habituation to startle is the difference between the mean startle response at the start of the test (block 1) and the end of the test (block 4). PPI, which we measured at three prepulse intensities , is the mean startle response during pulse-alone trials in blocks 2–3 normalized by the mean startle response during prepulse trials in blocks 2–3. Mice that exhibited a startle response in the absence of a pulse were excluded from GWAS, as were mice that did not exhibit a startle response during the first block of startle pulses  prepulsePhysiological traits: We measured body weight (g) on each testing day and at the time of death. One week after PPI, we measured blood glucose levels (mg/dL) after a four-hour fast. One week after glucose testing, we killed the mice, and measured tail length (cm from base to tip of the tail). We stored spleens in a 1.5 mL solution of 0.9% saline at −80 °C until DNA extraction. We removed the left hind limb of each mouse just below the pelvis; hind limbs were stored at −80 °C. Frozen hind limbs were phenotyped in the laboratory of Dr. Arimantas Lionikas at the University of Aberdeen. Phenotyped muscles include two dorsiflexors, TA and EDL, and three plantar flexors: gastrocnemius, plantaris, and soleus. We isolated individual muscles under a dissection microscope and weighed them to 0.1 mg precision on a Pioneer balance . After removing soft tissue from the length of tibia, we measured its length to 0.01 mm precision with a Z22855 digital caliper .Brain tissue: We collected HIP, PFC, and STR for RNA-seq from the brain of one mouse per cage. This allowed us to dissect each brain within five minutes of removing a cage from the colony room (rapid tissue collection was intended to limit stress-induced changes in gene expression). We preselected brain donors to prevent biased sampling of docile (easily caught) mice and to avoid sampling full siblings, which would reduce our power to detect eQTLs. Intact brains were extracted and submerged in chilled RNALater  for one minute before dissection. Individual tissues were stored separately in chilled 0.5-mL tubes of RNALater. All brain tissue was dissected by the same experimenter and subsequently stored at −80 °C until extraction.57 that we have adapted for use in mice and rats42. We extracted DNA from spleen using a standard salting-out protocol and prepared GBS libraries by digesting DNA with the restriction enzyme PstI. We sequenced 24 uniquely barcoded samples per lane of an Illumina HiSeq 2500 using single-end, 100 bp reads. We aligned 1110 GBS libraries to the mm10 reference genome before using GATK58 to realign reads around known indels in LG and SM21 (Supplementary Methods). We obtained an average of 3.2 million reads per sample. We discarded 32 samples with <1 million reads aligned to the main chromosome contigs  or with a primary alignment rate <77% , SNPs with Hardy–Weinberg Equilibrium (HWE) violations , determined from gene-dropping simulations as described in the Supplementary Methods), and SNPs with low imputation quality . We then pruned variants in high LD (r2 > 0.95) to obtain the 523,028 SNPs that we used for GWAS.After identifying and correcting sample mix-ups, we discarded 15 samples whose identities could not be resolved (Supplementary Methods). Next, we used Beagle65 to calculate r2 for all pairs of autosomal GWAS SNPs typed in G50–56 . Power to detect associations is reduced when the locus being tested is also included in the GRM because the effect of the locus is represented in both the fixed and random terms24. To address this issue, we calculated 19 separate GRMs, each one excluding a different chromosome. When testing SNPs on a given chromosome, we used the GRM that did not include markers from that chromosome as the polygenic effect in the model. Fixed covariates for each trait are listed in Supplementary Data We used a leave one chromosome out LMM (LOCO-LMM) implemented in GEMMA66 and SLIDE67 to obtain a genome-wide significance threshold that accounts for LD between nearby markers (Supplementary Methods). We obtained a significance threshold of p = 8.06 × 10−6 (α = 0.05) from 2.5 million samplings. Because the phenotypic data were quantile-normalized, we applied the same threshold to all traits. We converted p-values to LOD scores and used a 1.5-LOD support interval to approximate a critical region around each associated region, which enabled us to systematically identify overlap with eQTLs.We used a permutation-based approach implemented in MultiTransp-values for all pairs of traits measured in this study. Only mice with non-missing data for both traits in a pair were used to calculate the correlation. We also decomposed the trait covariance into genetic and environmental covariances using a multivariate LOCO-LMM in GEMMA68. This allowed us to evaluate the contribution of genetic and environmental factors to each pair of traits . We used Quant-iT kits to quantify RNA (Ribogreen) and cDNA . Barcoded sequencing libraries were prepared with the TruSeq RNA Kit , pooled in sets of 24, and sequenced on two lanes of an Illumina HiSeq 2500 using 100 bp, single-end reads.We extracted RNA from HIP, PFC, and STR using a standard phenol-chloroform procedure and prepared cDNA libraries