- Open Access
Multivariate generalized multifactor dimensionality reduction to detect gene-gene interactions
© Choi and Park; licensee BioMed Central Ltd. 2013
- Published: 13 December 2013
Recently, one of the greatest challenges in genome-wide association studies is to detect gene-gene and/or gene-environment interactions for common complex human diseases. Ritchie et al. (2001) proposed multifactor dimensionality reduction (MDR) method for interaction analysis. MDR is a combinatorial approach to reduce multi-locus genotypes into high-risk and low-risk groups. Although MDR has been widely used for case-control studies with binary phenotypes, several extensions have been proposed. One of these methods, a generalized MDR (GMDR) proposed by Lou et al. (2007), allows adjusting for covariates and applying to both dichotomous and continuous phenotypes. GMDR uses the residual score of a generalized linear model of phenotypes to assign either high-risk or low-risk group, while MDR uses the ratio of cases to controls.
In this study, we propose multivariate GMDR, an extension of GMDR for multivariate phenotypes. Jointly analysing correlated multivariate phenotypes may have more power to detect susceptible genes and gene-gene interactions. We construct generalized estimating equations (GEE) with multivariate phenotypes to extend generalized linear models. Using the score vectors from GEE we discriminate high-risk from low-risk groups. We applied the multivariate GMDR method to the blood pressure data of the 7,546 subjects from the Korean Association Resource study: systolic blood pressure (SBP) and diastolic blood pressure (DBP). We compare the results of multivariate GMDR for SBP and DBP to the results from separate univariate GMDR for SBP and DBP, respectively. We also applied the multivariate GMDR method to the repeatedly measured hypertension status from 5,466 subjects and compared its result with those of univariate GMDR at each time point.
Results from the univariate GMDR and multivariate GMDR in two-locus model with both blood pressures and hypertension phenotypes indicate best combinations of SNPs whose interaction has significant association with risk for high blood pressures or hypertension. Although the test balanced accuracy (BA) of multivariate analysis was not always greater than that of univariate analysis, the multivariate BAs were more stable with smaller standard deviations.
In this study, we have developed multivariate GMDR method using GEE approach. It is useful to use multivariate GMDR with correlated multiple phenotypes of interests.
- Generalize Estimate Equation
- Multifactor Dimensionality Reduction
- Generalize Estimate Equation Model
- Continuous Phenotype
- FBN1 Gene
Genome-wide association studies (GWAS) have been successfully conducted to detect disease susceptibility genes for common complex human diseases by focusing on associations between single-nucleotide polymorphisms (SNPs) and phenotypes . While traditional methods for GWAS consider only one SNP at a time, some common complex human diseases such as diabetes, hypertension, and various types of cancers are known to be influenced by multiple genetic variants . In addition, one of the greatest challenges in GWAS is to discover gene-gene and/or gene-environment interactions.
Classic logistic regression can be used to analyze the gene-gene interaction . However, logistic regression suffers from an overfitting problem in high-order interactions . Multifactor dimensionality reduction (MDR) method is a nonparametric, model-free, and combinatorial approach for interaction analysis by identification of a multi-locus model for association in case-control studies [5–9]. MDR method reduces multi-locus genotypes into two disease risk groups: high-risk and low-risk groups. If the ratio of cases and controls in a combination of genotypes is larger than a pre-assigned threshold T (e.g., T = 1), the cell of combination is labelled as "high risk", otherwise, "low risk". MDR method shows greater power for testing high-order interactions compared with logistic regression analysis . Several statistical methods have been extended from MDR approach [11–16]. One of the extended methods of MDR is a generalized MDR (GMDR) proposed by Lou et al. . GMDR method allows adjusting for covariates and applying to both dichotomous and continuous phenotypes; it uses the score-based statistic obtained from generalized linear model of phenotypes on the predictor-variable and covariates instead of the ratio of cases and controls in original MDR method.
These GWAS methods are generally implemented in a univariate framework analysing one phenotype at a time even though multiple phenotypes of interest are collected from a study population. In particular, pleiotropy that occurs due to potential genetic correlation between multiple phenotypic traits plays a role in pathogenesis of correlated human diseases . Jointly analysing correlated multivariate phenotypes may have more power to detect susceptible genes and gene-gene interactions by using more information from data. Classic multivariate methods such as likelihood based mixed effects model [18, 19] and generalize estimating equations (GEE) , and extended versions of these methods [21, 22] can be applied to multivariate phenotypes of GWAS.
