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Multifactor dimensionality reduction

## Summary

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

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

Z Y
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

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

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

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]

## Implementations

• www.epistasis.org provides an open-source and freely-available MDR software package.
• An R package for MDR.[68]
• An sklearn-compatible Python implementation.
• An R package for Model-Based MDR.[69]
• MDR in Weka.
• Generalized MDR.

## References

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5. ^ W., Hahn, Lance; H., Moore, Jason (1 January 2004). "Ideal Discrimination of Discrete Clinical Endpoints Using Multilocus Genotypes". In Silico Biology. 4 (2). ISSN 1386-6338.
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25. ^ Kim, Nora Chung; Andrews, Peter C.; Asselbergs, Folkert W.; Frost, H. Robert; Williams, Scott M.; Harris, Brent T.; Read, Cynthia; Askland, Kathleen D.; Moore, Jason H. (28 July 2012). "Gene ontology analysis of pairwise genetic associations in two genome-wide studies of sporadic ALS". BioData Mining. 5 (1): 9. doi:10.1186/1756-0381-5-9. ISSN 1756-0381. PMC 3463436. PMID 22839596.
26. ^ Cheng, Samantha; Andrew, Angeline S.; Andrews, Peter C.; Moore, Jason H. (1 January 2016). "Complex systems analysis of bladder cancer susceptibility reveals a role for decarboxylase activity in two genome-wide association studies". BioData Mining. 9: 40. doi:10.1186/s13040-016-0119-z. PMC 5154053. PMID 27999618.
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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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42. ^ Ma, D.Q.; Whitehead, P.L.; Menold, M.M.; Martin, E.R.; Ashley-Koch, A.E.; Mei, H.; Ritchie, M.D.; DeLong, G.R.; Abramson, R.K. (1 September 2005). "Identification of Significant Association and Gene-Gene Interaction of GABA Receptor Subunit Genes in Autism". The American Journal of Human Genetics. 77 (3): 377–388. doi:10.1086/433195. ISSN 0002-9297. PMC 1226204. PMID 16080114.
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47. ^ Williams, Scott M.; Ritchie, Marylyn D.; III, John A. Phillips; Dawson, Elliot; Prince, Melissa; Dzhura, Elvira; Willis, Alecia; Semenya, Amma; Summar, Marshall (1 January 2004). "Multilocus Analysis of Hypertension: A Hierarchical Approach". Human Heredity. 57 (1): 28–38. doi:10.1159/000077387. ISSN 0001-5652. PMID 15133310. S2CID 21079485.
48. ^ Sanada, Hironobu; Yatabe, Junichi; Midorikawa, Sanae; Hashimoto, Shigeatsu; Watanabe, Tsuyoshi; Moore, Jason H.; Ritchie, Marylyn D.; Williams, Scott M.; Pezzullo, John C. (1 March 2006). "Single-Nucleotide Polymorphisms for Diagnosis of Salt-Sensitive Hypertension". Clinical Chemistry. 52 (3): 352–360. doi:10.1373/clinchem.2005.059139. ISSN 0009-9147. PMID 16439609.
49. ^ Moore, Jason H.; Williams, Scott M. (1 January 2002). "New strategies for identifying gene-gene interactions in hypertension". Annals of Medicine. 34 (2): 88–95. doi:10.1080/07853890252953473. ISSN 0785-3890. PMID 12108579. S2CID 25398042.
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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52. ^ Duell, Eric J.; Bracci, Paige M.; Moore, Jason H.; Burk, Robert D.; Kelsey, Karl T.; Holly, Elizabeth A. (1 June 2008). "Detecting pathway-based gene-gene and gene-environment interactions in pancreatic cancer". Cancer Epidemiology, Biomarkers & Prevention. 17 (6): 1470–1479. doi:10.1158/1055-9965.EPI-07-2797. ISSN 1055-9965. PMC 4410856. PMID 18559563.
