EDGE accurately encodes SNPs according to simulated genetic models

Like traditional encodings, EDGE encodes the homozygous referent and homozygous alternate genotypes as 0 and 1, respectively. For additive, dominant, and recessive encodings, a predetermined heterozygous value is assigned (Table 2). In contrast, EDGE assigns a flexible heterozygous value (α) based on the genetic model each individual SNP demonstrates in the dataset (Table 2). For example, if SNP A’s underlying genetic model is additive, the heterozygous genotype bears 50% the risk of homozygous alternate genotype (Table 1), and EDGE will assign the heterozygous genotype as 0.5. If SNP B demonstrates a sub-additive underlying model, the heterozygous genotype incurs 25% the risk of the homozygous alternate genotype, and the EDGE-derived alpha will be assigned as 0.25. In the case of SNP B, the heterozygous risk does not fall within the traditional encoding methods that are listed in Table 2. EDGE also can detect cases in which SNP C’s heterozygous genotype bears more risk than the homozygous alternate genotype, where the underlying model is over-dominant (α > 1) or in which SNP D’s heterozygous genotype bears less risk than the homozygous referent genotype (under-recessive; α < 0). Encoding detection and assignment occurs prior to interaction analysis, allowing for an encoding that reflects the underlying model of each unique SNP. EDGE is available for download in PLATO software [31] on the Ritchie Lab website https://ritchielab.org/plato.

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Table 2. Recessive, additive, dominant, codominant and EDGE encoding schemes, scaled between 0 and 1.

https://doi.org/10.1371/journal.pgen.1009534.t002

The mechanisms of the EDGE method are described in the following steps:

1. Logistic or linear regression is run using a codominant (dummy) encoding with no intercept (so mean-centered trait; Y-E [Y] or prob (Y = 1)–pop prevalence). Note: if covariates are included in the analysis for adjustment, they are also included in the regression model and that, for consistency, we code the minor allele as the risk (alternate) allele.
   (1)
2. Using the beta values from the heterozygous genotype (Het) and homozygous alternate genotype (HA) dummy encodings, a weighted value (α) for the heterozygous genotype is calculated, whereby the α corresponds to the risk of the heterozygous genotype relative to homozygous alternate genotype when homozygous alternate risk is scaled to 1.
   (2)
3. These EDGE genotype encodings (homozygous referent = 0, heterozygous = α, homozygous alternate = 1) are then used for SNP-SNP interaction analysis. A common approach for genetic interaction is performing a likelihood ratio test (LRT) between a full and reduced model: Y = β₀ + β₁SNP1 + β₂SNP2 + β₃SNP1×SNP2 (full) and Y = β₀ + β₁SNP1 + β₂SNP2 (reduced). A significant LRT p-value indicates that including the interaction term (β₃ of the full model) significantly improves model fit when compared to a model containing only the main effects of the two SNPs (β₁ and β₂). It has been previously shown that if the encoding of SNP1 and SNP2 does not sufficiently represent the marginal effects of SNP1 and SNP2 in the full model, the full regression model will force some of the trait variance to be explained by the interaction term (even if in truth there is no interaction effect at all) [32]. Any overfitting that could be at play by employing EDGE parameter α is accounted for in the main effects of the models. Our expectation of low inflation was validated in the conserved false positive rates demonstrated by EDGE in the simulation studies further described below. Thus, the EDGE encoding reduces the false positive interaction terms because it maximizes the information from the data to code marginal effects in the optimal manner.

To ensure that EDGE assigns the expected heterozygous genotype value across different types of underlying genetic models, we simulated main effect SNPs with the following genetic models using the Biallelic Model Simulator (BAMS) (for a description of BAMS, see Methods and S1 Text; further information and download available at https://www.hall-lab.org): recessive (REC), sub-additive (SUB), additive (ADD), super-additive (SUP), and dominant (DOM) (1,000 simulated datasets for each model type) across varying minor allele frequencies (MAFs) (S1 Fig). Here, heterozygous genotypes are simulated to have 0% (REC), 25% (SUB), 50% (ADD), 75% (SUP), and 100% (DOM) the risk of homozygous alternate genotypes (see Table 1). Fig 1 displays the distribution of the alpha values for each SNP model type at simulated 30% MAF. For every model, the density peaks correspond to the simulated genetic model (REC≈0, SUB≈0.25, ADD≈0.5, SUP≈0.75, and DOM≈1), demonstrating that EDGE effectively assigns alpha values reflecting the simulated genetic model. Of note, a large portion (approximately half) of simulated recessive SNPs were assigned an alpha value between -0.5 and -0.001 and similarly, for simulated dominant SNPs, approximately half were assigned an alpha between 1.001 and 1.5. This is important to consider when natural SNPs are assigned alpha values below 0 or above than 1: while this could indicate a SNP’s action is under-recessive or over-dominant, if the value is close to 0 or 1, the SNP’s model may still be in the recessive or dominant range, respectively. When considering density peaks across MAFs, we observed variation in peak density across the allele frequencies (S1 Fig) with increased density at the expected alpha values for higher MAFs and greater variability for lower MAF (15% MAF or lower). One exception to this trend was for the simulated recessive acting SNPs, which showed the highest density for SNPs simulated with 5% MAF. Additionally, we observed differences in which simulated models showed the highest peak density at different MAFs. For example, as seen in Fig 1, at 30% MAF, the highest peak density was observed for simulated sub-additive SNPs. For MAF of 45%, EDGE assigned an alpha with the highest peak density for SNPs simulated with an additive model, and as described previously, recessive SNPs showed high peak density at 0 alpha with 5% MAF (S1 Fig). These results indicate that, while SNPs were assigned alpha values by EDGE that were consistent with the simulated model across all REC, SUB, ADD, SUP, and DOM models, EDGE performed most optimally at assigning an alpha value for SNPs whose simulated heterozygous risk relative to homozygous alternate is close in value to the MAF of the SNP.