In this study, we have proposed multivariate GMDR method by extending GMDR method for the multivariate phenotypes. We construct GEE model with multivariate phenotypes to extend generalized linear models. The GEE approach is exceptionally useful method for the analysis of longitudinal data, especially when the response variable is discrete . Using the score vectors from GEE, we discriminate high-risk from low-risk groups. The proposed multivariate GMDR method can also handle the repeatedly measured phenotypes.
We apply the proposed multivariate GMDR method to the Korean Association Resource study on blood pressure: systolic blood pressure (SBP) and diastolic blood pressure (DBP). A number of authors have investigated the genome-wide association studies on blood pressure and hypertension for Korean population [24–26] and for others [27–30]. However, not much work has been done for gene-gene interaction analyses. We compare the results of multivariate GMDR for SBP and DBP to the results from original univariate GMDR for SBP and DBP, respectively. We also apply the multivariate GMDR method to the repeated measured hypertension phenotypes and compare its result with those from univariate GMDR at each time point.
where and is estimator obtained from estimating equations under the null hypothesis . and are calculated using . Based on this residual score vector, each individual with phenotypes is discriminated between case and control status. From the residual score vector for individual, we propose the aggregation for elements of the score vector, , and use that as a prediction score for each individual. If the sum of prediction scores over those individuals who have the corresponding genotype combination is greater than a threshold value, assign 'high-risk' to the cell corresponding to the genotype combination. Otherwise, assign 'low-risk' to the cell.
Subject characteristics of the KARE.
Systolic blood pressure
Diastolic blood pressure
Body mass index (kg/m2)
(SBP ≥ 140 or
HP1 (Time 1)
DBP ≥ 90)
HP2 (Time 2)
HP3 (Time 3)
To compare multivariate analysis with univariate analysis, we first separately fit a logistic regression model for each dichotomized blood pressure measurement SBPB and DBPB with covariate adjustment for recruitment area, age, sex, and BMI. The correlation between SBPB and DBPB is 0.48. The multivariate analysis with two binary phenotypes (SBPB, DBPB) was conducted using the GEE approach. For the repeatedly measured hypertension status HP1, HP2, and HP3, we fit logistic models for each HPi and fit the GEE model for three HPs simultaneously. The pairwise correlations range from 0.32 to 0.36. In the GEE model, we assumed two types of genetic effect: homogeneous genetic effect and heterogeneous genetic effect for multivariate phenotypes. However, when we compared the effect sizes and p-values of homogeneous model with those of heterogeneous model, there was no strong evidence for supporting the homogeneous genetic effect. So, we present the results of the GEE model with heterogeneous genetic effects for multivariate phenotypes in both of blood pressures and repeatedly measured hypertension status.
To perform gene-gene interaction analysis using GMDR analyses, we first selected SNPs with strong marginal effects in univariate models and among those, we select the ones with strong effects in multivariate models. For SBPB and DBPB analysis, we selected the top 50 SNPs for each SBPB and DBPB. From these 100 SNPs, we chose 35 SNPs using a p-value criterion (< 1 × 10-4) from the GEE model. In a similar manner, we chose 34 SNPs for HP1, HP2, and HP3 by selecting the top 50 SNPs for each HPi using the same p-value criterion from their GEE model.
Univariate logistic and multivariate GEE analyses of SBPB and DBPB
Selected SNPs of SBP and DBP from univariate and multivariate analyses.
Univariate logistic and multivariate GEE analyses of HP1, HP2, and HP3
Selected SNPs of longitudinal hypertension from univariate and multivariate analyses.
Transition of hypertensive case over time.
HP1 Time 1 (716)
HP3 Time 3 (288)
HP3 Time 3 (410)
HP2 Time 2 (706)
Univariate GMDR and multivariate GMDR analyses of SBPB and DBPB
We present GMDR results to discover gene-gene and/or gene-environment interactions. For univariate GMDR analysis, logistic regression models for dichotomized SBPB and DBPB were constructed with area, age, sex, and BMI as covariates under the null hypothesis of no genetic effect. For multivariate GMDR analysis, the GEE model with same covariates was constructed. To reduce the computational burden, we focused on 35 SNPs selected from the preliminary analysis. All possible one and two locus models were fit for 35 SNPs. Through 10-fold-cross validation the best combination of loci with maximum train balanced accuracy (BA) which is average of sensitivity and specificity was chosen at each fold. To choose the final model, we considered cross-validation consistency (CVC) among a set of best combinations.