53. ^ Xu, Jianfeng; Lowey, James; Wiklund, Fredrik; Sun, Jielin; Lindmark, Fredrik; Hsu, Fang-Chi; Dimitrov, Latchezar; Chang, Baoli; Turner, Aubrey R. (1 November 2005). "The Interaction of Four Genes in the Inflammation Pathway Significantly Predicts Prostate Cancer Risk". Cancer Epidemiology, Biomarkers & Prevention. 14 (11): 2563–2568. doi:10.1158/1055-9965.EPI-05-0356. ISSN 1055-9965. PMID 16284379.
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.
56. ^ Collins, Ryan L.; Hu, Ting; Wejse, Christian; Sirugo, Giorgio; Williams, Scott M.; Moore, Jason H. (18 February 2013). "Multifactor dimensionality reduction reveals a three-locus epistatic interaction associated with susceptibility to pulmonary tuberculosis". BioData Mining. 6 (1): 4. doi:10.1186/1756-0381-6-4. PMC 3618340. PMID 23418869.
57. ^ Wilke, Russell A.; Reif, David M.; Moore, Jason H. (1 November 2005). "Combinatorial Pharmacogenetics". Nature Reviews Drug Discovery. 4 (11): 911–918. doi:10.1038/nrd1874. ISSN 1474-1776. PMID 16264434. S2CID 11643026.
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62. ^ Pendergrass, Sarah A.; Frase, Alex; Wallace, John; Wolfe, Daniel; Katiyar, Neerja; Moore, Carrie; Ritchie, Marylyn D. (30 December 2013). "Genomic analyses with biofilter 2.0: knowledge driven filtering, annotation, and model development". BioData Mining. 6 (1): 25. doi:10.1186/1756-0381-6-25. PMC 3917600. PMID 24378202.
63. ^ Moore, Jason H. (1 January 2015). "Epistasis analysis using ReliefF". Epistasis. Methods in Molecular Biology. Vol. 1253. pp. 315–325. doi:10.1007/978-1-4939-2155-3_17. ISBN 978-1-4939-2154-6. ISSN 1940-6029. PMID 25403540.
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65. ^ Greene, Casey S.; Sinnott-Armstrong, Nicholas A.; Himmelstein, Daniel S.; Park, Paul J.; Moore, Jason H.; Harris, Brent T. (1 March 2010). "Multifactor dimensionality reduction for graphics processing units enables genome-wide testing of epistasis in sporadic ALS". Bioinformatics. 26 (5): 694–695. doi:10.1093/bioinformatics/btq009. ISSN 1367-4811. PMC 2828117. PMID 20081222.
66. ^ Bush, William S.; Dudek, Scott M.; Ritchie, Marylyn D. (1 September 2006). "Parallel multifactor dimensionality reduction: a tool for the large-scale analysis of gene-gene interactions". Bioinformatics. 22 (17): 2173–2174. doi:10.1093/bioinformatics/btl347. ISSN 1367-4811. PMC 4939609. PMID 16809395.
67. ^ Sinnott-Armstrong, Nicholas A.; Greene, Casey S.; Cancare, Fabio; Moore, Jason H. (24 July 2009). "Accelerating epistasis analysis in human genetics with consumer graphics hardware". BMC Research Notes. 2: 149. doi:10.1186/1756-0500-2-149. ISSN 1756-0500. PMC 2732631. PMID 19630950.
68. ^ Winham, Stacey J.; Motsinger-Reif, Alison A. (16 August 2011). "An R package implementation of multifactor dimensionality reduction". BioData Mining. 4 (1): 24. doi:10.1186/1756-0381-4-24. ISSN 1756-0381. PMC 3177775. PMID 21846375.
69. ^ Calle, M. Luz; Urrea, Víctor; Malats, Núria; Van Steen, Kristel (1 September 2010). "mbmdr: an R package for exploring gene-gene interactions associated with binary or quantitative traits". Bioinformatics. 26 (17): 2198–2199. doi:10.1093/bioinformatics/btq352. ISSN 1367-4811. PMID 20595460.

## Further reading

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