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Fig 1. Distribution of the estimated heterozygous action (α).

Single SNP simulations were performed with five distinct underlying models: recessive (red), sub-additive (yellow), additive (green), super-additive (blue), and dominant (purple). All SNPs shown in this figure were simulated with a 30% MAF. 1,000 datasets for each SNP model were generated and the distribution of the alpha parameter was plotted. Along the x-axis is the EDGE assigned alpha value for simulated SNPs of each genetic model and along the y-axis is the density.

https://doi.org/10.1371/journal.pgen.1009534.g001

The additive encoding demonstrates inflation of type I error

To assess type I error, two categories of main effect SNP-SNP models were simulated: One-SNP Main Effect and Two-SNP Main Effect. For One-SNP Main Effect models, only one SNP exhibited a main effect while the other did not have an effect on the phenotype. The main effect SNPs were simulated with ADD, DOM, REC, SUB, and SUP genetic models. An additional genetic model was included, the heterozygous (HET) model, in which heterozygous genotypes were simulated to have full risk relative to both homozygous genotypes, which had 0 risk. For Two-SNP Main Effect models, two SNPs with main effects were simulated with no interaction effect. We simulated all pairwise combinations between REC, SUB, ADD, SUP, DOM, and HET. Null samples with no main or interaction effects were also simulated. We simulated 1,000 datasets for each SNP-SNP model to calculate a false positive rate (FPR) as the percentage of the 1000 datasets with a likelihood ratio test (LRT) p-value < 0.05 per model using the additive, dominant, recessive, codominant, and EDGE encodings. FPR for the simulated main effect models represents the frequency at which the encoding identified a significant interaction term when no simulated interaction existed. As shown in Fig 2 with MAF of 30% as an example, all of the encodings demonstrated a FPR near 5% for the null data. For the main effect models, the majority of encodings showed conserved FPR. A deviation from this was seen for the additive encoding, where inflation was observed, including the scenarios under which: 1) both SNPs exhibit a main effect with a simulated dominant genetic model (FPR≅16%); 2) one SNP exhibits a main effect with a simulated heterozygous model (FPR≅11%), and 3) SNP1 exhibits a main effect with a simulated dominant model and SNP2 exhibits a main effect with a simulated heterozygous model (FPR≅9%). According to Bradley’s liberal criteria [32], the estimated type I error should fall within the interval (0.025, 0.075). This is the case for the majority of encoding schemes except for additive encoding, which confirms the observation by Mahachie et al [32].

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Fig 2. Type I error in simulated main effect and null data.

Power to detect two-SNP (above the top solid horizontal line) and One-SNP (below top solid line) main effect simulations are displayed. Null model simulations are shown below the bottom solid horizontal line. The dashed horizontal line separates models involving HET SNPs (below dashed line) from non-HET SNPs (above the dashed lines) because the homozygous genotypes in HET SNPs were simulated to have 0 risk relative to the heterozygous genotype, while the homozygous alternate genotypes for all other SNPs were simulated to have full risk comparted to heterozygotes. Main Effect Only models and null data were simulated (1,000 datasets each) and LRT p-values were calculated using additive (red circle), dominant (yellow square), recessive (green diamond), codominant (blue triangle), and EDGE (purple inverted triangle) encodings to identify the percentage of time each encoding identified a significant interaction when there was none (false positive rate). The red vertical line marks a 5% false positive rate and the dashed black lines mark where the estimated type I error should fall within according to Bradley’s liberal criteria [32]. All SNPs in this figure were simulated with a 30% minor allele frequency.

https://doi.org/10.1371/journal.pgen.1009534.g002

To compare the average false positive rate for the encodings among type I error datasets including the main effect and null datasets across 10%, 30%, and 50% MAF, we standardized signal to noise ratios. Fig 3 depicts the type I error at standardized, increasing signal to noise ratios using the different encodings. Each of the models from Fig 2 across the varying MAFs were averaged (solid lines in Fig 3). We observed low average type I error for the codominant encoding and EDGE (following the 5% FPR level) as well as the recessive encoding (below the 5% level at high signal to noise ratios). However, at a higher signal to noise ratios, we again observed inflation of type I error for the additive and dominant encodings. Further inspection of each model’s maximum type I error (dashed lines in Fig 3) at high signal to noise ratios revealed exceedingly high type I error for additive and dominant encodings.

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Fig 3. Average and maximum type I error across all main effect and null simulated datasets.