from 741 samples with RNA integrity scores ≥8.0 70, read mapping in an AIL may be biased toward the reference strain (C57BL/6J)71. We addressed this concern by aligning RNA-seq reads to custom genomes created from LG and SM using whole-genome sequence data21. We used default parameters in HISAT72 for alignment and GenomicAlignments73 for assembly, assigning each read to a gene as defined by Ensembl (Mus_musculus.GRCm38.85)74. We required that each read overlap one unique disjoint region of the gene. If a read contained a region overlapping multiple genes, genes were split into disjoint intervals, and any shared regions between them were hidden. If the read overlapped one of the remaining intervals, it was assigned to the gene that the interval originated from; otherwise, it was discarded. Next, we reassigned the mapping position and CIGAR strings for each read to match mm10 genome coordinates and combined the LG and SM alignment files for each sample by choosing the best mapping. Only uniquely mapped reads were included in the final alignment files. We then used DESeq75 to obtain normalized read counts for each gene in HIP, PFC, and STR. We excluded genes detected in <95% of samples within each tissue. We retained a total of 16,533 genes in HIP, 16,249 genes in PFC, and 16,860 genes in STR.Because mapping quality tends to be higher for reads that closely match the reference genomeWe also excluded 30 samples with <5 M mapped reads or with an alignment rate <91.48% . We used all SNPs for permutation; therefore, we expect these thresholds to be slightly conservative. We define master eQTLs as 5 Mb regions that contain ten or more trans-eQTLs. To identify master eQTLs, we divided chromosomes into 5 Mb bins and assigned each trans-eQTL gene to the bin containing its most significant eQTL SNP.SNPs on chromosomes that did not contain the gene being tested were considered potential Csmd1 mutants were created by Lexicon Genetics by inserting a Neomycin cassette into the first exon of Csmd1 using embryonic stem cells derived from 129S5 mice78. The mice we used were the result of a C57BL/6×129S5 intercross designated B6;129S5-Csmd1tm1Lex/Mmucd (the exact C57BL/6 substrain is unknown). We bred heterozygous males and females and tested littermate offspring to account for their mixed genetic background. Csmd1 spans 1.6 Mb and has 70 exons. Its four major transcripts, termed Csmd1-1 to Csmd1-4, are expressed in the central nervous system51. Distler et al. (ref. Csmd1 homozygous mutant mice express <30% of wild-type Csmd1 levels in the brain, and heterozygous mice show a 54% reduction in Csmd1 expression. Residual expression of Csmd1 in homozygous mutant mice is derived from Csmd1-4, the only transcript that does not include the first exon. We analyzed locomotor behavior on two days following a saline injection in 31 wild-type, 59 heterozygous, and 48 mutant mice.Example code for running the analyses in this study are provided in Supplementary Note Supplementary InformationPeer Review FileDescription of Additional Supplementary FilesSupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Data 5Supplementary Data 6Source Data


Grid-LMM (https://github.com/deruncie/GridLMM), an extendable algorithm for repeatedly fitting complex linear models that account for multiple sources of heterogeneity, such as additive and non-additive genetic variance, spatial heterogeneity, and genotype-environment interactions. Grid-LMM can compute approximate (yet highly accurate) frequentist test statistics or Bayesian posterior summaries at a genome-wide scale in a fraction of the time compared to existing general-purpose methods. We apply Grid-LMM to two types of quantitative genetic analyses. The first is focused on accounting for spatial variability and non-additive genetic variance while scanning for QTL; and the second aims to identify gene expression traits affected by non-additive genetic variation. In both cases, modeling multiple sources of heterogeneity leads to new discoveries.Linear mixed effect models are powerful tools used to account for population structure in genome-wide association studies (GWASs) and estimate the genetic architecture of complex traits. However, fully-specified models are computationally demanding and common simplifications often lead to reduced power or biased inference. We describe  The goal of quantitative genetics is to characterize the relationship between genetic variation and variation in quantitative traits such as height, productivity, or disease susceptibility. A statistical method known as the linear mixed effect model has been critical to the development of quantitative genetics. First applied to animal breeding, this model now forms the basis of a wide-range of modern genomic analyses including genome-wide associations, polygenic modeling, and genomic prediction. The same model is also widely used in ecology, evolutionary genetics, social sciences, and many other fields. Mixed models are frequently multi-faceted, which is necessary for accurately modeling data that is generated from complex experimental designs. However, most genomic applications use only the simplest form of linear mixed methods because the computational demands for model fitting can be too