Abstract
Background
One strategy for addressing missing heritability in genomewide association study is genegene interaction analysis, which, unlike a single gene approach, involves highdimensionality. The multifactor dimensionality reduction method (MDR) has been widely applied to reduce multilevels of genotypes into high or low risk groups. The CoxMDR method has been proposed to detect genegene interactions associated with the survival phenotype by using the martingale residuals from a Cox model. However, this method requires a crossvalidation procedure to find the best SNP pair among all possible pairs and the permutation procedure should be followed for the significance of genegene interactions. Recently, the unified model based multifactor dimensionality reduction method (UMMDR) has been proposed to unify the significance testing with the MDR algorithm within the regression model framework, in which neither crossvalidation nor permutation testing are needed. In this paper, we proposed a simple approach, called Cox UMMDR, which combines CoxMDR with the key procedure of UMMDR to identify genegene interactions associated with the survival phenotype.
Results
The simulation study was performed to compare Cox UMMDR with CoxMDR with and without the marginal effects of SNPs. We found that Cox UMMDR has similar power to CoxMDR without marginal effects, whereas it outperforms CoxMDR with marginal effects and more robust to heavy censoring. We also applied Cox UMMDR to a dataset of leukemia patients and detected genegene interactions with regard to the survival time.
Conclusion
Cox UMMDR is easily implemented by combining CoxMDR with UMMDR to detect the significant genegene interactions associated with the survival time without crossvalidation and permutation testing. The simulation results are shown to demonstrate the utility of the proposed method, which achieves at least the same power as CoxMDR in most scenarios, and outperforms CoxMDR when some SNPs having only marginal effects might mask the detection of the causal epistasis.
Background
Many statistical methods in genomewide association studies (GWAS) have been developed to identify susceptibility genes by considering a single SNP at a time. Since the first published GWAS on agerelated macular degeneration [1], the GWAS Catalog has come to contain 60,000 unique SNPtrait associations based on 3300 publications as of February of 2018 (www.ebi.ac.uk/gwas). However, the effective sizes of the loci identified via GWAS are relatively small, and these individual loci may not be useful in assessing risk in personal genetics, as pointed out by Moore and Williams [2] and Manolio [3]. Furthermore, only a small proportion of heritability has been explained, leading to the missing heritability problem [4].
In order to overcome the missing heritability, the singlelocus approach has been moved into genegene interaction analysis because complex diseases might be associated with multiple genes and their interactions [3]. However, the study of genegene interactions in GWAS involves the challenge of higherorder dimensionality, which Ritchie et al. [5] proposed circumventing using the multifactor dimensionality reduction (MDR) method, now commonly used to analyze genegene interactions in genetic studies [6, 7]. MDR reduces multidimensional genotypes into onedimensional binary attributes, in which multilevel genotypes of SNPs are classified into either high or low risk groups, using a ratio of cases and controls. The MDR algorithm then finds the best pair of SNPs among all possible SNP combination, yielding the maximum balanced accuracy through crossvalidation. The MDR mechanism can apply higherorder interactions such as twoway, threeway and so forth because all combinations of multiway interactions can be reduced to either high or low risk groups using the appropriate classification rules. Many modifications and extensions to MDR have been published by generalizing the classification rules and phenotypes, including the use of odds ratios [8], loglinear models [9], a generalized multifactor dimensionality reduction method (GMDR) for generalized linear models [10], methods for imbalanced data [11], modelbased multifactor dimensionality reduction methods (MBMDR) [12] and quantitative multifactor dimensionality reduction (QMDR) for the continuous response variables [13].
On the other hand, for a prospective cohort study, the MDR concept has been also extended to investigate those genegene interactions associated with the survival time. Since the first approach, called SurvMDR, was proposed by Gui et al. [14], both CoxMDR [15] and AFTMDR [16] have been developed for the survival phenotype. These methods extend the MDR algorithm to the survival time by using alternative classification rules, which are more applicable to survival data. For example, the classification rule for SurvMDR corresponds to a logrank test statistic whereas those for CoxMDR and AFTMDR correspond to a martingale residual of a Cox model and a standardized residual of an accelerated failure time (AFT) model, respectively. In addition, comparing the performance of these three methods, both CoxMDR and AFTMDR have greater power in identifying genegene interactions than SurvMDR when there is a confounding covariate, whose confounding effect can be adjusted for under CoxMDR and AFTMDR in the frame of regression model. Whereas SurvMDR is nonparametric and no covariate effect can be adjusted for [17].