Average false positive rates (solid lines) and the false positive rate for the maximum inflated model (dashed lines) were calculated for standardized signal to noise ratios. Values are plotted at increasing signal to noise ratios for each encoding: additive (red), dominant (yellow), recessive (green), codominant (blue), and EDGE (purple). The black line denotes a 5% false positive rate.

https://doi.org/10.1371/journal.pgen.1009534.g003

Power to detect interaction varies widely across genetic encoding and SNP model types

Once EDGE demonstrated accuracy at encoding individual SNPs based on the simulated genetic model and low type I error, we evaluated EDGE’s power to detect SNP-SNP interaction models compared to the traditional encoding methods. We simulated 29 SNP-SNP interactions with no main effect. The first type of simulated SNP-SNP interaction models involved SNPs with comprehensive pairwise combinations of underlying genetic models: ADD, DOM, REC, SUB, SUP, and HET. Additionally, we evaluated simulated interactions using genotype-based interaction models that include penetrance functions (e.g., XOR, Hyperbolic) and scenarios in which only one of the 9 interaction penetrance cells confers risk while the other 8 demonstrate no risk (e.g., Homozygous Referent-Homozygous Referent–HR-HR). Power calculations for all 29 SNP-SNP interaction models using each of the encodings as the percentage of simulated datasets (out of 1,000) with an LRT p-value < 0.05. As an example, Fig 4 shows each encoding’s power to detect the 29 interacting models we simulated, where MAF for both SNPs was 30%. Results demonstrated that additive, dominant, and recessive encodings outperformed the other encodings for the interaction models for which they are designed: ADDxADD, DOMxDOM, and RECxREC, respectively. EDGE showed over 80% power for every model in which at least one traditional encoding had over 80% power. The power to detect different models varied widely. For models in which one or two SNPs had an underlying REC model, no encoding identified signal with 80% or more power. Conversely, models including SNP(s) with DOM or SUP action demonstrated consistently high power across encodings. The recessive encoding demonstrated consistently low power and did not exceed 80% power for any model.

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Fig 4. Results of power analysis across 29 simulated SNP-SNP interaction models.

Power was calculated as the percentage of simulated models (out of 1,000 datasets for each model) for which the encoding detected an interaction signal at LRT p-value < 0.05: additive (red circle), dominant (yellow square), recessive (green diamond), codominant (blue triangle), and EDGE (purple inverted triangle). Twenty-nine SNP-SNP interaction models were simulated. Above the solid horizontal line are interaction models with comprehensive two-SNP combinations between SNPs with REC, SUB, ADD, SUP, and DOM action (between the solid and dotted line are models including SNPs with HET action). Below the solid horizontal line are genotype-based interaction models. The blue and red vertical lines marks 5% and 80% power, respectively.

https://doi.org/10.1371/journal.pgen.1009534.g004

EDGE demonstrates robust power compared to additive, dominant, recessive and codominant encodings

The power for an encoding to detect a given interaction model varies depending on the type of underlying genetic model (Fig 4). To assess average power for each encoding across the interacting models with consistency, signal to noise ratio was standardized across the different models. We compared the average power of each encoding across all traditional interacting models from Fig 4 and at increasing signal to noise ratios (10%, 30%, and 50% MAF). At 30% MAF (Fig 5), EDGE demonstrated the highest average power across traditional models with increasing signal to noise ratio; the codominant encoding showed comparable power to EDGE until the signal to noise reached a threshold at which increasing numbers of models diverged, reducing power; additive and dominant encodings demonstrated diminished power compared to EDGE; and recessive encoding showed low power to detect models even at high signal to noise ratio levels. Increasing the MAF to 50% resulted in greater average power for EDGE, codominant, and recessive, and reduced average power for dominant and additive when compared to 30% MAF. At 10% MAF, high average power for EDGE, additive, and dominant encodings was observed, while codominant and recessive showed exceedingly low power due to model non-convergence. Similar patterns were observed for the genotype-based models (Fig 6). However, the codominant encoding only demonstrated reduced average power for the datasets simulated with 10% MAF, and additive, dominant, and EDGE showed diminished power at 50% MAF as compared to Fig 5.

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Fig 5.

Average power for standardized signal to noise ratio across all pairwise interaction models at 10%, 30% and 50% MAF. Average power of each encoding was calculated for standardized signal to noise ratios and plotted by increasing signal to noise ratios for the pairwise model SNP-SNP interaction models using each encoding: additive (red), dominant (yellow), recessive (green), codominant (blue), and EDGE (purple).

https://doi.org/10.1371/journal.pgen.1009534.g005

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Fig 6.

Average power for standardized signal to noise ratio across all genotype-based interaction models at 10%, 30% and 50% MAF. Average power was calculated for standardized increasing signal to noise ratios and plotted at increasing signal to noise ratios for the genotype-based SNP-SNP interaction models using each encoding: additive (red), dominant (yellow), recessive (green), codominant (blue), and EDGE (purple).

https://doi.org/10.1371/journal.pgen.1009534.g006

Allele frequency and sample size influence power to detect models across all encodings

To test the impact of allele frequency, sample size, baseline risk, and case-control ratio on power, we performed a parameter test whereby all 29 interacting models were simulated with combinations of variation in each of the four parameters. We used ANOVA to test for significant effects of each parameter on the encodings’ power to detect these simulated models. As shown in Fig 7, minor allele frequency and sample size demonstrated a significant effect on power for a large number of models for each encoding, while case-control ratio and baseline risk affected very few models. The differences observed across varying MAFs and sample sizes revealed similar trends: with increasing MAF and sample size, the power for each encoding to detect the interaction signal increased.