great. We develop a flexible approach for fitting linear mixed models to genome scale data that greatly reduces their computational burden and provides flexibility for users to choose the best statistical paradigm for their data analysis. We demonstrate improved accuracy for genetic association tests, increased power to discover causal genetic variants, and the ability to provide accurate summaries of model uncertainty using both simulated and real data examples. Population stratification, genetic relatedness, ascertainment, and other sources of heterogeneity lead to spurious signals and reduced power in genetic association studies –5. When Random effect terms are used in LMMs to account for specific correlations among observations. Fitting an LMM requires estimating the importance of each random effect, called its variance component. General-purpose tools for this are too slow to be practical for genome-scale datasets with thousands of observations and millions of genetic markers . This laTo this end, several specialized approaches have been developed to improve the speed of LMMs, including the use of sparse matrix operations , 21, speGrid-LMM takes a different approach to fitting LMMs: rather than directly optimizing the variance components separately for each test, we define a grid spanning all valid values of the variance components and fit simple linear models at each grid location. Each evaluation involves a single Cholesky decomposition of the random effect covariance matrix, which is then reused to calculate closed-form ML solutions or Bayesian posterior summaries . Since the variance components must be non-negative, their proportions are restricted to the unit interval  and sum to 1, forming a simplex. Therefore, a finite grid can span all valid parameter values. While the size of the grid increases rapidly with the number of random effects, for a small number of random effects (∼1-8) and a moderate grid resolution, the size of the grid remains tiny relative to the number of models in a typical GWAS. As we show below, highly accurate test statistics are achieved even with a coarse grid in most reasonable situations, and further improvements to efficiency are possible by using heuristics to adaptively sample the grid or reduce the number of grid locations computed for the majority of tests. This strategy of conditioning on variance components over a grid and then combining solutions can be applied to many other tools in quantitative genetics including set tests for rare variants ⊤ and an arbitrary q × (c + p) matrix M uses the general F-statistic:q and (n − c − p) degrees of freedom, where θ using the REML estimate of V ⊤, σ2) ~ NIG, and then integrate over the prior to calculate the marginal likelihood of the data given h2. This results in the following:a* = a0 + n/2, and b* = b0 + RSSy*,Ψ/2 with RSSy*,Ψ having the same form as RSSy* in p(y | h2) at each vertex of the grid as described above. Assuming a discrete-valued prior p(h2), we can then compute the posterior distribution of h2 as:h2 are denoted with a subscript. Continuous target densities π(h2) can be approximated as p(h2) by assigning each grid vertex a probability equal to the integral of π(h2) over the L-dimensional rectangle centered at the corresponding value h2. We assume a uniform prior over our grid for all analyses presented in the main text. Bayes factors are computed by comparing models under the alternative and null hypotheses as the ratios in α and σ2 can be used, which results in scale-independence [For Bayesian inference, rather than working with the profile likelihoods, we instead use a conditional normal-inverse-gamma prior ([pendence .g < p) and can easily be parallelized across multiple computers. However, g grows rapidly as the grid resolution and number of random effects increases:m divisions per The full grid search described above is dramatically faster than the naive algorithm for mixed-model GWAS analyses, as long as the vertices of the grid is less than the number of genetic markers . We calculate the scores for each test at each vertex of the grid, and then compare the maximum scores to the scores at p2 markers, we select the set g2 around all of them, dropping vertices already tested. This procedure is repeated t7 times until the new grid no-longer increases scores for any test and we are confident that all (approximate) maximums have been found. This accelerated search has total complexity t6 the number of iterations needed to optimize variance components under the null model, and p1 = p  over an initial grid of resolution 1/m1 with discrete prior m2 = 2m1 and a new prior m2 distance from any mi+1 = 2mi and repeat the grid-filling steps again. We note that this procedure is only appropriate if the posterior is convex and, therefore, is limited to the case of uniform priors on h2. A similar procedure was proposed for Bayesian inference in Gaussian process models in the GPstuff toolbox, although it is not optimized for parallel inference in GWAS [Analogously, to accelerate evaluations of posterior distributions, we evaluate  in GWAS .m that is sufficiently fine such that we expect each posterior to be distributed across multiple vertices. We then calculate p(y | h2) for each