However, the MDR algorithm requires crossvalidation to identify the best multilocus model among all possible combinations of SNPs and further implements computationally intensive permutation testing to check the significance of the selected multilocus model. A variety of classification rules has been proposed but the intensive computational procedure for crossvalidation and permutation testing should be implemented as done in the original MDR method.
Recently, the UMMDR method has been proposed to address this issue by unifying the significance test with the MDR algorithm using regression model [17]. UMMDR provides the significance test for the multilocus model by introducing an indicator variable for the high risk after classification. It also allows a variety of classification rules and phenotypes.
In this paper, we proposed a simple approach, called Cox UMMDR, which combines CoxMDR with UMMDR. We compared it with CoxMDR by simulation studies. We also applied the proposed method to a real dataset of Korean leukemia patients and concluded with a discussion.
Methods
As described in Yu et al. [18], the UMMDR method was proposed to avoid the intensive computing procedure for achieving the significance of a multilocus model. To this, they proposed a twostep unified model based MDR approach, in which multigenetic levels were classified into high and low risk groups and an indicator variable for high risk group was defined in the first step, and then the significance of multilocus model was achieved in the regression model with an indicator variable as well as adjusting covariates in the second step. The key idea of UMMDR is to unify the algorithm of MDR and the significance testing of multilocus model by using an indicator variable for high risk group. UMMDR allows different types of traits and evaluation of the significance of existing MDR methods.
In this paper, we extended UMMDR to the survival phenotype using CoxMDR. In the first step of Cox UMMDR, we classify the multilevel genotypes into high or low risk groups by using the martingale residual of a Cox model with only the baseline hazard function. We then define an indicator variable, S, taking 1 for the highrisk group and 0 for the lowrisk group. In the second step, we fit a Cox model given as follows:
Here λ_{0}(t) is a baseline hazard function, S is an indicator variable for the highrisk group and Z is the vector coding for the adjusting covariates, β and γ are the corresponding parameters to S and Z, respectively. By testing the null hypothesis of H_{0} : β = 0, we investigate whether the corresponding multilocus is associated with the survival time after adjusting for covariate effects. In order to test the significance of multilocus model, we used the Waldtype test statistic, \( W={\widehat{\beta}}^2/V\widehat{a}r\left(\widehat{\beta}\right) \), whose asymptotic distribution is not the central chisquare distribution under the null hypothesis [12]. This is because the expected value of the estimate of β is not equal to zero under the null hypothesis, which is owing to the fact that S represents a highrisk group classified by the martingale residual in the first step. In other words, the asymptotic null distribution of \( \sqrt{W} \) has a nonzero mean due to the classification step and the asymptotic distribution of its squared statistic, W, follows the noncentral chisquare distribution with one degree of freedom and the noncentrality parameter q. Since the mean of the noncentral chisquare distribution is q + 1, we can estimate the noncentrality parameter as, \( \widehat{q}=\max \left(0,\widehat{\mu}1\right) \) where \( \widehat{\mu} \) is the estimator for the mean of W under the null distribution. In order to estimate q, we permuted the trait a few times, say 5 times, and took the sample mean for statistic W as \( \widehat{\mu} \). We can estimate the noncentrality parameter for each multilocus model or pool all the statistics and then estimate the common noncentrality parameter for all multiloci models as mentioned in [18].
Through intensive simulation studies, we compared the performance of Cox UMMDR with that of CoxMDR without and with adjusting for marginal effects. We considered two diseasecausal SNPs among 10 unlinked diallelic loci with the assumption of HardyWeinberg equilibrium and linkage equilibrium. For the covariate adjustment, we consider only the one covariate which is associated with the survival time but has no interactions with any SNPs. We generated simulation datasets from different penetrance functions [11], which define a probabilistic relationship between the high or low risk status of groups and SNPs. We then considered 14 different combinations of two different minor allele frequencies of (0.2, 0.4) and seven different heritability of (0.01, 0.025, 0.05, 0.1, 0.2, 0.3, 0.4). For each of 14 heritabilityallele frequency combinations, a total of five models were generated, yielding 70 epistasis models with various penetrance functions, as described in [11] (supplemental Table 1).