Comparison of results for SBP and DBP by GMDR and multivariate GMDR.
No. of Loci
Univariate GMDR and multivariate GMDR analyses of HP1, HP2, and HP3
Comparison of results for longitudinal hypertension by GMDR and multivariate GMDR.
No. of Loci
Comparison of univariate GMDR and multivariate GMDR
Comparison of results for SBP and DBP by multivariate GMDR and hypertension at time 1 (HP1) by GMDR.
No. of Loci
Multivariate GMDR with BPs
GMDR with HP1
Multivariate GMDR with BPs
GMDR with HP1
In this paper, we have developed multivariate analysis for discovering gene-gene interaction, namely multivariate GMDR. Our multivariate GMDR analysis was developed by utilizing a GEE approach to multivariate phenotypes. Many studies emphasized the importance and the increase of power for multivariate analysis in GWAS [33–35]. Although MDR method has been developed in variety of manners [5–9], there have been no extensions to the multivariate analysis. We proposed multivariate GMDR analysis by utilizing the GEE model to calculate the prediction score to be a tool for reducing the multifactor dimensionality. The GEE approach is an extension of generalized linear models to the longitudinal data and handles both discrete and continuous phenotypes. Thus, our multivariate GMDR can be applicable to both discrete and continuous phenotypes.
Though real GWAS data analysis, we investigated the properties of multivariate GMDR. Firstly, the result of multivariate GMDR does not always coincide with that of GEE approach. That is, the best SNP set selected by multivariate GMDR does not always have the smallest p-value from GEE model. In our analysis, note that the SNP set selected by multivariate GMDR still tends to have quite a small p-value. Secondly, the test BAs of the multivariate GMDR is not always larger than those of univariate GMDR. As shown in Figures 3 to 5, the distribution of test BAs from the multivariate GMDR is different from those of univariate GMDR. The test BAs of multivariate GMDR are more densely distributed with a smaller standard deviation than those of univariate GMDR. Thus, a direct comparison of test BAs between multivariate GMDR and univariate GMDR may lead a misleading conclusion.
The proposed multivariate GMDR can be extended in many different ways. The modified version BAs which takes account for the distributional difference is expected to improve the performance of multivariate GMDR. The testing procedure using the modified BAs under the null distribution would enable us to demonstrate the increase of power of multivariate GMDR. A prediction score is defined as the sum of elements of the score vector from GEE model. We are currently working on several different weighting schemes for accounting various relationships between phenotypes. The weighted prediction score is also expected to improve the performance of multivariate GMDR. In the future studies, all these extensions will be evaluated through extensive simulation studies.
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP)(2012R1A3A2026438, 2008-0062618).
The publication cost for this article was supported by the Seoul National University.
This article has been published as part of BMC Systems Biology Volume 7 Supplement 6, 2013: Selected articles from the 24th International Conference on Genome Informatics (GIW2013). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S6.
- Foulkes AS: Applied Statistical Genetics with R: For Population-based association Studies. 2009, SpringerView ArticleGoogle Scholar
- Davies JL, Kawaguchi Y, Bennett ST, Copeman JB, Cordell HJ, Pritchard LE, Reed PW, Gough SC, Jenkins SC, Palmer SM: A genome-wide search for human type 1 diabetes susceptibility genes. Nature. 1994, 371: 130-136. 10.1038/371130a0.View ArticlePubMedGoogle Scholar