test, starting from p(y | h2) for that marker. As for the ML and REML solutions above, markers with little to no association with the phenotype will lead to posteriors of h2 that are concentrated close to p(h2 | y) strongly to new regions of the grid.To accelerate evaluations of GWAS Bayes Factors, we combine the two previously described algorithms. In particular, we define a grid centered on Grid-LMM analyses presented here with grid step sizes of 0:01 Unless otherwise specified, we used accelerated grid searches for all Arabidopsis thaliana traits and 216,130 genetic markers were downloaded from https://github.com/Gregor-Mendel-Institute/atpolydb/wiki. We follow practices suggested by the original authors of these data and log-transformed a subset of the phenotypes prior to analyses  [sommer package in R to calculate an additive relationship matrix from all 216,130 markers [DZKZ⊤D where K is the 175 × 175 additive genomic relationship matrix, Z is a 350 × 175 incidence matrix linking observations to accessions, and D is a 350 × 350 diagonal matrix with elements equal to −1 or 1 corresponding to observations measured under “7_FT10” and “57_FT Field”, respectively. Plasticities for each accession were calculated as the difference between “57_FT Field” and “7_FT10”. We ran GEMMA (version 0.97) with the “-lmm1” option for Wald tests using the K matrix described above. We emulated EMMAX and pylmm functionality by estimating variance components using a null model with no marker effects, and either a single K matrix  or 3 random effects . Wald test statistics were computed for each marker using our GridLMM R function with a grid consisting of only a single vertex. For G×E analyses, we fit a model with a main effect of the environment, a main effect for each marker of interest, and an interaction between the marker and the environment—and then calculated the Wald-F statistic only for the interaction effect.Genotype and phenotype data on 107 to zero) . For the markers , and thehttp://mtweb.cs.ucl.ac.uk/mus/www/mouse/index.shtml) consists of 1,814 individuals from 85 families, all descending from eight inbred progenitor strains [BGLR R package [sommer. Wald test statistics and Bayes Factors were calculated using the heuristic (accelerated) grid search of Grid-LMM. Bayes Factors were calculated assuming flat priors on the intercept and residual variance term, a standard normal prior N on the marker effects—similar to the previously proposed D2 prior [The heterogeneous stock of mice data from the Wellcome Trust Centre for Human Genetics  were calculated with Arabidopsis accessions was downloaded from http://signal.salk.edu/1001.php and subsetted to genes with average counts ≥ 10. A genomic relationship matrix (KA) for 1,135 accessions was downloaded from http://1001genomes.org/data/GMI-MPI/releases/v3.1/SNP_matrix_imputed_hdf5/. Both sets of data were subsetted to an overlapping set of 665 accessions. KA was then centered by projecting out the intercept and normalized to have mean diagonal elements equal to one. The pairwise-epistasis genomic relationship matrix KE was calculated with element-wise multiplication as KA ☉ KA, and then also normalized to have mean diagonal elements equal to one. The gene expression matrix was normalized and variance-stabilized using the varianceStabilizingTransformation function of the DEseq2 package [Grid-LMM REML estimates were compared to exact REML estimates from the LDAK program with variance components constrained to be non-negative. Grid-LMM posterior distributions estimates were compared to those estimated using Stan [rstan R package [KA covariance matrix by calculating the singular value decomposition KA = USU⊤, and pre-multiplying both sides of the LMM by U⊤. We used a half-Student-t prior distribution with 3 degrees of freedom and a scale of 10 for the three standard deviation parameters. We ran four MCMC chains each of length 10,000, with the first 5,000 as warmup and a thinning rate of 20. Because of poor mixing, this resulted in an “neff” of approximately 200-400 per variance component parameter for each trait.Gene expression data on 24,175 genes from 728  package . Grid-LMing Stan  with the package . To speeLDAK, (2) the approximate two-random effect model fit with Grid-LMM with a grid size of 0.1 h2-units, and (3) the approximate two-random effect model pylmm that conditions on variance components estimated under the null model [GEMMA: (4) a model that only included the additive genetic relationships and ignored the G×E covariance, and (5) a model that only included the G×E covariance and ignored the additive genetic relationships.We compared the power of one and two-random effect models for detecting G×E markers based on the Arabidopsis flowering time data. We calculated additive genetic and G×E relationship matrices as described above using the actual genotypes. We then simulated phenotypes by summing together a G×E effect of a particular marker, (co)variance from the two random effects, and normally distributed error. For each combination of marker effect sizes , and random effect proportions . We then “corrected” all F-statistics by dividing by the appropriate inflation factor, and calculated statistical power for each GWAS method as