Let f_{ik} be an element from the i^{th} row and the k^{th} column of a penetrance function. Assuming that SNP1 and SNP2 are the two diseasecausal SNPs, we have the following penetrance function:
We generated 400 patients from each of 70 penetrance models to create one simulated dataset and repeated this procedure 100 times. We simulated the survival time from a Cox model specified as follows:
Here x is an indicator variable with value 1 for the highrisk group and 0 for the lowrisk group. We set α = 1.0,γ = 1.0 and z as an adjusting covariate generated from N(0, 1). In addition, the baseline hazard function follows a Weibull distribution with a shape parameter of 5 and a scale parameter of 2, the censoring time being generated from a uniform distribution, U(0,c) depending on the censoring fractions which have four different censoring fractions of (0.0, 0.1, 0.3, 0.5).
For the power comparison, we consider two different scenarios for the simulation study. First, we conducted the power comparison when there is no marginal effect of SNPs. Under this scenario, the survival times are generated from the Cox model as follows:
where α = 1.0, γ = 1.0.
Secondly, we compared the power of Cox UMMDR with that of CoxMDR when there is a marginal effect of SNPs. Under this scenario, the survival times are generated from the given Cox model as follows:
where α = 1.0,γ = 1.0,δ = 0.5, and δ denotes the marginal main effect of SNP3 on the hazard rate.
Results
Simulation results
We first considered whether the type I error is controlled under the null hypothesis. For type I error, the simulation data sets were iteratively generated 1000 times under the null hypothesis of no genetic effect model across 5 different MAFs and 4 different censoring fractions. The raw type I error was calculated without adjusting the noncentrality of the asymptotic chisquare distribution while the corrected type I error was calculated by adjusting the noncentrality. As shown in Table 1, the raw type I error is not controlled and increases as the minor allele frequency increases while the corrected type I error is wellcontrolled regardless of MAF. Here, we tried 5 times permutation for estimating the noncentrality because the number of permutations did not affect the test statistics, W. In addition, Fig. 1 displays QQ plots for the uncorrected (raw) and corrected type I errors, respectively.
For the power comparison, 100 simulated datasets for each of the 70 models were generated including two diseasecausal SNPs. The power of Cox UMMDR is defined as the percentage of times that the corrected (after Bonferroni correction) pvalue for testing the significance of the indicator variable S is less than or equal to the nominal size, called PBonf, as referred in [18]. On the other hand, the power of CoxMDR is defined as the percentage of times that CoxMDR correctly chooses the two diseasecausal SNPs as the best model out of each set of 100 datasets for each model. This is because the significance of the best pair of SNPs selected by CoxMDR can only be obtained by permutation testing. Therefore, the power of CoxMDR may not be comparable with PBonf in terms of the evaluation measure. For a fair comparison, the alternative power of Cox UMMDR is defined similarly as that of CoxMDR, being the percentage of times that the causal model is ranked first by the corrected pvalue, called PRank, as referred in [18]. We compared the PBonf and PRank of Cox UMMDR with the power of CoxMDR.
As mentioned in the previous section, we considered two different scenarios, with and without the marginal SNP effects.
Under the first scenario in which no marginal SNP effect is considered, we classified the multigenetic genotypes into high and low risk groups using this martingale residual of a Cox model with only the baseline hazard function and define S as 1 for a highrisk group and 0 otherwise. Next, we fit the following Cox model:
In the model above, we tested the null hypothesis H_{0} : β = 0 which means that there is no significant multilocus effect associated with the survival phenotype. If this null hypothesis is rejected, it implies that there is a significant genegene interaction associated with the survival time. We overlaid the three different power curves related to Cox UMMDR and CoxMDR as shown in Fig. 2, in which the xaxis represents 70 models ordered by the values of 2 different MAF and 7 different heritabilities. Since there are 5 models available for each combination of MAF and heritability, a total of 70 different powers are plotted consecutively on the xaxis, in which 14 different points represents the heritability within each MAF. The power results show a consistent trend in that PRank of Cox UMMDR is always greater than PBonf of Cox UMMDR and the power of CoxMDR. Under no censoring, the PRank of Cox UMMDR is similar to the power of CoxMDR but the PRank of Cox UMMDR is greater than the power of CoxMDR as the censoring fraction increases. In general, the power trend is consistent in the sense that it is more powerful for MAF = 0.2 than MAF = 0.4. The power increases as the heritability increases but decreases as the censoring fraction increases. However, the PRank of Cox UMMDR seems robust even under heavier censoring than 0.5 whereas the power of CoxMDR decreases rapidly when the censoring fraction is heavier than 0.5.