- Kraft P, Yen YC, Stram DO, Morris J, Gauderman WG: Exploiting gene environment interaction to detect genetic associations. Hum Hered. 2007, 63: 111-119. 10.1159/000099183.View ArticlePubMedGoogle Scholar
- Marchini J, Donnelly P, Cardon LR: Genome-wide strategies for detecting multiple loci that influence complex diseases. Nature genetics. 2005, 37: 413-417. 10.1038/ng1537.View ArticlePubMedGoogle Scholar
- Ritchie MD, Hahn LW, Roodi N, Bailey LR, Dupont WD, Parl FF, Moore JH: Multifactor-dimensionality reduction reveals high-order interactions among estrogen-metabolism genes in sporadic breast cancer. Am J Hum Genet. 2001, 69: 138-147. 10.1086/321276.PubMed CentralView ArticlePubMedGoogle Scholar
- Moore JH, Williams SM: New strategies for identifying gene-gene interactions in hypertension. Ann Med. 2002, 34: 88-95. 10.1080/07853890252953473.View ArticlePubMedGoogle Scholar
- Hahn LW, Ritchie MD, Moore JH: Multifactor dimensionality reduction software for detecting gene-gene and gene-environment interactions. Bioinformatics. 2003, 19: 376-382. 10.1093/bioinformatics/btf869.View ArticlePubMedGoogle Scholar
- Ritchie MD, Hahn LW, Moore JH: Power of multifactor dimensionality reduction for detecting gene-gene interactions in the presence of genotyping error, missing data, phenocopy, and genetic heterogeneity. Genet Epidemiol. 2003, 24: 150-157. 10.1002/gepi.10218.View ArticlePubMedGoogle Scholar
- Hahn LW, Moore JH: Ideal discrimination of discrete clinical endpoints using multilocus genotypes. In Silico Biol. 2004, 4: 183-194.PubMedGoogle Scholar
- Heidema AG, Boer JM, Nagelkerke N, Mariman EC, van der A DL, Feskens EJ: The challenge for genetic epidemiologists: how to analyze large numbers of SNPs in relation to complex diseases. BMC Genet. 2006, 7: 23-PubMed CentralView ArticlePubMedGoogle Scholar
- Martin ER, Ritchie MD, Hahn L, Kang S, Moore JH: A novel method to identify gene-gene effects in nuclear families: the MDR-PDT. Genet Epidemiol. 2006, 30: 111-123. 10.1002/gepi.20128.View ArticlePubMedGoogle Scholar
- Chung Y, Lee SY, Elston RC, Park T: Odds ratio based multifactor-dimensionality reduction method for detecting gene-gene interactions. Bioinformatics. 2007, 23: 71-76. 10.1093/bioinformatics/btl557.View ArticlePubMedGoogle Scholar
- Lee SY, Chung Y, Elston RC, Kim Y, Park T: Log-linear model-based multifactor dimensionality reduction method to detect gene-gene interactions. Bioinformatics. 2007, 23: 2589-2595. 10.1093/bioinformatics/btm396.View ArticlePubMedGoogle Scholar
- Calle ML, Urrea V, Vellalta G, Malats N, Van Steen K: Improving strategies for detecting genetic patterns of disease susceptibility in association studies. Stat Med. 2008, 27: 6532-6546. 10.1002/sim.3431.View ArticlePubMedGoogle Scholar
- Oh S, Lee J, Kwon MS, Weir B, Ha K, Park T: A novel method to identify high order gene-gene interactions in genome-wide association studies: Gene-based MDR. BMC Bioinformatics. 2012, 13 (Suppl 9): S5-10.1186/1471-2105-13-S9-S5.PubMed CentralView ArticlePubMedGoogle Scholar
- Lou XY, Chen GB, Yan L, Ma JZ, Zhu J, Elston RC, Li MD: A generalized combinatorial approach for detecting gene-by-gene and gene-by-environment interactions with application to nicotine dependence. Am J Hum Genet. 2007, 80: 1125-1137. 10.1086/518312.PubMed CentralView ArticlePubMedGoogle Scholar
- Manolio TA, Collins FS, Cox NJ, Goldstein DB, Hindorff LA: Finding the missing heritability of complex diseases. Nature. 2009, 461: 747-753. 10.1038/nature08494.PubMed CentralView ArticlePubMedGoogle Scholar
- Laird NM, Ware JH: Random-Effects Models for Longitudinal Data. Biometrics. 1982, 38: 963-974. 10.2307/2529876.View ArticlePubMedGoogle Scholar
- Fitzmaurice GM, Laird NM: A Likelihood-Based Method for Analysing Longitudinal Binary Responses. Biometrika. 1993, 80: 141-151. 10.1093/biomet/80.1.141.View ArticleGoogle Scholar