the proportion of corrected Wald test p-values exceeding the genome-wide significance level at the conventional Bonferroni corrected threshold P = 2 × 10−7 -values is greatly inflated when G×E-variation is large. On the other hand, when only G×E variance is modeled and yet the additive genetic variance is large, -log10(p)-values are greatly deflated. These results all confirm our expectations that modeling covariance accurately, but not necessarily precisely, is important for accurate association mapping in GWAS.The results show that Grid-LMM in a reasonable amount of time without the confounding issue of changes in population structure across different populations. These simulations are based on the WTCHG body weight data described above, creating simulations with similar levels of structure and random effect covariance as in the real analysis, and comparing the accuracy of marker tests using the Grid-LMM and null-LMM methods to the Exact-LMM method.We developed a simulation strategy to evaluate the effect of sample size on the accuracy of n = 1814 WTCHG heterogeneous stock mice), randomly selected 300 markers, and calculated the three covariance matrices for additive, addtive-additive epistasis, and cage random effects. Next, we created a larger dataset of 5× the original size by repeating the original marker data five times and similarly created three block-diagonal covariance matrices by repeating each original covariance matrix five times. While not completely realistic, this created a population similar to what we would expect if the heterogeneous stock population was created five separate times from five independent sets of progenitors; therefore, it has a similar level of structure relative to its size as the original n = 1, 814 population. Finally, we created a smaller dataset of 1/5× the size by subsampling the first 362 individuals from this dataset, their corresponding marker data, and the corresponding partitioned subsets of the three covariance matrices.We began with our largest dataset (i.e. the Exact-LMM and null-LMM methods. We then simulated Grid-LMM results by perturbing each of the three variance component proportions from Exact-LMM: ±0.05 for Grid-LMM(0.1) and ±0.005 Grid-LMM(0.01), respectively. Lastly, we selected the p-value from the model with the highest REML score. This represented a “worst-case scenario” for Grid-LMM where the optimal variance components were maximally far from the grid vertices.For each simulation, we selected a single marker and assigned it an effect size between 0 and 0.15 in 16 steps. We then added four random vectors corresponding to the three random effects and “iid” error. To be realistic, we used the variance component proportions 0.23, 0.29, and 0.25, respectively, as weights for the random vectors . This choice was based on the observed variance components in the real bodyweight data. We repeated this simulation strategy for each of the 300 markers in the three populations and each of the 16 effect sizes. Within each simulation, we calculated marker p-values using the Exact-LMM for a single marker using LDAK (T1), the time to perform a single Cholesky decomposition (T2), and the time to calculate p-values for a set of p markers given a pre-calculated Cholesky decomposition pT3. We then calculated the total time for a GWAS with p markers as:Exact-LMM: pT1null-LMM: T1 + T2 + pT3Grid-LMM(0.1): g1(T2 + pT3)Grid-LMM-Fast(0.01): T1 + g2(T2 + pT3) + g3(T2 + T3)where g1 = 220 is the size of a complete grid for three random effects with resolution h2 = 0.1, g2 = 27 is the size of a ball around the null-LMM variance component estimates in the Grid-LMM-fast heuristic, and g3 is the number of additional grid vertices that must be traversed by the Grid-LMM-fast algorithm. For the Grid-LMM-fast calculations, we assumed that only a single marker had a non-zero effect (and so only this marker would need to be taken through multiple iterations of the heuristic search), and that the effect size of this marker was approximately at the power threshold given the sample size  Computational times for the most costly steps of typical mixed model fitting algorithms: inverting an n × n covariance matrix , and multiplying the inverse matrix by an n-vector (i.e. a vector of marker genotypes). The red curve shows the time required for a Cholesky decomposition using the base R function chol as a function of n. The green curve shows the time required for a Cholesky matrix-by-marker matrix multiplication with 1 × 105 markers, as a function of n. The blue curve is the sum of the Cholesky decomposition and matrix-vector multiplication operations for a single grid-cell in Grid-LMM with 1 × 105 markers. The green and blue curves are barely distinguishable across most of the range because the Cholesky decomposition is generally not limiting. The purple curve would be the expected time for a separate Cholesky decomposition and matrix-vector multiplication for each marker in a GWAS with 1 × 105 markers . Both the Grid-LMM and LDAK times are per-iteration. Grid-LMM requires this time at each grid cell. LDAK requires multiple iterations for the REML optimization separately for each marker. Generally, Grid-LMM will evaluate more grid cells than LDAK requires iterations per test. However this will not cause a reversal in the relative time requirements unless a very large grid is used. (b) Memory requirements for storing an n × n Cholesky matrix as a function of sample size. In both panels, the curves were extrapolated based on tests with n between 256 and 4096 . All timings were estimated using base R functions.