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Strong Heredity Models in High Dimensional Data

Presentation at IBC 2016 in Victoria, BC

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A Model for Interpretable High Dimensional
Interactions
.
Sahir Rai Bhatnagar
Joint work with Yi Yang, Mathieu Blanchette and Celia Greenwood
McGill University
sahirbhatnagar.com
Motivation
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one predictor variable at a time
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Predictor Variable Phenotype
one predictor variable at a time
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Predictor Variable Phenotype
Test 1
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Test 3
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Test 5
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a network based view
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Predictor Variable
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Phenotype
a network based view
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Predictor Variable
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Phenotype
Ad

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Strong Heredity Models in High Dimensional Data

  • 1. A Model for Interpretable High Dimensional Interactions . Sahir Rai Bhatnagar Joint work with Yi Yang, Mathieu Blanchette and Celia Greenwood McGill University sahirbhatnagar.com
  • 3. one predictor variable at a time . . . . . . Predictor Variable Phenotype
  • 4. one predictor variable at a time . . . . . . Predictor Variable Phenotype Test 1 Test 2 Test 3 Test 4 Test 5 1/25
  • 5. a network based view . . . . . . . . . . . . .. Predictor Variable . Phenotype
  • 6. a network based view . . . . . . . . . . . . .. Predictor Variable . Phenotype
  • 7. a network based view . . . . . . . . . . . . .. Predictor Variable . Phenotype . Test 1 2/25
  • 8. system level changes due to environment . . . . . . . . . . . . .. Predictor Variable . Phenotype . Environment . . . . . . . . . .. A . B
  • 9. system level changes due to environment . . . . . . . . . . . . .. Predictor Variable . Phenotype . Environment . . . . . . . . . .. A . B . Test 1 3/25
  • 10. Motivating Dataset: Newborn epigenetic adaptations to gesta- tional diabetes exposure (Luigi Bouchard, Sherbrooke) ... Environment Gestational Diabetes .. Large Data Child's epigenome (p ≈ 450k) . . . Phenotype Obesity measures 4/25
  • 11. Differential Correlation between environments (a) Gestational diabetes affected pregnancy (b) Controls 5/25
  • 13. formal statement of initial problem • n: number of subjects 7/25
  • 14. formal statement of initial problem • n: number of subjects • p: number of predictor variables 7/25
  • 15. formal statement of initial problem • n: number of subjects • p: number of predictor variables • Xn×p: high dimensional data set (p >> n) 7/25
  • 16. formal statement of initial problem • n: number of subjects • p: number of predictor variables • Xn×p: high dimensional data set (p >> n) • Yn×1: phenotype 7/25
  • 17. formal statement of initial problem • n: number of subjects • p: number of predictor variables • Xn×p: high dimensional data set (p >> n) • Yn×1: phenotype • En×1: environmental factor that has widespread effect on X 7/25
  • 18. formal statement of initial problem • n: number of subjects • p: number of predictor variables • Xn×p: high dimensional data set (p >> n) • Yn×1: phenotype • En×1: environmental factor that has widespread effect on X Objective • Which elements of X that are associated with Y, depend on E? 7/25
  • 20. ECLUST - our proposed method: 3 phases ... Original Data
  • 21. ECLUST - our proposed method: 3 phases ... Original Data .. E = 0 . 1) Gene Similarity .. E = 1
