Multifactor dimensionality reduction

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Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches,[1] for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable.[2][3][4][5][6][7][8] MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression.

The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute.[9] This process of constructing a new attribute changes the representation space of the data.[10] The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data.

Illustrative example[edit]

Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between the attributes (X1 and X2) and the class variable (Y) is defined by the XOR function such that Y = X1 XOR X2.

Table 1

X1 X2 Y
0 0 0
0 1 1
1 0 1
1 1 0

A machine learning algorithm would need to discover or approximate the XOR function in order to accurately predict Y using information about X1 and X2. An alternative strategy would be to first change the representation of the data using constructive induction to facilitate predictive modeling. The MDR algorithm would change the representation of the data (X1 and X2) in the following manner. MDR starts by selecting two attributes. In this simple example, X1 and X2 are selected. Each combination of values for X1 and X2 are examined and the number of times Y=1 and/or Y=0 is counted. In this simple example, Y=1 occurs zero times and Y=0 occurs once for the combination of X1=0 and X2=0. With MDR, the ratio of these counts is computed and compared to a fixed threshold. Here, the ratio of counts is 0/1 which is less than our fixed threshold of 1. Since 0/1 < 1 we encode a new attribute (Z) as a 0. When the ratio is greater than one we encode Z as a 1. This process is repeated for all unique combinations of values for X1 and X2. Table 2 illustrates our new transformation of the data.

Table 2

0 0
1 1
1 1
0 0

The machine learning algorithm now has much less work to do to find a good predictive function. In fact, in this very simple example, the function Y = Z has a classification accuracy of 1. A nice feature of constructive induction methods such as MDR is the ability to use any data mining or machine learning method to analyze the new representation of the data. Decision trees, neural networks, or a naive Bayes classifier could be used in combination with measures of model quality such as balanced accuracy[11][12] and mutual information.[13]

Machine learning with MDR[edit]

As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine learning algorithms are good at finding patterns in completely random data. It is often difficult to determine whether a reported pattern is an important signal or just chance. One approach is to estimate the generalizability of a model to independent datasets using methods such as cross-validation.[14][15][16][17] Models that describe random data typically don't generalize. Another approach is to generate many random permutations of the data to see what the data mining algorithm finds when given the chance to overfit. Permutation testing makes it possible to generate an empirical p-value for the result.[18][19][20][21] Replication in independent data may also provide evidence for an MDR model but can be sensitive to difference in the data sets.[22][23] These approaches have all been shown to be useful for choosing and evaluating MDR models. An important step in a machine learning exercise is interpretation. Several approaches have been used with MDR including entropy analysis[9][24] and pathway analysis.[25][26] Tips and approaches for using MDR to model gene-gene interactions have been reviewed.[7][27]

Extensions to MDR[edit]

Numerous extensions to MDR have been introduced. These include family-based methods,[28][29][30] fuzzy methods,[31] covariate adjustment,[32] odds ratios,[33] risk scores,[34] survival methods,[35][36] robust methods,[37] methods for quantitative traits,[38][39] and many others.

Applications of MDR[edit]

MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation,[40][41] autism,[42] bladder cancer,[43][44][45] breast cancer,[46] cardiovascular disease,[14] hypertension,[47][48][49] obesity,[50][51] pancreatic cancer,[52] prostate cancer[53][54][55] and tuberculosis.[56] It has also been applied to other biomedical problems such as the genetic analysis of pharmacology outcomes.[57][58][59] A central challenge is the scaling of MDR to big data such as that from genome-wide association studies (GWAS).[60] Several approaches have been used. One approach is to filter the features prior to MDR analysis.[61] This can be done using biological knowledge through tools such as BioFilter.[62] It can also be done using computational tools such as ReliefF.[63] Another approach is to use stochastic search algorithms such as genetic programming to explore the search space of feature combinations.[64] Yet another approach is a brute-force search using high-performance computing.[65][66][67]


See also[edit]


