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.

Contents

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

X1X2Y
000
011
101
110

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

ZY
00
11
11
00

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

See also

References

  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. doi:10.3233/ISB-00126. 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. 1 2 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   978-1-4939-2154-6. PMID   25403539.
  8. ↑ Moore, Jason H. (1 January 2010). "Detecting, Characterizing, and Interpreting Nonlinear Gene–Gene Interactions Using Multifactor Dimensionality Reduction". Computational Methods for Genetics of Complex Traits. Advances in Genetics. Vol. 72. pp. 101–116. doi:10.1016/B978-0-12-380862-2.00005-9. ISBN   978-0-12-380862-2. ISSN   0065-2660. PMID   21029850.
  9. 1 2 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. Bibcode:2006JThBi.241..252M. 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.
  11. ↑ Velez, Digna R.; White, Bill C.; Motsinger, Alison A.; Bush, William S.; Ritchie, Marylyn D.; Williams, Scott M.; Moore, Jason H. (1 May 2007). "A balanced accuracy function for epistasis modeling in imbalanced datasets using multifactor dimensionality reduction". Genetic Epidemiology. 31 (4): 306–315. doi:10.1002/gepi.20211. ISSN   0741-0395. PMID   17323372. S2CID   28156181.
  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. 1 2 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.
  15. ↑ Motsinger, Alison A.; Ritchie, Marylyn D. (1 September 2006). "The effect of reduction in cross-validation intervals on the performance of multifactor dimensionality reduction". Genetic Epidemiology. 30 (6): 546–555. doi:10.1002/gepi.20166. ISSN   1098-2272. PMID   16800004. S2CID   20573232.
  16. ↑ Gory, Jeffrey J.; Sweeney, Holly C.; Reif, David M.; Motsinger-Reif, Alison A. (5 November 2012). "A comparison of internal model validation methods for multifactor dimensionality reduction in the case of genetic heterogeneity". BMC Research Notes. 5: 623. doi: 10.1186/1756-0500-5-623 . ISSN   1756-0500. PMC   3599301 . PMID   23126544.
  17. ↑ Winham, Stacey J.; Slater, Andrew J.; Motsinger-Reif, Alison A. (22 July 2010). "A comparison of internal validation techniques for multifactor dimensionality reduction". BMC Bioinformatics. 11: 394. doi: 10.1186/1471-2105-11-394 . ISSN   1471-2105. PMC   2920275 . PMID   20650002.
  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). "Enabling personal genomics with an explicit test of epistasis". Biocomputing 2010: Pacific Symposium on Biocomputing. World Scientific. pp. 327–336. doi:10.1142/9789814295291_0035. ISBN   978-981-4299-47-3. PMC   2916690 . PMID   19908385.
  20. ↑ Dai, Hongying; Bhandary, Madhusudan; Becker, Mara; Leeder, J. Steven; Gaedigk, Roger; Motsinger-Reif, Alison A. (22 May 2012). "Global tests of P-values for multifactor dimensionality reduction models in selection of optimal number of target genes". BioData Mining. 5 (1): 3. doi: 10.1186/1756-0381-5-3 . ISSN   1756-0381. PMC   3508622 . PMID   22616673.
  21. ↑ Motsinger-Reif, Alison A. (30 December 2008). "The effect of alternative permutation testing strategies on the performance of multifactor dimensionality reduction". BMC Research Notes. 1: 139. doi: 10.1186/1756-0500-1-139 . ISSN   1756-0500. PMC   2631601 . PMID   19116021.
  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.
  23. ↑ Piette, Elizabeth R.; Moore, Jason H. (19 April 2017). "Improving the Reproducibility of Genetic Association Results Using Genotype Resampling Methods". Applications of Evolutionary Computation. Lecture Notes in Computer Science. Vol. 10199. pp. 96–108. doi:10.1007/978-3-319-55849-3_7. ISBN   978-3-319-55848-6.
  24. ↑ Moore, Jason H.; Hu, Ting (1 January 2015). "Epistasis Analysis Using Information Theory". Epistasis. Methods in Molecular Biology. Vol. 1253. pp. 257–268. doi:10.1007/978-1-4939-2155-3_13. ISBN   978-1-4939-2154-6. ISSN   1940-6029. PMID   25403536.
  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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  33. ↑ Chung, Yujin; Lee, Seung Yeoun; Elston, Robert C.; Park, Taesung (1 January 2007). "Odds ratio based multifactor-dimensionality reduction method for detecting gene-gene interactions". Bioinformatics. 23 (1): 71–76. doi: 10.1093/bioinformatics/btl557 . ISSN   1367-4811. PMID   17092990.
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  38. ↑ Gui, Jiang; Moore, Jason H.; Williams, Scott M.; Andrews, Peter; Hillege, Hans L.; van der Harst, Pim; Navis, Gerjan; Van Gilst, Wiek H.; Asselbergs, Folkert W. (1 January 2013). "A Simple and Computationally Efficient Approach to Multifactor Dimensionality Reduction Analysis of Gene-Gene Interactions for Quantitative Traits". PLOS ONE. 8 (6) e66545. Bibcode:2013PLoSO...866545G. doi: 10.1371/journal.pone.0066545 . ISSN   1932-6203. PMC   3689797 . PMID   23805232.
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  41. ↑ Asselbergs, Folkert W.; Moore, Jason H.; van den Berg, Maarten P.; Rimm, Eric B.; de Boer, Rudolf A.; Dullaart, Robin P.; Navis, Gerjan; van Gilst, Wiek H. (1 January 2006). "A role for CETP TaqIB polymorphism in determining susceptibility to atrial fibrillation: a nested case control study". BMC Medical Genetics. 7: 39. doi: 10.1186/1471-2350-7-39 . ISSN   1471-2350. PMC   1462991 . PMID   16623947.
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  43. ↑ Andrew, Angeline S.; Nelson, Heather H.; Kelsey, Karl T.; Moore, Jason H.; Meng, Alexis C.; Casella, Daniel P.; Tosteson, Tor D.; Schned, Alan R.; Karagas, Margaret R. (1 May 2006). "Concordance of multiple analytical approaches demonstrates a complex relationship between DNA repair gene SNPs, smoking and bladder cancer susceptibility". Carcinogenesis. 27 (5): 1030–1037. doi: 10.1093/carcin/bgi284 . ISSN   0143-3334. PMID   16311243.
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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.
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