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Fig 7. Significance of each parameter on power.

ANOVA tests were performed to determine the impact of parameters MAF, sample size, baseline (penetrance), and case-control ratio on power for each interaction model for the additive (red), dominant (yellow), recessive (green), codominant (blue), and EDGE (purple) encodings.

https://doi.org/10.1371/journal.pgen.1009534.g007

EDGE assigns heterozygous values that are consistent with traditional encoding methods across five phenotypes

We applied EDGE to biomedical data from the Electronic Medical Records and Genomics (eMERGE) Network using age-related macular degeneration (AMD), age-related cataract, glaucoma, type 2 diabetes (T2D), and resistant hypertension.

To see how EDGE-derived alpha values corresponded to traditional encoding main effect results, four GWAS were performed for each of the phenotypes (a multi-encoding GWAS), one for each traditional encoding: additive, dominant, recessive, and codominant. While genome-wide significant results were only found for T2D and AMD, all five phenotypes demonstrated similar patterns when the results of their five GWAS were plotted along the x-axis by each SNP’s EDGE-derived heterozygous alpha value (Figs 8 and S2): For SNPs assigned an alpha value near 0, which indicates an underlying recessive genetic model, the recessive encoding tended to demonstrate the lowest p-values. Additive encoding predominated for SNPs with an alpha near 0.5 (indicating SNPs with an underlying additive model). Dominant encoding produced the lowest p-values for SNPs with an alpha near 1 (indicating SNPs with a dominant genetic model).

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Fig 8.

Manhattan plot of results from a multi-encoding GWAS for T2D (A) and AMD (B). GWAS were performed using the additive (red circle), dominant (yellow square), recessive (green diamond) and codominant (blue triangle) encodings for A) 358,569 SNPs and 20,341samples (7,101 cases and 13,240 controls, 53% female) for type 2 diabetes and B) 311,161 SNPs and 13,153 samples (961 cases and 4,129, 56% female) for AMD. SNPs with EDGE-derived alpha values between 0 and 1 were sorted by the alpha value along the x-axis. Along the y-axis is the–log10 of the uncorrected p-value. The black line denotes the genome-wide significance threshold (5×10⁻⁸).

https://doi.org/10.1371/journal.pgen.1009534.g008

For T2D, 10 SNPs were identified at the genome-wide significance level (5×10⁻⁸) for at least one traditional encoding (Fig 8A and Sheets A-D in S1 Table). Six SNPs were significant for three out of the four traditional encodings, including the top three, and no SNP was genome-wide significant for all four encodings. rs4132670 (alpha: 0.68; MAF: 0.40; intron, transcription factor 7 like 2 (TCF7L2)) was the T2D top result with the additive encoding (Bonferroni p: 1.82×10⁻¹³) followed by codominant encoding (Bonferroni p: 1.96x10^-13) and dominant encoding (Bonferroni p: 1.26x10^-13). It was not significant using the recessive encoding. rs12255372 (alpha: 0.53; MAF: 0.28; intron, transcription factor 7 like 2 (TCF7L2)) demonstrated the same order of significance for the encodings as rs4132670 (1.45×10⁻¹⁰, 1.50×10⁻¹⁰, and 2.29×10⁻⁸ for additive, codominant, and dominant Bonferroni adjusted p-values, respectively), and was also not genome-wide significant using the recessive encoding. The third top SNP result, rs2308953 (alpha: 0.13; MAF: 0.084; intron, RAD1 checkpoint DNA exonuclease (RAD1)), demonstrated the lowest p-value with the codominant encoding (Bonferroni p: 1.06x10^-5) followed by the recessive encoding (Bonferroni p: 3.39x10^-4) and additive (Bonferroni p: 9.93x10^-4) and was not genome-wide significant using the dominant encoding.

Eighteen SNPs were genome-wide significant for association with AMD using at least one of the traditional encoding methods (Fig 9 and Sheets A-D in S2 Table). Six SNPs were identified as genome-wide significant for all four traditional methods. The top result was for rs10801558 (alpha: 0.54; MAF: 0.41; intron, complement factor H (CFH)) using the codominant encoding (Bonferroni p: 8.21x10^-32), followed by additive encoding (Bonferroni p: 2.07x10^-31), dominant encoding (Bonferroni p: 5.41x10^-26), and recessive encoding (Bonferroni p: 2.73x10^-13). Three other SNPs in the CFH gene demonstrated genome-wide significance across all traditional encodings: rs399469 (alpha: 0.51; MAF: 0.46), rs10733086 (alpha: 0.50; MAF: 0.43;), and rs380390 (alpha: 0.50; MAF: 0.43;). Another top SNP was genome-wide significant for all four encodings, rs3750846 (alpha: 0.40; MAF: 0.23; age-related maculopathy susceptibility 2 (ARMS2)), with Bonferroni p: 5.76x10^-20 for additive, Bonferroni p: 2.47x10^-19 for codominant, Bonferroni p: 3.61x10^-14 for dominant, and Bonferroni p: 1.44x10^-9 for recessive. rs12042442 (alpha: 0.020, MAF: 0.15; intron of assembly factor for spindle microtubules (ASPM)) was genome-wide significant using the codominant encoding (Bonferroni p: 0.0035). This SNP was not genome-wide significant using the additive, recessive, and dominant encodings. No results from the multi-encoding GWAS for age related cataract (Sheets A-D in S3 Table), resistant hypertension (Sheets A-D in S4 Table), and glaucoma (Sheets A-D in S5 Table) met a genome-wide significance threshold.

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Fig 9.

Pairwise comparisons of the -log10 p-values for the multi-encoding GWAS of (A) T2D and (B) AMD. For each SNP, we plotted the -log10 p-values for pairwise combination of encodings in six scatterplots for (A) T2D and (B) AMD compared to the identity line (1:1 line) such that pairs of encodings having p-values that fall closer to the line are more concordant.

https://doi.org/10.1371/journal.pgen.1009534.g009

To visualize the concordance of results between each pair of encodings for the multi-encoding GWAS for T2D and AMD, we generated pairwise scatterplots (Fig 9). Similar trends were seen for both phenotypes: we observed high concordance for -log10 p-values between codominant and additive, codominant and dominant, and additive and dominant (though for AMD additive and codominant encodings had higher -log10 p-values for some of the top SNPs compared to dominant). Recessive consistently showed lowered concordance with results from the other three encodings.

Genetic interaction analysis of eMERGE results vary by phenotype and genetic encoding

After performing the multi-encoding GWAS for the five phenotypes, the results were used as a main effect filter for SNP-SNP interactions analysis. SNP-SNP interactions were assessed using the five encodings: additive, dominant, recessive, codominant, and EDGE by computing an LRT p-value (see Methods).

One SNP-SNP interaction model demonstrated a significant LRT p-value for T2D when adjusting for the number of tests: rs117537110 (alpha: 0.35; MAF: 0.45; intron, protein phosphatase 1 regulatory subunit 18 (PPP1R18))–rs4149477 (alpha: -0.055; MAF: 0.48; intron, tyrosylprotein sulfotransferase 2 (TPST2)) using the recessive encoding (Bonferroni adjusted LRT p: 0.00051; 5,671 SNP-SNP models tested; r²: 0.00015) (Fig 10 and Sheets A-E in S6 Table). Although it did not meet Bonferroni significance, EDGE identified this interaction with the next lowest p-value compared to the other methods (unadjusted LRT p: 8.91× 10⁻⁵). To determine if the SNP-SNP interaction models replicated in a separate dataset, we used data from the UK Biobank (UKB). Interaction replication analysis of these SNP pairs in UKB yielded no significant results for any encoding. We further evaluated potential replication of SNPs within a 50kb region of the original SNPs, and no significant epistasis models were identified for any encoding in this region-based replication analysis after multiple test correction.

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Fig 10. Interaction plot of the top results from the main effect filtered SNP-SNP interaction analysis for T2D.

The top 50 SNPs from the multi-encoding T2D GWAS for each traditional encoding were considered for a pairwise SNP-SNP interaction analysis using additive (red), dominant (green), recessive (blue), codominant (yellow) and EDGE (purple) encoding methods. Track one displays the EDGE-derived heterozygous alpha value of SNP 1, track 2 displays the alpha value for SNP 2, and track 3 displays the–log10 of the unadjusted LRT p-value of each SNP-SNP interaction model. Vertical blue lines denote a 0 alpha value (indicative of recessive genetic model), vertical red lines denote a 0.5 alpha value (indicative of additive genetic model), vertical green lines denote an alpha value of 1 (indicative of dominant genetic model). Note that some of the SNPs demonstrated alpha values outside the 0–1 range, indicative of under-recessive (α < 0) and over-dominant (α > 1) genetic models. The vertical black line denotes the Bonferroni significance threshold for this analysis (5,671 SNP-SNP models tested).

https://doi.org/10.1371/journal.pgen.1009534.g010

For age-related cataract, one SNP-SNP interaction met a Bonferroni corrected significance threshold, rs7787286 (alpha: -0.037; MAF: 0.041; intergenic region of chromosome 7)–rs4695885 (alpha: 0.62; MAF: 0.34; intergenic region of chromosome 4), and this was found using EDGE (Bonferroni LRT p: 0.018; 9,591 SNP-SNP models tested; r²: 0.00056) (Fig 11 and Sheets A-E in S7 Table). Interaction between these two SNPs was not replicated in UKB, where the MAFs were 0.030 and 0.34 for rs7787286 and rs4695885, respectively. The recessive encoding identified signal in a region-based interaction replication in UKB between SNPs rs60374751 (MAF: 0.030 and alpha: -0.65; 17,434 bp from rs7787286) and rs6843594 (MAF: 0.34 and alpha: -0.54; 24,084 bp from rs4695885) (Bonferroni LRT p: 0.026; r²: 0.00000033) (Sheets A-E in S8 Table).

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Fig 11. Interaction plot of the top results from the main effect filtered SNP-SNP interaction analysis for age-related cataract in the eMERGE Network.