  1. ^ McKinney, Brett A.; Reif, David M.; Ritchie, Marylyn D.; Moore, Jason H. (1 January 2006). "Machine learning for detecting gene-gene interactions: a review". Applied Bioinformatics. 5 (2): 77–88. doi:10.2165/00822942-200605020-00002. ISSN 1175-5636. PMC 3244050. PMID 16722772.
  2. ^ Ritchie, Marylyn D.; Hahn, Lance W.; Roodi, Nady; Bailey, L. Renee; Dupont, William D.; Parl, Fritz F.; Moore, Jason H. (1 July 2001). "Multifactor-Dimensionality Reduction Reveals High-Order Interactions among Estrogen-Metabolism Genes in Sporadic Breast Cancer". The American Journal of Human Genetics. 69 (1): 138–147. doi:10.1086/321276. ISSN 0002-9297. PMC 1226028. PMID 11404819.
  3. ^ Ritchie, Marylyn D.; Hahn, Lance W.; Moore, Jason H. (1 February 2003). "Power of multifactor dimensionality reduction for detecting gene-gene interactions in the presence of genotyping error, missing data, phenocopy, and genetic heterogeneity". Genetic Epidemiology. 24 (2): 150–157. doi:10.1002/gepi.10218. ISSN 1098-2272. PMID 12548676. S2CID 6335612.
  4. ^ Hahn, L. W.; Ritchie, M. D.; Moore, J. H. (12 February 2003). "Multifactor dimensionality reduction software for detecting gene-gene and gene-environment interactions". Bioinformatics. 19 (3): 376–382. doi:10.1093/bioinformatics/btf869. ISSN 1367-4803. PMID 12584123.
  5. ^ W., Hahn, Lance; H., Moore, Jason (1 January 2004). "Ideal Discrimination of Discrete Clinical Endpoints Using Multilocus Genotypes". In Silico Biology. 4 (2): 183–194. ISSN 1386-6338. PMID 15107022.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  6. ^ Moore, Jason H. (1 November 2004). "Computational analysis of gene-gene interactions using multifactor dimensionality reduction". Expert Review of Molecular Diagnostics. 4 (6): 795–803. doi:10.1586/14737159.4.6.795. ISSN 1473-7159. PMID 15525222. S2CID 26324399.
  7. ^ a b Moore, JasonH.; Andrews, PeterC. (1 January 2015). "Epistasis Analysis Using Multifactor Dimensionality Reduction". In Moore, Jason H.; Williams, Scott M. (eds.). Epistasis. Methods in Molecular Biology. Vol. 1253. Springer New York. pp. 301–314. doi:10.1007/978-1-4939-2155-3_16. ISBN 9781493921546. PMID 25403539.
  8. ^ Moore, Jason H. (1 January 2010). Detecting, characterizing, and interpreting nonlinear gene-gene interactions using multifactor dimensionality reduction. pp. 101–116. doi:10.1016/B978-0-12-380862-2.00005-9. ISBN 978-0-12-380862-2. ISSN 0065-2660. PMID 21029850. {{cite book}}: |journal= ignored (help)
  9. ^ a b Moore, Jason H.; Gilbert, Joshua C.; Tsai, Chia-Ti; Chiang, Fu-Tien; Holden, Todd; Barney, Nate; White, Bill C. (21 July 2006). "A flexible computational framework for detecting, characterizing, and interpreting statistical patterns of epistasis in genetic studies of human disease susceptibility". Journal of Theoretical Biology. 241 (2): 252–261. doi:10.1016/j.jtbi.2005.11.036. PMID 16457852.
  10. ^ Michalski, R (February 1983). "A theory and methodology of inductive learning". Artificial Intelligence. 20 (2): 111–161. doi:10.1016/0004-3702(83)90016-4.
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  12. ^ Namkung, Junghyun; Kim, Kyunga; Yi, Sungon; Chung, Wonil; Kwon, Min-Seok; Park, Taesung (1 February 2009). "New evaluation measures for multifactor dimensionality reduction classifiers in gene-gene interaction analysis". Bioinformatics. 25 (3): 338–345. doi:10.1093/bioinformatics/btn629. ISSN 1367-4811. PMID 19164302.
  13. ^ Bush, William S.; Edwards, Todd L.; Dudek, Scott M.; McKinney, Brett A.; Ritchie, Marylyn D. (1 January 2008). "Alternative contingency table measures improve the power and detection of multifactor dimensionality reduction". BMC Bioinformatics. 9: 238. doi:10.1186/1471-2105-9-238. ISSN 1471-2105. PMC 2412877. PMID 18485205.
  14. ^ a b Coffey, Christopher S.; Hebert, Patricia R.; Ritchie, Marylyn D.; Krumholz, Harlan M.; Gaziano, J. Michael; Ridker, Paul M.; Brown, Nancy J.; Vaughan, Douglas E.; Moore, Jason H. (1 January 2004). "An application of conditional logistic regression and multifactor dimensionality reduction for detecting gene-gene Interactions on risk of myocardial infarction: The importance of model validation". BMC Bioinformatics. 5: 49. doi:10.1186/1471-2105-5-49. ISSN 1471-2105. PMC 419697. PMID 15119966.