- Liang K, Zeger SL: Longitudinal Data Analysis Using Generalized Linear Models. Biometrika. 1986, 73: 13-22. 10.1093/biomet/73.1.13.View ArticleGoogle Scholar
- Fitzmaurice GM, Laird NM: Regression Models for Mixed Discrete and Continuous Responses with Potentially Missing Values. Biometrics. 1997, 53: 110-122. 10.2307/2533101.View ArticlePubMedGoogle Scholar
- Liu J, Pei Y, Papasian CJ, Deng HW: Bivariate association analyses for the mixture of continuous and binary traits with the use of extended generalized estimating equations. Genet Epidemiol. 2009, 33: 217-227. 10.1002/gepi.20372.PubMed CentralView ArticlePubMedGoogle Scholar
- Fitzmaurice GM, Davidian M, Verbeke G, Molenberghs G: Longitudinal Data Analysis. 2009, Boca Raton, FL: Chapman and Hall/CRC PressGoogle Scholar
- Hong KW, Jin HS, Cho YS, Lee JY, Lee JE, Cho NH: Replication of the Wellcome Trust Genome-Wide Association Study on Essential Hypertension in a Korean population. Hypertens Res. 2009, 32: 570-574. 10.1038/hr.2009.68.View ArticlePubMedGoogle Scholar
- Hong KW, Go MJ, Jin HS, Lim JE, Lee JY, Han BG, Hwang SY, Lee SH, Park HK, Cho YS, Oh B: Genetic variations in ATP2B1, CSK, ARSG and CSMD1 loci are related to blood pressure and/or hypertension in two Korean cohorts. J Hum Hypertens. 2010, 24: 367-372. 10.1038/jhh.2009.86.View ArticlePubMedGoogle Scholar
- Hong KW, Jin HS, Lim JE, Lim JE, Kim S, Go MJ, Oh B: Recapitulation of two genomewide association studies on blood pressure and essential hypertension in the Korean population. J Hum Genet. 2010, 55: 336-341. 10.1038/jhg.2010.31.View ArticlePubMedGoogle Scholar
- Wang Y, O'Connell JR, McArdle PF, Wade JB, Dorff SE, Shah SJ: From the cover: whole-genome association study identifies STK39 as a hypertension susceptibility gene. Proc Natl Acad Sci USA. 2009, 106: 226-231. 10.1073/pnas.0808358106.PubMed CentralView ArticlePubMedGoogle Scholar
- Newton-Cheh C, Johnson T, Gateva V, Tobin MD, Bochud M, Coin L: Genome-wide association study identifies eight loci associated with blood pressure. Nat Genet. 2009, 41: 666-676. 10.1038/ng.361.PubMed CentralView ArticlePubMedGoogle Scholar
- Levy D, Ehret GB, Rice K, Verwoert GC, Launer LJ, Dehghan A: Genome-wide association study of blood pressure and hypertension. Nat Genet. 2009, 41: 677-687. 10.1038/ng.384.PubMed CentralView ArticlePubMedGoogle Scholar
- Xi B, Shen Y, Reilly KH, Wang X, Mi J: Recapitulation of four hypertension susceptibility genes (CSK, CYP17A1, MTHFR, and FGF5) in East Asians. Metabolism. 2013, 62: 196-203. 10.1016/j.metabol.2012.07.008.View ArticlePubMedGoogle Scholar
- Shen C, Lu X, Wang L, Chen S, Li Y, Liu X, Li J, Huang J, Gu D: Novel genetic variation in exon 28 of FBN1 gene is associated with essential hypertension. Am J Hypertens. 2011, 24: 687-693. 10.1038/ajh.2011.21.View ArticlePubMedGoogle Scholar
- Yang HC, Liang YJ, Chen JW, Chiang KM, Chung CM, Ho HY: Identification of IGF1, SLC4A4, WWOX, and SFMBT1 as hypertension susceptibility genes in Han Chinese with a genomewide gene-based association study. PLoS One. 2012, 7: e32907-10.1371/journal.pone.0032907.PubMed CentralView ArticlePubMedGoogle Scholar
- Schmitz S, Cherny SS, Fulker DW: Increase in power through multivariate analyses. Behav Genet. 1998, 5: 357-363.View ArticleGoogle Scholar
- Pei YF, Zhang L, Liu J, Deng HW: Multivariate association test using haplotype trend regression. Ann Hum Genet. 2009, 73: 456-464. 10.1111/j.1469-1809.2009.00527.x.PubMed CentralView ArticlePubMedGoogle Scholar
- Liu YZ, Pei YF, Liu JF, Yang F, Guo Y, Zhang L, Liu XG, Yan H, Wang L, Zhang YP, Levy S, Recker RR, Deng HW: Powerful bivariate genome-wide association analyses suggest the SOX6 gene influencing both obesity and osteoporosis phenotypes in males. PLoS One. 2009, 4: e6827-10.1371/journal.pone.0006827.PubMed CentralView ArticlePubMedGoogle Scholar
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