The top 50 SNPs from the multi-encoding cataract GWAS in the eMERGE Network were considered for a pairwise SNP-SNP interaction analysis using additive (red), dominant (green), recessive (blue), codominant (yellow) and EDGE (purple) encoding methods. Track one displays the EDGE-derived heterozygous alpha value of SNP 1, track 2 displays the alpha value for SNP 2, and track 3 displays the–log10 of the unadjusted LRT p-value of each SNP-SNP interaction model. Vertical blue lines denote a 0 alpha value (indicative of recessive genetic model), vertical red lines denote a 0.5 alpha value (indicative of additive genetic model), vertical green lines denote an alpha value of 1 (indicative of dominant genetic model). Note that some of the SNPs demonstrated alpha values outside the 0–1 range, indicative of under-recessive (α < 0) and over-dominant (α > 1) genetic models. The vertical black line denotes the Bonferroni significance threshold for this analysis (9,591 SNP-SNP models tested).

https://doi.org/10.1371/journal.pgen.1009534.g011

One SNP-SNP model met a corrected significance threshold for resistant hypertension using the dominant encoding: rs3801888 (alpha: 0.028; MAF: 0.28) in sorting nexin 10 (SNX10) and rs2858808 (alpha: -1.39; MAF: 0.28) (Bonferroni adjusted LRT p: 0.047; 10,296 SNP-SNP models tested; r²: 0.00037) (Fig 12 and Sheets A-E in S9 Table). The resistant hypertension phenotype was not available in UKB for replication. We tested an interaction between these SNPs for hypertension in UKB due to its relation to the resistant hypertension phenotype and availability in the dataset. A significant interaction was found between rs3801888 (MAF: 0.27 and alpha: 1.097) and rs2858808 (MAF: 0.30 and alpha: 0.23) in UKB using the recessive encoding (LRT p: 0.025; r²: 0.0000021) (Sheets A-E in S10 Table).

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Fig 12. Interaction plot of the top results from the main effect filtered SNP-SNP interaction analysis for resistant hypertension.

The top 50 SNPs from the multi-encoding resistant hypertension GWAS were considered for a pairwise SNP-SNP interaction analysis using additive (red), dominant (green), recessive (blue), codominant (yellow) and EDGE (purple) encoding methods. Track one displays the EDGE-derived heterozygous alpha value of SNP 1, track 2 displays the alpha value for SNP 2, and track 3 displays the–log10 of the unadjusted LRT p-value of each SNP-SNP interaction model. Vertical blue lines denote a 0 alpha value (indicative of recessive genetic model), vertical red lines denote a 0.5 alpha value (indicative of additive genetic model), vertical green lines denote an alpha value of 1 (indicative of dominant genetic model). Note that some of the SNPs demonstrated alpha values outside the 0–1 range, indicative of under-recessive (α < 0) and over-dominant (α > 1) genetic models. The vertical black line denotes the Bonferroni significance threshold for this analysis (10,296 SNP-SNP models tested).

https://doi.org/10.1371/journal.pgen.1009534.g012

In the AMD interaction analysis, two SNP-SNP interaction models were significant when adjusting for the number of tests run (4,278 SNP-SNP models tested) (Fig 13 and Sheets A-E in S11 Table). The top SNP-SNP model for this phenotype was rs2336502 (alpha: 0.39; MAF: 0.33; pseudogene LOC100996886)–rs6695321 (alpha: 0.52; MAF: 0.36; intron, CFH), which was identified by the additive encoding (Bonferroni LRT p: 4.3×10⁻¹²), EDGE encoding (Bonferroni LRT p: 6.9×10⁻¹²), codominant encoding (Bonferroni LRT p: 6.8×10⁻¹⁰), dominant encoding (Bonferroni LRT p: 3.2×10⁻⁸), and recessive encoding (Bonferroni LRT p: 0.026) (r²: 0.0030). Another significant interaction model also included rs2336502, which was identified as interacting with rs5993 (alpha: 0.28; MAF: 0.26; intron, coagulation factor XIII B chain (F13B)) using the additive encoding (Bonferroni LRT p: 0.021) and EDGE (Bonferroni LRT p: 0.047) (r²: 0.00010). In the UKB replication, the interaction between rs2336502 (alpha: 0.55; MAF: 0.31) and rs6695321 (alpha: 0.52; MAF: 0.38) was significant (Sheets A-E in S12 Table) for the additive (LRT p: 1.7×10⁻⁴), codominant (LRT p: 0.0021), dominant (LRT p: 2.4×10⁻⁴), and EDGE (LRT p: 1.2×10⁻⁴) encodings (r²: 0.000047). Replication interaction for SNPs rs2336502 and rs5993 (alpha: 0.60; MAF: 0.16) was significant for the dominant encoding (LRT p: 0.016; r²: 0.040). There were no interaction results that met a Bonferroni corrected LRT p-value threshold for the interaction analysis of glaucoma.

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Fig 13. Interaction plot of the top results from the main effect filtered SNP-SNP interaction analysis for AMD.

