First, we compared MDR, FMDR, GMDR, and FGMDR in terms of their success rates (power) for causal SNPs detection, using various simulation data consisting of a continuous phenotype category and two case-control categories. In simulation data experiments, the FGMDR showed higher power than the others; consequently, we applied FGMDR to two real datasets for illustrations.
Simulation study
Simulation data consists of three categories of data, one continuous phenotype and two case-control categories. In the continuous phenotype categories (scenario 1), a phenotype variable is calculated by a linear sum of genetic effects, covariates, and error terms, to simulate a continuous phenotype such as blood pressure. In the first case-control category (scenario 2), a binary variable is 1 or 0, depending on whether or not a continuous value exceeds a certain threshold value. This type of data is generated for simulation of diseases whose status is determined by continuous variables, such as obesity. In the second case-control category (scenario 3), a binary variable is determined as a probability, based on a logit model with genetic effects, covariates, and error terms, for simulation of binary type diseases such as cancer.
Common to all three scenarios, genetic effects are based on 70 penetrance models (7 heritability values: 0.01~ 0.4, 2 minor allele frequency values: 0.2, 0.4 and 5 interaction models), without marginal effect [29], and power is defined as a proportion of how many times the true causal SNPs were selected as the model with the highest BAs (BA for MDR and GMDR, BAFUZZY for FMDR and FGMDR) among 100 replicates for a given model. Each replicate consists of 2000 samples (1000 cases and 1000 controls for case-control types), with genotype information for 100 SNPs, a covariate, and a phenotype. Genotype values of two causal SNPs are then determined by the minor allele frequency (MAF: 0.2, 0.4) of the penetrance models, and genotype values of non-causal SNPs, are randomly selected from a Hardy-Weinberg equilibrium, based on MAF values in [0.05, 0.5]. We tested various coefficients of disease models. In the results from tests, consistent patterns were seen. Therefore, in each scenario, coefficients of disease models are adjusted to about a 50~ 60% average success rate for all the methods.
Scenario 1
Since scenario 1 simulates a continuous phenotype, we used the following model with an identity link function Yi = α + XiTβ + ZiTγ + εi, where Yi represents a phenotype value, Xi represents a genetic effect, Zi represents covariates of the ith individual, and εi represents the error. Xi is randomly selected based on normal distribution of the mean: a penetrance value corresponding to the genotype value of the ith individual and standard deviation 0.1. Zi is randomly selected on a normal distribution of mean 0, and standard deviation 0.7. εi is randomly selected on a normal distribution of mean 0, and standard deviation 1. All values of coefficients (β, γ) are the same as one. Simulation results of scenario 1 are summarized in Fig. 1.
In Fig. 1, the powers of MDR and FMDR are lower than that of others, because they cannot use information in covariates. In other words, covariates are useful for causal SNP detections in genetic association studies. Then, FGMDR shows higher power than that of GMDR, in some penetrance models, or similar power. In terms of the average power, MDR was 0.427, FMDR was 0.433, GMDR was 0.611 and FGMDR was 0.621. Additionally, we performed significance testing of power comparisons, using the Wilcoxon signed-rank test. The power of FGMDR is significantly higher than that of MDR (p-value: 2.82e-10), FMDR (p-value: 5.99e-11 and GMDR (p-value: 2.59e-02).
Scenario 2
Since scenario 2 simulates a binary phenotype determined by a continuous value, we calculated a continuous value, and discretized as 1 for higher than a specific threshold (a median of the continuous values), or 0 for the others. For a continuous value calculation, we used the same identity link function in GLM as scenario 1. All other parameter values, except the standard deviation of error (0.5), are the same as in scenario 1. The simulation results of scenario 2 are summarized in Fig. 2.
In Fig. 2, similar patterns as in Fig. 1, are shown. The powers of the MDR and FMDR, for scenario 2, are lower than that of others in many penetrance models, and it means importance of covariates in case-control association studies. Among the other two methods, the FGMDR showed higher than that of GMDR, in some penetrance models. In terms of the average power, MDR was 0.545, FMDR was 0.555, GMDR was 0.606 and FGMDR was 0.616. Wilcoxon signed-rank tests showed that the mean power of FGMDR was significantly higher than that of MDR (p-value: 1.35e-07), FMDR (p-value: 2.26e-06) and also higher than that of GMDR, but not significantly (p-value: 5.15e-02).