  • 22. ECLUST - our proposed method: 3 phases ... Original Data .. E = 0 . 1) Gene Similarity .. E = 1
  • 23. ECLUST - our proposed method: 3 phases ... Original Data .. E = 0 . 1) Gene Similarity .. E = 1 . 2) Cluster Representation .
  • 24. ECLUST - our proposed method: 3 phases ... Original Data .. E = 0 . 1) Gene Similarity .. E = 1 . 2) Cluster Representation .. n × 1 . n × 1
  • 25. ECLUST - our proposed method: 3 phases ... Original Data .. E = 0 . 1) Gene Similarity .. E = 1 . 2) Cluster Representation .. n × 1 . n × 1 . 3) Penalized Regression . Yn×1 . ∼ . + . ×E 8/25
  • 26. the objective of statisti- cal methods is the reduction of data. A quantity of data . . . is to be replaced by relatively few quantities which shall adequately represent . . . the relevant information contained in the original data. - Sir R. A. Fisher, 1922 8/25
  • 27. Underlying model Y = β0 + β1U + β2U · E + ε (1) X ∼ F(α0 + α1U, ΣE) (2) • U: unobserved latent variable • X: observed data which is a function of U • ΣE: environment sensitive correlation matrix 9/25
  • 28. Measure of similarity: topological overlap matrix (TOM) 10/25
  • 29. Method to detect gene clusters Table 1: Method to detect gene clusters General Approach Formula TOM Scoring |TOME=1 − TOME=0| 11/25
  • 30. Cluster Representation Table 2: Methods to create cluster representations General Approach Type Unsupervised average 1st principal component 12/25
  • 31. Model g(µ) =β0 + β1X1 + · · · + βpXp + βEE main effects + α1E(X1E) + · · · + αpE(XpE) interactions 1Choi et al. 2010, JASA 2Chipman 1996, Canadian Journal of Statistics 13/25
  • 32. Model g(µ) =β0 + β1X1 + · · · + βpXp + βEE main effects + α1E(X1E) + · · · + αpE(XpE) interactions Reparametrization1 : αjE = γjEβjβE. 1Choi et al. 2010, JASA 2Chipman 1996, Canadian Journal of Statistics 13/25
  • 33. Model g(µ) =β0 + β1X1 + · · · + βpXp + βEE main effects + α1E(X1E) + · · · + αpE(XpE) interactions Reparametrization1 : αjE = γjEβjβE. Strong heredity principle2 : ˆαjE ̸= 0 ⇒ ˆβj ̸= 0 and ˆβE ̸= 0 1Choi et al. 2010, JASA 2Chipman 1996, Canadian Journal of Statistics 13/25
  • 34. Strong Heredity Model with Penalization arg min β0,β,γ 1 2 ∥Y − g(µ)∥ 2 + λβ (w1β1 + · · · + wqβq + wEβE) + λγ (w1Eγ1E + · · · + wqEγqE) wj = 1 ˆβj , wjE = ˆβj ˆβE ˆαjE 14/25
  • 37. TOM based on all subjects (a) TOM(Xall) 16/25
  • 38. TOM based on unexposed subjects (a) TOM(XE=0) 17/25
  • 39. TOM based on exposed subjects (a) TOM(XE=1) 18/25
  • 40. Difference of TOMs (a) |TOM(XE=1) − TOM(XE=0)| 19/25
  • 41. Results: Test set MSE 20/25
  • 43. Open source software • Software implementation in R: http://sahirbhatnagar.com/eclust/ • Allows user specified interaction terms • Automatically determines the optimal tuning parameters through cross validation • Can also be applied to genetic data 22/25
  • 45. Conclusions and Contributions • Large system-wide changes are observed in many environments 23/25
  • 46. Conclusions and Contributions • Large system-wide changes are observed in many environments • Dimension reduction is achieved through leveraging the environmental-class-conditional correlations 23/25
  • 47. Conclusions and Contributions • Large system-wide changes are observed in many environments • Dimension reduction is achieved through leveraging the environmental-class-conditional correlations • R software: http://sahirbhatnagar.com/eclust/ 23/25
  • 48. Limitations • There must be a high-dimensional signature of the exposure 24/25
  • 49. Limitations • There must be a high-dimensional signature of the exposure • Clustering is unsupervised 24/25
  • 50. Limitations • There must be a high-dimensional signature of the exposure • Clustering is unsupervised • Two tuning parameters 24/25
  • 51. Limitations • There must be a high-dimensional signature of the exposure • