Supplemental Material
S1. Derivation of meta-analyzed Levene’s test statistics using summary statistics
Let ni denote the count in the ith genotype group. Levene’s test statistics to assess whether the genotype groups share a common variance is:
Where, the th observation in the th genotype group and the group mean of. is the group mean of and the overall mean of. Without loss of generality, assume the quantitative trait conditional on a genotype is centered about its group mean (i.e. no main effect). is then reduced to.
Let , , be the genotype counts summed over all studies, N the overall sample size. Calculation of Levene’s test statistics by simply combining samples assumes the following natural weights: () and ().
Let superscript “+” denote the statistics calculated under a hypothetical situation where individual-level data from all studies are used:
the overall within genotype mean
the grand mean
We then express the test statistics L+ by mathematical equivalence using only the summary statistics and weight:
Table S1. Study-specific Quanlity Control Results
Study / TraitHeight / BMI
Number of Samples / Number of SNPs / Number of Samples / Number of SNPs
MESA / 2,358 / 695,368 / 2,168 / 692,326
NHS / 3,307 / 687,532 / 2,449 / 671,098
HPFS / 2,449 / 667,108 / 1,275 / 647,506
Combined / 8,114 / 660,716 / 5,892 / 642,600
Table S2 SNPs associated with other traits/disease reported in the published GWAS catalogue (at p-value < 5E-08) that are within 500kb distance away from the Variance heterogeneity SNPs.
Variance HeterogeneousSNP / Chr / Position
(KB) / Nearest Gene / Known Associated Disease/Trait / Associated SNPs / R2 / D’ / Distance (kb) / PubMed ID (References)
Between Variance Heterogeneous SNP and Known SNP
BMI
rs12132044 / 1 / 72306264 / NEGR1 / BMI / rs2568958 / 0.325 / 0.888 / 231.440
/ 20935630
19079260
Height
rs11224301 / 11 / 99959951 / ARHGAP42 / Blood Pressure / rs633185 / 0.012 / 1.000 / 138.797 / 21909115
rs12919408 / 16 / 13831900 / ERCC4 / Menarche / rs1659127 / 0.047 / 0.697 / 463.906 / 21102462
rs7153476 / 14 / 68102983 / RAD51L1 / Diabetes Mellitus, Type 1 / rs1465788 / 0.000 / 0.016 / 230.369 / 19430480
Multiple Sclerosis / rs4902647 / 0.007 / 0.092 / 220.961 / 21833088
rs857179 / 16 / 13838131 / ERCC4 / Menarche / rs1659127 / 0.047 / 0.697 / 457.675 / 21102462
Table S3. Power to detect a gene-environment and gene-gene interaction, respectively
Each condition in the corresponding cell was simulated 5,000 times with n (= 1, 000, 2, 500, 5, 000, 7, 500) individuals. Four studies were individually analyzed to generate summary statistics for meta-analysis of Levene’s test. SNPs with nominally significant meta-analyze Levene’s test p-values were then selected by Variance Prioritization for interaction with either a continuous environmental covariate or a second SNP. The beta-coefficients represents the main effect of the prioritized SNP, main effect of the interacting covariate (or a second SNP, not necessarily selected by Variance Prioritization) and the interaction effect. Exhaustive search power represents the power to detect an interaction with linear regression after correcting for M = 500,000 SNPs (p-value < 0.05/M). VP power represents the power of Variance Prioritization at the optimal p-value threshold. Increase in power relative to exhaustive search is computed as a ratio and approaches infinity when the exhaustive power approaches 0. Variance explained (VE) by covariate and interaction was calculated using beta-coefficients and also to reflect effect sizes as a function of minor allele frequencies.
Table S3.1 Power to detect a gene-environment interaction
Beta-Coefficients / MAF = 10% / MAF = 20% / MAF = 30% / MAF = 40% / MAF = 50%β1= 0
β2 = 0.3
β3 = 0.05 / Exhaustive Search Power / 0.0032 / 0.0372 / 0.1066 / 0.1784 / 0.1942
Proportion of Prioritized SNPs / 6% / 6% / 30% / 21% / 19%
VP Power / 0.0098 / 0.052 / 0.1352 / 0.2096 / 0.2276
Relative Increase / 2.063 / 0.398 / 0.268 / 0.175 / 0.172
VE Covariate (%) / 8.253 / 8.251 / 8.249 / 8.2478 / 8.2474
VE Interaction (%) / 0.0412 / 0.0733 / 0.0962 / 0.11 / 0.1145
β1 = 0
β2 = 0.3
β3 = 0.08 / Exhaustive Search Power / 0.1474 / 0.6558 / 0.8874 / 0.95 / 0.966
Proportion of Prioritized SNPs / 21% / 50% / 69% / 64% / 73%
VP Power / 0.1764 / 0.6732 / 0.8928 / 0.9532 / 0.9684
Relative Increase / 0.197 / 0.027 / 0.006 / 0.003 / 0.002
VE Covariate (%) / 8.248 / 8.241 / 8.2365 / 8.234 / 8.233
VE Interaction (%) / 0.1056 / 0.1875 / 0.246 / 0.281 / 0.2927
β 1 = 0.1
β 2 = 0.3
β3 = 0.05 / Exhaustive Search Power / 0.0038 / 0.0412 / 0.1168 / 0.1724 / 0.1934
Proportion of Prioritized SNPs / 19% / 9% / 21% / 18% / 24%
VP Power / 0.0086 / 0.063 / 0.14 / 0.2046 / 0.2322
Relative Increase / 1.263 / 0.529 / 0.199 / 0.187 / 0.201
VE Covariate (%) / 8.24 / 8.227 / 8.2173 / 8.212 / 8.21
VE Interaction (%) / 0.0412 / 0.0731 / 0.0959 / 0.1095 / 0.114
β1 = 0.1
β2 = 0.3
β3 = 0.08 / Exhaustive Search Power / 0.1536 / 0.6434 / 0.8898 / 0.9582 / 0.9614
Proportion of Prioritized SNPs / 22% / 63% / 79% / 78% / 71%
VP Power / 0.183 / 0.661 / 0.8932 / 0.9604 / 0.9644
Relative Increase / 0.191 / 0.027 / 0.004 / 0.002 / 0.003
VE Covariate (%) / 8.234 / 8.217 / 8.205 / 8.198 / 8.195
VE Interaction (%) / 0.1054 / 0.187 / 0.2451 / 0.28 / 0.2914
Table S3.2. Power to detect a gene-gene interaction
Beta-Coefficients / MAF1 = 10% MAF2 = 10% / MAF1 = 10% MAF2 = 20% / MAF1 = 20% MAF2 = 20% / MAF1 = 20% MAF2 = 40% / MAF1 = 30% MAF2 = 30% / MAF1 = 30% MAF2 = 50%β1=0.4
β2=0.4
β3=0.256 / Exhaustive Search Power / 0.0774 / 0.6972 / 0.999 / 1 / 1 / 1
Proportion of Prioritized SNPs / 56% / 56% / 87% / 17% / 10% / 69%
VP Power / 0.0912 / 0.7164 / 0.999 / 1 / 1 / 1
Relative Increase / 0.178 / 0.028 / 0 / 0 / 0 / 0
VE SNP1 (%) / 2.7177 / 2.6573785 / 4.61631 / 4.4989 / 5.864076 / 5.7883
VE SNP2 (%) / 2.7177 / 4.7242284 / 4.61631 / 6.7483 / 5.864076 / 6.8908
VE Interaction (%) / 0.2004 / 0.3483079 / 0.605069 / 0.8845 / 1.008809 / 1.1854
β1 = 0.4
β2 = 0.4
β3 = 0.1 / Exhaustive Search Power / 0 / 0 / 0.001 / 0.011 / 0.026 / 0.0682
Proportion of Prioritized SNPs / 1% / 2% / 10% / 5% / 8% / 7%
VP Power / 0 / 2.00E-04 / 0.0036 / 0.0278 / 0.0648 / 0.1408
Relative Increase / 0 / Inf / 2.6 / 1.527 / 1.492 / 1.065
VE SNP1 (%) / 2.722 / 2.6652452 / 4.64010208 / 4.5328 / 5.914639 / 5.847
VE SNP2 (%) / 2.722 / 4.7382137 / 4.64010208 / 6.7993 / 5.914639 / 6.9608
VE Interaction (%) / 0.0306 / 0.0533049 / 0.09280204 / 0.136 / 0.155259 / 0.1827
β1 = 0.1
β2 = 0.4
β3 = 0.1 / Exhaustive Search Power / 0 / 0 / 4.00E-04 / 0.0088 / 0.025 / 0.0708
Proportion of Prioritized SNPs / 1% / 1% / 33% / 28% / 36% / 67%
VP Power / 0 / 4.00E-04 / 0.0022 / 0.0144 / 3.00E-02 / 0.071
Relative Increase / 0 / Inf / 4.50 / 0.636 / 0.2 / 0.003
VE SNP1 (%) / 0.1746 / 0.17084672 / 0.30319568 / 0.2959 / 0.391366 / 0.3866
VE SNP2 (%) / 2.7936 / 4.85963993 / 4.85113092 / 7.101 / 6.261858 / 7.3644
VE Interaction (%) / 0.0314 / 0.05467095 / 0.09702262 / 0.142 / 0.164374 / 0.1933
Figure S1 – Quantile-quantile plot of Levene’s test P-value using individual-level data and aggregated-level data (meta-analyzed Levene’s test P-value)
We showed with simulated data, under the null of variance homogeneity, the p-values given by meta-Levene test statistics using both individual-level and summary-level data were identical by mathematical equivalence.
Figure S2. Quantile-quantile plots of Levene’s test p-value distribution in individual studies and meta-analysis for log(height) and log(BMI)
Illustrated in (A-D) are the quantile-quantile plots of meta-analyzed Levene's test P-values for log (height) in individual studies (MESA, NHS, HPFS) and combined analysis (8,114 individuals combined), respectively. Illustrated in (E-H) is the quantile-quantile plot of meta-analyzed Levene's test P-values for log (BMI) in individual studies (MESA, NHS, HPFS) and combined analysis (5,892 individuals combined) , respectively.
Figure S3. NEGR1 Gene Region
Regional plot of the Levene’s test p-values obtained from meta-analysis of Body Mass Index variance heterogeneity. The purple diamond represents the rs12132044 variant with Levene’s test p-value of 4.28e-6. Whereas the light blue circle to the far right (r-squared estimated from HapMap CEU panel [The International HapMap 3 Consortium (2010). Integrating common and rare genetic variation in diverse human populations Nature: 10.1038/nature09298] is 0.325) represents the rs2815752 variant known to be associated with BMI and extreme obesity. This plot is generated using LocusZoom (Pruim RJ, Welch RP, Sanna S, Teslovich TM, Chines PS, Gliedt TP, Boehnke M, Abecasis GR, Willer CJ. (2010) LocusZoom: Regional visualization of genome-wide association scan results. Bioinformatics 2010 September 15; 26(18): 2336.2337).
Figure S4.1 - Variance Prioritization power to detect a gene-environment interaction
Consider a hypothetic genetic consortium comprising 4 studies, with 1,000, 2,500, 5,000 and 7,500 participants and a common set of 500,000 genotyped SNPs. Four studies were individually analyzed using Variance Prioritization and meta-analyzed to generate summary statistics for meta-analysis (Left panel). Each condition on the right panel was simulated 5,000 times. Minor allele frequency of the SNP was set at 20%. For each condition, assume that the environmental exposure explained 13.8% of the quantitative trait variance, the horizontal line represents the power to detect an interaction that explained 0.16% of the quantitative trait variance with linear regression after correcting for M = 500,000 SNPs (p-value < 0.05/M). Black curves represent the power of Variance Prioritization when Levene’s test p-value thresholds range from 0.001 to 1 with 0.001 incremental increase. The power of Variance Prioritization was maximized at the optimal p-value threshold.
Figure S4.2 - Variance Prioritization power to detect a gene-gene interaction
Each condition on the right panel was simulated 5,000 times with n (= 1, 000, 2, 500, 5, 000, 7, 500) individuals. Four studies were individually analyzed using Variance Prioritization and meta-analyzed to generate summary statistics for meta-analysis (Left panel). Minor allele frequency of both interacting SNPs was set at 20%. For each condition, assume individual SNPs each explained 4.6% of the quantitative trait variance, the horizontal line represents the power to detect a gene-gene interaction explaining 0.2% of the quantitative trait variance with linear regression after correcting for M = 500,000 SNPs (p-value < 0.05/M2). Black curves represent the power of Variance Prioritization when Levene’s test p-value thresholds range from 0.001 to 1 with 0.001 incremental increase. The power of Variance Prioritization was maximized at the optimal p-value threshold.
We compared the performance of VP using Meta-Levene to a conventional method (i.e. exhaustive search with correction for all SNPs tested). When considering a gene-environment interaction explaining 0.16% of the quantitative trait variance (Figure S4.1), power to detect the interaction using both Meta-Levene and Levene’s test on individual-level data was estimated at 58.2%, as compared to 52.2% using exhaustive search (left panel). Further, when Levene’s test was not meta-analyzed, the VP powers to detect interactions under the same condition for individual studies were0.002% (n = 1, 000), 0.4% (n = 2, 500), 2.5% (n = 5, 000) and 9.54% (n = 7, 500), corresponding to even lower conventional powers of 0%, 0.008%, 0.56% and 6.38%, respectively (right panel). For a gene-gene interaction model, we considered the simplified situation where the two interacting loci had the same minor allele frequency (p) and main effect (β1) on the quantitative trait. When the proportion of variance explained by a gene-gene interaction was 0.2% and individual SNPs explained 4.6% of the quantitative trait variance(Figure S4.2), power to detect gene-gene interactions using exhaustive search was 8.90%, while Variance Prioritization led to an improved power of 14.04% (left panel). When Levene’s test was not meta-analyzed, statistical powers of Variance Prioritization to detect interactions under the same conditions for individual studies were 0% (n = 1, 000), 0% (n = 2, 500), 0.08% (n = 5, 000) and 0.42% (n = 7, 500), corresponding to conventional powers of 0%, 0%, 0% and 0.02%, respectively (right panel).