Under the second scenario which marginal SNP effect is considered, we classified the multigenetic genotypes into high and low risk groups using this martingale residual of a Cox model with only the baseline hazard function and define S as 1 for a highrisk group and 0 otherwise. Next, we fitted the following Cox model:
where SNP1 and SNP2 represent the main effects of SNPs attributed to the definition of S. In the model above, we tested the null hypothesis of H_{0} : β = 0, which means that there is no significant multilocus effect associated with the survival phenotype. Figure 3 displays the three different power curves overlaid. As shown in Fig. 3, the PRank of Cox UMMDR is always largest and the PBonf of Cox UMMDR is rank second in relation to the power of CoxMDR. The general trend of these three power curves is the same as that without considering the marginal effect in terms of MAF, heritability and the censoring fraction. However, it is noted that the power of CoxMDR is very low for almost all cases, which implies that the twoway interaction effect between SNPs can hardly be discriminated from the main marginal effect. On the other hand, Cox UMMDR can detect the twoway interaction effect in the unified model by controlling the main effects of SNPs. As shown in Figs. 2 and 3, the PRank of Cox UMMDR has reasonable power when the heritability is larger than 0.2 and seems to be robust to the censoring fraction regardless of considering the main effect of SNPs. In addition, when we compared the CPU time for the power calculation, Cox UMMDR takes 146 s for fitting one model whereas CoxMDR takes 1600 s, which implies that Cox UMMDR is almost 10 times faster than CoxMDR.
Real data analysis
We applied the Cox UMMDR procedure to analyze real leukemia patient data and compared the results with those obtained by CoxMDR. This real dataset of 97 AML patients who had been followedup which have age, sex and genetic information of 139 SNPs. At the end of the study, there were 40 deaths and 57 patients still alive. We considered two adjusting covariates, age and sex, in detecting genegene interaction associated with the survival time.
To take into account the marginal effect of SNP, we first fitted a univariate Cox model with each SNP adjusting for age and sex. We found that 21 SNPs had a significant marginal effect on the survival time. To summarize the marginal effects of 21 SNPs, we implemented the principal component analysis and took the two principal components (PC) as a covariate, which account for 78% of variation. We considered the four different models in identifying genegene interactions by Cox UMMDR as follows:

(1)
PC unadjusted and main effects of SNP1 and SNP2 unadjusted:

(2)
PC adjusted and main effects of SNP1 and SNP2 unadjusted:

(3)
PC unadjusted and main effects of SNP1 and SNP2 adjusted:

(4)
PC adjusted and main effects of SNP1 and SNP2 adjusted:
The Venn diagram in Fig. 4 shows the number of SNP pairs that have a pvalue less than 0.05 for testing H_{0} : β = 0 without adjusting multiple testing by the four models above. As shown in the Venn diagrams, 640 pairs, 279 pairs, 492 pairs and 432 pairs are detected by models (1), (2), (3) and (4), respectively. More SNP pairs are detected when the PC effect is unadjusted rather than adjusted in the model, for example, (640, 492) vs. (279, 432). The adjusting effect of PC seems more substantial when the main effect of SNPs is unadjusted since the number of SNP pairs decreases from 640 to 279. However, the adjusting effect of PC is not critical when the main effect of SNPs is adjusted because the number of SNP pairs decreases from 492 to 432. As shown in Figs. 4, 68 SNP pairs are overlapped by all four models, which imply that 68 multilocus models might be significant with the survival phenotype regardless of the adjusting factors. For these 68 multilocus we investigate whether the interaction effect of the corresponding SNP pairs was statistically significant or not by testing the interaction coefficient under the Cox model given as follows:
Among the 68 SNP pairs, only 16 pairs provided statistically significant interaction effects with a pvalue less than 0.05, which implies that Cox UMMDR may yield more false positive results than Cox regression model. The genegene interaction effect can be described in various terms, for example, using a semiparametric model like a Cox model or a nonparametric approach like Cox UMMDR and so forth. The more important point is that the interaction effect detected by the statistical method should be interpreted from a biological point of view. However, it is not easy to connect the statistical significance directly to the biological findings.