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  18. ^ Pattin, Kristine A.; White, Bill C.; Barney, Nate; Gui, Jiang; Nelson, Heather H.; Kelsey, Karl T.; Andrew, Angeline S.; Karagas, Margaret R.; Moore, Jason H. (1 January 2009). "A computationally efficient hypothesis testing method for epistasis analysis using multifactor dimensionality reduction". Genetic Epidemiology. 33 (1): 87–94. doi:10.1002/gepi.20360. ISSN 1098-2272. PMC 2700860. PMID 18671250.
  19. ^ Greene, Casey S.; Himmelstein, Daniel S.; Nelson, Heather H.; Kelsey, Karl T.; Williams, Scott M.; Andrew, Angeline S.; Karagas, Margaret R.; Moore, Jason H. (1 October 2009). Biocomputing 2010. pp. 327–336. doi:10.1142/9789814295291_0035. ISBN 9789814299473. PMC 2916690. PMID 19908385. {{cite book}}: |journal= ignored (help)
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  22. ^ Greene, Casey S.; Penrod, Nadia M.; Williams, Scott M.; Moore, Jason H. (2 June 2009). "Failure to Replicate a Genetic Association May Provide Important Clues About Genetic Architecture". PLOS ONE. 4 (6): e5639. Bibcode:2009PLoSO...4.5639G. doi:10.1371/journal.pone.0005639. ISSN 1932-6203. PMC 2685469. PMID 19503614.
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  28. ^ Martin, E. R.; Ritchie, M. D.; Hahn, L.; Kang, S.; Moore, J. H. (1 February 2006). "A novel method to identify gene-gene effects in nuclear families: the MDR-PDT". Genetic Epidemiology. 30 (2): 111–123. doi:10.1002/gepi.20128. ISSN 0741-0395. PMID 16374833. S2CID 25772215.
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  30. ^ Cattaert, Tom; Urrea, Víctor; Naj, Adam C.; De Lobel, Lizzy; De Wit, Vanessa; Fu, Mao; Mahachie John, Jestinah M.; Shen, Haiqing; Calle, M. Luz (22 April 2010). "FAM-MDR: a flexible family-based multifactor dimensionality reduction technique to detect epistasis using related individuals". PLOS ONE. 5 (4): e10304. Bibcode:2010PLoSO...510304C. doi:10.1371/journal.pone.0010304. ISSN 1932-6203. PMC 2858665. PMID 20421984.
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  32. ^ Gui, Jiang; Andrew, Angeline S.; Andrews, Peter; Nelson, Heather M.; Kelsey, Karl T.; Karagas, Margaret R.; Moore, Jason H. (1 January 2010). "A simple and computationally efficient sampling approach to covariate adjustment for multifactor dimensionality reduction analysis of epistasis". Human Heredity. 70 (3): 219–225. doi:10.1159/000319175. ISSN 1423-0062. PMC 2982850. PMID 20924193.
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  50. ^ De, Rishika; Verma, Shefali S.; Holzinger, Emily; Hall, Molly; Burt, Amber; Carrell, David S.; Crosslin, David R.; Jarvik, Gail P.; Kuivaniemi, Helena (1 February 2017). "Identifying gene-gene interactions that are highly associated with four quantitative lipid traits across multiple cohorts" (PDF). Human Genetics. 136 (2): 165–178. doi:10.1007/s00439-016-1738-7. ISSN 1432-1203. PMID 27848076. S2CID 24702049.
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  54. ^ Lavender, Nicole A.; Rogers, Erica N.; Yeyeodu, Susan; Rudd, James; Hu, Ting; Zhang, Jie; Brock, Guy N.; Kimbro, Kevin S.; Moore, Jason H. (30 April 2012). "Interaction among apoptosis-associated sequence variants and joint effects on aggressive prostate cancer". BMC Medical Genomics. 5: 11. doi:10.1186/1755-8794-5-11. ISSN 1755-8794. PMC 3355002. PMID 22546513.
  55. ^ Lavender, Nicole A.; Benford, Marnita L.; VanCleave, Tiva T.; Brock, Guy N.; Kittles, Rick A.; Moore, Jason H.; Hein, David W.; Kidd, La Creis R. (16 November 2009). "Examination of polymorphic glutathione S-transferase (GST) genes, tobacco smoking and prostate cancer risk among men of African descent: a case-control study". BMC Cancer. 9: 397. doi:10.1186/1471-2407-9-397. ISSN 1471-2407. PMC 2783040. PMID 19917083.
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Further reading[edit]

  • Michalski, R. S., "Pattern Recognition as Knowledge-Guided Computer Induction," Department of Computer Science Reports, No. 927, University of Illinois, Urbana, June 1978.