((PDF)Click here for additional data file.S2 FigGEMMA to p-values calculated by the approximate methods EMMAX, and Grid-LMM using either the naive approach with a complete grid of size 0.1 h2-units, or the fast heuristic algorithm Grid-LMM-fast with a fine grid size of 0.01-h2 units. Grid-LMM p-values were always at least as accurate  as those calculated by EMMAX. Specifically, p-values calculated with a fine grid-size of 0.01  were nearly indistinguishable from those of GEMMA, except in the rare cases where the REML surface was not unimodal. This was generally restricted to a small subset of rare markers with small-moderate effect sizes, and only occurred for a few traits. In these cases, the complete—but more coarse—grid search of Grid-LMM with step sizes of 0.1 was more accurate.GWASs were run for each phenotype using 216,130 markers and up to 199 accessions, with a single random effect controlling for additive genetic relationships among lines. For each phenotype (represented as a single point in the plots), we compared the exact Wald-test -(PDF)Click here for additional data file.S3 FigF-statistics for an interaction between the marker and the environment. Bars show an estimate of genomic control inflation factors [exact-LMM-G+GxE is an exact LMM algorithm fit with LDAK. This model included both random effects and the marker effect. At a genome-wide scale, it is very slow, with computational complexity exact-LMM-G and exact-LMM-GxE are exact LMM algorithms, similar to GEMMA, which included only one random effect and the marker effect. null-LMM is an approximate method similar to pylmm that conditions on variance components estimated under a null model with no marker effect. It was run with both random effects. Grid-LMM was run with a grid size of 0.1 h2-units and included both random effects and the marker effect. The λ values were calculated as the ratio between the median value of the the F-statistics returned by each model and the median value of a F1,316−4 distribution. The horizontal line shows the expected value λ = 1 under the true model.Simulated datasets were created based on the Atwell genotype data and the G×E analysis. We randomly selected 10,000 markers, generated simulated data with different proportions of additive (G) and gene-environment interaction (G× E) variation for each marker, and calculated Wald (PDF)Click here for additional data file.S4 FigF-statistics from each method were normalized by dividing by the genomic control inflation factor computed for Figure, and then p-values were calculated and compared to the Bonferroni corrected threshold P = 2 × 10−7 to determine significance.Bars show the genome-wide power for randomly selected SNPs in the Atwell genotype data under simulations with different proportions of additive (G) and gene-environment interaction (G×E) variation with different marker effect sizes. Simulations were generated as described in (PDF)Click here for additional data file.S5 Figa) REML estimates for the additive genetic variance (variance component for KA). (b) REML estimates for the epistatic genetic variance (variance component for KE).((PDF)Click here for additional data file.S6 FigGrid-LMM with a grid size of 0.05 h2 units. Point positions are jittered for clarity.Both models were fit using (PDF)Click here for additional data file.S7 Figb) and (c) of KA and KE for the random effects was applied to each grid vertex. The prior was approximated by simulating 1 × 104 independent draws for σA, σE and σe, converting these to prior draws for Panels ((PDF)Click here for additional data file.S8 FigKA and KE as random effects plus uncorrelated random error). (a)-(b) Each standard deviation parameter was assigned a half-Student-t prior with 3 degrees of freedom and scale parameter of 10. (c)-(d) Each variance parameter was assigned an inverse-Gamma prior with shape parameter 2 and scale parameter 1. (e)-(g) A uniform prior was applied to the 2-dimensional simplex of a)-(c)-(e) 2D-density plots for the two variance component proportions. Lighter blue denotes higher prior density. (b)-(d)-(f) Marginal densities for the KA variance component proportion under each prior. The half-Student-t prior implies high probability that only one variance component is important. The inverse-Gamma prior implies high probability that all variance component proportions are non-zero.We compare the implied prior distributions on variance component proportions for three classes of priors in a two-random effect model (e.g. (PDF)Click here for additional data file.S1 Table10(p) > 3 on chromosomes 4 and 11 are shown. Position information for each marker is derived from [9 and F10 generation mice from of an intercross between LG/J and SM/J strains selected for differences in body size.Markers with -logved from . Other s(XLSX)Click here for additional data file.S1 Text(PDF)Click here for additional data file.