The top 50 SNPs from the multi-encoding AMD GWAS were considered for a pairwise SNP-SNP interaction analysis using additive (red), dominant (green), recessive (blue), codominant (yellow) and EDGE (purple) encoding methods. Track one displays the EDGE-derived heterozygous alpha value of SNP 1, track 2 displays the alpha value for SNP 2, and track 3 displays the–log10 of the unadjusted LRT p-value of each SNP-SNP interaction model. Vertical blue lines denote a 0 alpha value (indicative of recessive genetic model), vertical red lines denote a 0.5 alpha value (indicative of additive genetic model), vertical green lines denote an alpha value of 1 (indicative of dominant genetic model). Note that some of the SNPs demonstrated alpha values outside the 0–1 range, indicative of under-recessive (α < 0) and over-dominant (α > 1) genetic models. The vertical black line denotes the Bonferroni significance threshold for this analysis (4,278 SNP-SNP models tested).

https://doi.org/10.1371/journal.pgen.1009534.g013

Simulated datasets

To assess the ability of EDGE to accurately assign heterozygous genotype values and identify SNP-SNP interactions across different genetic models with high power and low type I error, we developed the Biallelic Model Simulator (available at https://www.hall-lab.org/). This script generates two independent, biallelic SNPs in Hardy-Weinberg equilibrium according to given minor allele frequencies (for a further description of the simulation method, see S1 Text).

To evaluate multiple scenarios for epistasis to occur between a pair of SNPs [48], we simulated interactions with no main effect between SNP1 and SNP2 across pairwise combinations of underlying genetic models, including all two-way combinations between the following models: additive (ADD), dominant (DOM), recessive (REC), sub-additive (SUB), super-additive (SUP), and heterozygous (HET), referred to as “pairwise SNP-SNP” interaction models. Here, heterozygous genotypes are simulated to have half (ADD), the same (DOM), zero (REC), 25% (SUB), and 75% (SUP) the risk of homozygous alternate genotypes. Heterozygous genotypes for HET SNPs were simulated to have full risk relative to both homozygous genotypes. Additionally, we evaluated simulated interactions using genotype-base interaction models that include penetrance functions (e.g., XOR, Hyperbolic) and scenarios in which only one of the 9 interaction penetrance cells confers risk while the other 8 demonstrate no risk (e.g., Homozygous Referent-Homozygous Referent–HR-HR) (Table 3 lists all genotype-based models). To assess the impact of the aforementioned parameters, we performed a parameter test by simulating all models with every combination of the following: MAF: 0.05, 0.1, 0.3, 0.5; sample size: 2,000, 10,000, and 20,000; case-control ratio: 1:1 (balanced), 1:3, and 3:1 (unbalanced), and baseline risk.

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Table 3. Genotype-based simulated models.

https://doi.org/10.1371/journal.pgen.1009534.t003

To test for type I error, 1,000 null signal datasets were simulated for each combination of the parameters described above. In addition, to assess inflation driven by the encoding when models demonstrate main effects without interaction signal, we simulated datasets under two scenarios. The first involved simulating two SNPs, both exhibiting a main effect but having no interaction effect (“Two-SNP Main Effect”). The second involved simulating two SNPs in which only one SNP exhibited a main effect and with no interaction effect (“One-SNP Main Effect”). For the Two-SNP Main Effect models, underlying biological model combinations between REC, SUB, ADD, SUP, DOM, and HET were simulated. For the One-SNP Main Effect models, we simulated datasets allowing for only one SNP to exhibit the above biological models.

eMERGE datasets

Genome-wide genotyping was performed on approximately 55,000 samples (397 of Asian ancestry, 11,109 of African ancestry, 40,243 of European ancestry, 108 of Native American ancestry, and 3,167 of unknown ancestry) across the eMERGE II study sites at the Broad Institute and at the Center for Inherited Disease Research (CIDR) using the Illumina 660W-Quad or 1M-Duo BeadChips. In eMERGE, genetic data is imputed to 1000 genomes reference panel (March 2012 release). Imputation was performed on datasets separated by site and platform using IMPUTE2 on the phased genotyped data (SHAPEIT2 was used for phasing). The following sites were included for analysis: Geisinger (AMD, Glaucoma, RH, T2D), Group Health (AMD, Cataract, Glaucoma, RH, T2D), Marshfield (AMD, Cataract, Glaucoma, RH, T2D), Mayo (AMD, Cataract, Glaucoma, RH, T2D), Mount Sinai (RH, T2D), Northwestern (AMD, Glaucoma, RH, T2D), Vanderbilt (AMD, Cataract, Glaucoma, RH, T2D). Data were cleaned using the eMERGE QC pipeline developed by the eMERGE Genomics Working Group [28]. This process includes evaluation of sample and marker call rate, sex mismatch, duplicate and HapMap concordance, batch effects, Hardy-Weinberg equilibrium, sample relatedness, and population stratification. For all phenotypes, QC filters with 99% marker and sample call rates were applied. SNPs were LD pruned using an r-square threshold of 0.7.

For the T2D phenotype, there were 20,341 total samples (7,101 cases and 13,240 controls, 53% female), and following QC and LD pruning, 358,569 SNPs were available for analysis. The age-related cataract analysis included 6,815 total samples (5,104 cases and 1,711 controls, 56% females) and 333,898 post-QC SNPs. For glaucoma, 5,090 total samples (961 cases and 4,129 controls, 56% female) and 348,329 post-QC SNPs were considered. AMD included 13,153 total samples (2,167 cases and 10,986 controls, 54% female) and 311,161 post-QC SNPs. Finally, for resistant hypertension, 3,706 total samples (2,830 cases and 876 controls, 58% female) and 368,528 post-QC SNPs were included for analysis.