Scenario 3
Since scenario 3 simulates a binary phenotype with a probability using a logit model given below:
$$ \ln \left(\frac{p\left({Y}_i=1\right)}{1-p\left({Y}_i=1\right)}\right)=\upalpha +{X_i}^T\beta +{Z_i}^T\gamma +{\varepsilon}_i. $$
In this scenario, the value of β is reduced to 0.5, and the standard deviation of the error increased to 2. The simulation results of scenario 3 are summarized in Fig. 3.
The simulation results of scenario 3 in Fig. 3 show some interesting patterns, compared to the previous results. Here, the order of power (MDR < FMDR < GMDR < FGMDR) was consistently similar with previous results, and the power of all the methods increased in both the heritability and MAF values. In terms of the average power, MDR was 0.473, FMDR was 0.487, GMDR was 0.519, and FGMDR was 0.533. In the Wilcoxon signed-rank tests, the power of FGMDR was significantly higher than that of MDR (p-value: 1.31e-07), FMDR (1.36e-07) and GMDR (p-value: 1.94e-03).
Real data experiments
Crohn’s disease (CD)
The CD data in Wellcome Trust Case Control Consortium [28] dataset, consists of 1949 cases and 3004 controls. For each individual, genetic information for about 500,000 SNPs, age information (in decades), and sex were provided. However, all values of the age information in the case samples, were the same value. Therefore, we used only sex as a covariate. For adapting our FGMDR method to analyze CD data, residuals were calculated, using the logistic regression model, with sex as a covariate and odds ratio of sex is 1.47 (95% confidence interval: 1.31–1.65, p-value of likelihood ratio test: 6.93E-11). Among SNPs, we selected 30 SNPs reported to associate with the CD phenotype [28, 30, 31] for illustration, and the basic characteristics of those SNPs, are summarized in Table 1. P-values and their rank of Table 1 were calculated by likelihood ratio test, under a codominant model with two degrees of freedom.
Table 1 Basic characteristics of each SNPs for CD We next performed FGMDRs with/without covariate adjustment, with 10-fold cross validation, from two to five-locus SNP combinations, as summarized in Table 2. FGMDR without covariate adjustment is performed to investigate the effect of covariate adjustment. SNP5 was consistently included the best SNP combinations from two to five-locus SNP combinations in the results of FGMDR with covariate adjustment, while SNP5 was included only for three and four-locus SNP combinations in the results of FGMDR without covariate adjustment. SNP combinations in three and four-locus models are the same in results of FGMDR with/without covariate adjustment but they are different in two and five-locus models. CVC values in the FGMDR with covariate adjustment are higher than or similar to that of FGMDR without covariate adjustment. BA values are similar in FGMDR with covariate adjustment and FGMDR without covariate adjustment regardless of training or testing data.
Table 2 Results of CD data analysis Among these results, we selected a four-locus (order: 4) SNP combination as the best SNP combination, based on the best BAFUZZY in testing data, and its relatively high cross-validation consistency (CVC) value. Interaction of this SNP combination is represented in Fig. 4.
In Fig. 4, upper case letters denote major alleles, while and lower case letters denote minor alleles. ‘A’ or ‘a’ represent the genotypes of the first SNP in the SNP combination; ‘B’ and ‘b’ represent the genotypes of the second SNP, and so on. The left bar-labeled value represents the sum of the positive residuals, while the right bar-labeled value represents sum of the negative residuals. The green background colored cells mean the membership value that cells are close to 0, and the red background colored cells mean the membership value that cell are close to 1. The dark background color means the value is far from 0.5 (i.e., closer to 1 or 0), while the white background color denotes a 0.5 membership value.
Figure 4 shows some interesting patterns for interpretation of the interaction. First, with respect to the diagonal line, most of the cells in the right top quadrant had red background, while most of the cells in the lower-left quadrant cells were green background. Based on these observations, it seems that an additive risk pattern increased from left to right, and from top to bottom. However, genotype patterns represent combinations of two SNPs with vertical and horizontal genotypes. For example, in vertical genotypes, there are genotype combinations of SNP1 and SNP2. However, note that the order of the SNP combination is important for interpretation. For example, (SNP1, SNP2) seemed to be additive in effect, while (SNP2, SNP1) didn’t suggest an additive effect. A possible interpretation of interaction between SNP1 and SNP2 is that the risk of CD is dominated by SNP1 minor allele at first and SNP2 then affects CD risk for each genotype sample of SNP1. Second, interaction patterns of SNP2 and SNP8 are not consistent for each genotype combination of SNP1 and SNP 5. The interaction patterns of SNP2 and SNP8 are represented by separated blocks, consisting of 3 × 3 cells. For example, the color pattern of the top left block (SNP1, SNP5) = (AA, CC) is different from all other blocks.