Among the 16 pairs, we selected the top two SNP pairs and compared these with the top two SNP pairs detected by CoxMDR. Since the Cox model is commonly used to explain the association between risk factors and survival time, we compared the power of both Cox UMMDR and CoxMDR by significance testing for the interaction effects in a Cox model. Table 2 shows the pvalues for testing the interaction effects for the selected SNP pairs by both Cox UMMDR and CoxMDR methods, respectively. The comparison is valid since both Cox UMMDR and CoxMDR share a common algorithm for classifying the multilevel genotypes into high and low risk groups. Cox UMMDR shows greater power in detecting the epistasis between two SNPs than does CoxMDR. As shown in Table 2, the top two pairs of (rs747199, rs2847153) and (rs1960207, rs1004474) selected from Cox UMMDR have significant interaction effects (p = 0.008 and 0.005), respectively. On the other hand, the top two pairs of (rs12404655, rs1004474) and (rs532545, rs2847153) show no significant interaction effect (p = 0.098 and 0.591), respectively. It is interesting to note that both (rs2847153) and (rs1004474) are selected simultaneously as one part of a SNP pair by both Cox UMMDR and CoxMDR but the interaction effect for the corresponding SNP pairs is determined by the other part of SNP pair such as (rs747199) and (rs1960207) by Cox UMMDR. Although CoxMDR selects the two pairs of (rs12404655, rs1004474) and (rs532545, rs2847153) as the best, the interaction effect of these pairs is not found to be significant in the Cox regression model. This is one of drawbacks of CoxMDR method, in which it cannot be guaranteed that the best SNP pairs are statistically significant without permutation testing.
In addition, we investigated how well the high and low risk groups can be classified by the SNP pairs attributed by Cox UMMDR. To this end, we fitted a Cox model given in (1) with the attributed SNP pairs and calculated a risk score from the fitted model. We then classified all subjects into high and low risk groups based on the median risk score and tested the equivalence of the survival curves of these two groups by a logrank test. We found significant logrank test results with very low pvalues for all 68 SNP pairs. Figure 5 displays four plots which include the survival curves of highrisk and lowrisk groups attributed by (rs747199, rs2847153), (rs1960207 and rs1004474), (rs12404655, rs1004474) and (rs532545, rs2847153), respectively. As shown in Fig. 5, the two survival curves of high and low risk groups are significantly separated, with almost zero pvalues. Furthermore, we investigated the effect of SNPs on the significant separation of these two survival curves by comparing the change of the logrank test statistics. As displayed in Table 3, the logrank test statistic of the no SNP effect model is 12.798, which means that two survival curves are significantly separated by both age and sex. By adding the SNP pairs attributed by Cox UMMDR, the logrank test statistic increases to 18.341 and 20.672, respectively, which show more powerful result. However, for the SNP pairs attributed by CoxMDR, the logrank test does not guarantee more powerful result because one of SNP pairs yields the lower logrank test statistic of 8.976 whereas the other case provides the logrank test statistic of 17.278. For all 16 SNP pairs showing the significant interaction effects, which are attributed by Cox UMMDR, the power of the logrank test is always greater than that under the model only with age and sex (data not given here). It would be said that the multilocus model identified by Cox UMMDR performs better in detecting the high risk group.
Discussion
In this paper, we proposed a simple approach, called Cox UMMDR, which combines the classification rule of CoxMDR with the testing procedure of UMMDR for the survival phenotype. Through the intensive simulation study, we compared the power of Cox UMMDR with that of CoxMDR. The simulation results show that Cox UMMDR is more powerful than CoxMDR and is robust to the censoring fraction. Furthermore, we applied the proposed method to a real dataset of Korean leukemia patients and compared the results with those of CoxMDR. We found that the results from Cox UMMDR are more consistent than those from CoxMDR in that the interaction effect of SNP pairs identified by Cox UMMDR is statistically significant, whereas those identified by CoxMDR show no significant interaction effect in a Cox model. In addition, the multilocus model identified by Cox UMMDR improves the power in detecting the high risk group by a logrank test.
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Funding
This research was supported by the Basic Science Research Program through the National Research Foundation (NRF) funded by the Ministry of Science, ICT & Future Planning (2016R1D1A1B03934908) and (2017M3A9C4065964), and by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (HI16C2037).
Availability of data and materials
The real data of Korean leukemia patients is not available. The Rprogram for Cox UMMDR is available at http://github.com/leesy80/Seungylee.
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SYL and TSP conceived the study and designed the simulation study. SYL wrote the manuscript. DHS, YKK and WBY implemented the simulation program and analysed a real data. All authors read the paper and approved the final manuscript.
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Lee, S., Son, D., Kim, Y. et al. Unified Cox model based multifactor dimensionality reduction method for genegene interaction analysis of the survival phenotype. BioData Mining 11, 27 (2018). https://doi.org/10.1186/s1304001801891
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DOI: https://doi.org/10.1186/s1304001801891
Keywords
 Survival time
 Cox model
 Multifactor dimensionality reduction method
 Genegene interaction
 Unified model based method