Replication datasets

Candidate replication SNP-SNP interaction analyses were performed on data from the UK Biobank (UKB) [30]. The UK Biobank contains genetic and phenotypic data on approximately 500,000 individuals. For genotypic quality, we removed 35,785 poor quality samples determined to be outliers for heterozygosity and/or missing rate as well as individuals found to be related based on a Pi-hat of 0.25. Originally genotyped to 96 million variants, we excluded variants that had an imputation info score less than 0.3 and pruned the data at an r-square threshold of 0.7. Additionally, we retained only individuals of European ancestry as inferred by UKB due to the skew in sample size of this ancestry group compared to others in UKB.

The PheWAS R package [49] was used to determine ICD-9 and ICD-10 based phecode phenotypes. The following phecodes were used in analysis: 362.2 degeneration of macula and posterior pole of retina (AMD), 366 cataract, and 401.1 essential hypertension, and 250.2 type 2 diabetes. Note that data was not available for resistant hypertension; however, hypertension was considered in UKB to explore potential similarities of epistatic signal across the two related phenotypes. A subject was considered a case for the phecode if they had one instance of an ICD code mapping to that phecode and a control if they had 0 instances of an ICD code mapping to that phecode and did not meet the exclusion criteria as defined in the PheWAS R package. The resulting datasets included 1,992 with AMD (286,169 controls), 16,574 with age-related cataract (264,905 controls), 16,621 with T2D (276,718 controls), and 70,089 with hypertension (219,326 controls) in UKB. If the exact SNP-SNP interaction did not replicate in UKB (as was the case for T2D and age-related cataract), we performed a region-based SNP-SNP interaction analysis, whereby interactions were considered between SNPs within a 50kb window upstream and downstream of the original SNP-SNP interaction models to allow for differences in LD structure and MAF across the datasets. To control for multiple tests in the region-based replication analyses, we applied a Bonferroni adjustment for the number of SNP-SNP models.

Statistical analyses

For all simulated and eMERGE datasets, regression modelling was performed using PLATO software [31], which employs EDGE, additive, dominant, recessive, and dominant encodings with user specification. Multi-encoding GWAS: In the eMERGE dataset, we performed four GWAS for each phenotype: each GWAS employing one of the traditional encodings (i.e., additive-encoded GWAS, dominant-encoded GWAS, recessive-encoded GWAS, and codominant-encoded GWAS) using logistic regression. We created pairwise scatterplots of the -log10 p-values from the results of each of the four traditional genetic encodings, additive, codominant, dominant, and recessive, to better visualize how concordant the p-values were across these encodings for the AMD and T2D phenotypes.

Interaction analysis: For both simulated and natural data, to determine the significance of a SNP-SNP interaction model above and beyond the main effects of both SNPs combined, we performed a likelihood ratio test (LRT) between the full (Y = β₀ + β₁SNP1 + β₂SNP2 + β₃SNP1×SNP2) and reduced (Y = β₀ + β₁SNP1 + β₂SNP2) models and derived an LRT p-value using each of the five encodings separately: additive, dominant, recessive, codominant, and EDGE. Select covariates were included in the regression models as well, as described below. In order to reduce computational and multiple testing burden in the eMERGE interaction analysis, a main effect filter was applied. So as to ensure that SNPs with diverse underlying genetic models were selected, the top 50 SNPs from each encoding were selected for subsequent interaction analysis. We chose this approach over a specified significance threshold so as not to bias any one encoding type if one encoding yielded more results from each multi-encoding GWAS. Pairwise combinations of interactions were tested between SNPs within the union of the top 50 most significant SNPs, and SNP-SNP LRT tests were performed with additive, dominant, recessive, codominant, and EDGE encodings separately.

Covariate adjustment. In eMERGE, for all phenotypes, regression models were tested while adjusting for sex, decade of birth, eMERGE site, genotyping platform, and BMI. To adjust for population stratification, PCA analysis was performed using Eigenstrat [50] and we included as covariates the first 6 (T2D and glaucoma), 3 (cataract and AMD), and 10 (resistant hypertension) principal components (number of principal components included as covariates varied across phenotypes due to differences in the amount of variance explained by principal components for each individual phenotype).

Testing the impact of parameters on power. To test the impact of allele frequency, sample size, penetrance, and case-control ratio on power, we performed a parameter test. All 29 interacting models were simulated with comprehensive combinations of each of the four parameters. After compiling all power results from simulated datasets of our parameter sweep, we performed ANOVA tests of the power by model for each encoding to identify effects of each parameter on power for every genetic model type.

Replication analyses

For replication analyses in UKB, we extracted SNP-SNP interaction models found to be significant in eMERGE for AMD, T2D, age-related cataract, and hypertension phenotypes. PLATO software was used to run logistic regression for each of these models, again considering each of the additive, codominant, dominant, recessive, and EDGE encodings separately. Each model was adjusted for age, sex, BMI, and principal components (first 10 UKB-generated PCs).