Additionally, the interaction between IR23R (SNP1) and ATG16L1 (SNP5) for CD, was reported and in a case-control study within a cohort study [32], and reviewed for explanation of CD mechanism [33]. However, we cannot find direct evidence of interactions of these particular SNPs in the four-SNP combination.
Homeostatic model assessment of insulin resistance (HOMA-IR)
We next analyzed HOMA-IR data from the Korea Association REsource project (KARE) to illustrate FGMDR in the context of quantitative traits. A total of 8577 samples are available, after removing subjects with at least one missing phenotype value. The genomic DNAs were genotyped using Affymetrix Genome-Wide Human SNP Array 5.0. For GGI analysis using our FGMDR, we used only 10 candidate SNPs identified in earlier studies [34,35,36] from the single SNP GWAS analysis. The basic characteristics of these SNPs are summarized in Table 3. P-values and their rank in Table 3, were calculated by likelihood ratio test, under a codominant model with two degrees of freedom. Since the distribution of HOMA-IR is skewed, many researchers perform a log-transformation before applying the regression analysis [34], and we did likewise. Sex, age, area, and BMI were used as environmental covariates. Then, the regression model for FGMDR is given by
$$ \log \left(\mathrm{HOMA}-\mathrm{IR}\right)={\beta}_0+{\beta}_1{Sex}_i+{\beta}_2{Age}_i+{\beta}_3{Area}_i+{\beta}_4{BMI}_i+{\varepsilon}_i. $$
(2)
Using the residuals calculated from (2), FGMDR was then performed.
Table 3 Basic characteristics of each SNPs for HOMA-IR We performed FGMDRs with/without covariate adjustment, with 10-fold cross validation from two to five-locus SNP combinations and summarized the results of HOMA-IR in Table 4. All best SNP combinations included SNP 5 except 5 locus SNP combination with covariate adjustment, consistent with its p-value in Table 3 (the lowest p-value and rank: 1). In the results of FGMDR without covariate adjustment, SNPs in lower order SNP combination models are included in higher order SNP combination models. In addition, BA values of FGMDR with covariate adjustment are higher than those of FGMDR without covariate adjustment in both training and testing data. These differences may be caused by covariate adjustment. Similar to the results of CD data analysis, SNP combinations identified by FGMDR with covariate adjustment are different from those by FGMDR without covariate adjustment. While a further biological investigation is required, we expect that the covariate adjustment makes not only a performance improvement but also a more accurate identification of true causal SNP interactions.
Table 4 Results of HOMA-IR data analysis Among the results of FGMDR, we selected the four-locus SNP combination as the best SNP combination based on BAFUZZY in testing data and CVC. Three SNPs in the selected SNP combinations except SNP10 are located in ROR1, JAK1, and nearby SOCS5 (about 19.8 k BP). For these three genes, several biological evidences of interactions are pre-reported: 1) ‘Jak1 has previously been implicated in adipocyte insulin resistance.’ [37], 2) ‘Most of the known SOCS proteins are involved in the modulation of the development of insulin resistance.’ [38], 3) ‘When JAK1 and SOCS5 are co-expressed in cells, JAK1 is continually being phosphorylated and de-phosphorylated during the course of the transfection, and SOCS5 presumably interacts with active (phosphorylated) JAK1 to inhibit further enzymatic activity’ [39], 4) ‘ROR1 was shown to interact with and be inhibited by resistin.’ [40], 5) ‘Resistin is also correlated with insulin resistance.’ [41].
CVCs decreased by increase of order, in both Tables 2 and 4. This is a general phenomenon in multi-locus association tests. For example, among 30 SNPs, there are 435 possible two-locus SNP combinations and 2610 possible three-locus SNP combinations. An interesting point is relatively low BAFUZZY in testing. These BAFUZZY values are not directly comparable to ordinary BA because the Fuzzy set theory has been implemented. BAFUZZY is more concentrated near 0.5, compared to ordinary BA. Nevertheless, BAFUZZY values in testing HOMA-IR data were lower than those of CD. This seems to be caused by heritability differences. The heritability of CD is 53% [42] but the heritability of HOMA-IR is 8% in black and Spanish populations [43], and 22% in Asian Indian families [44].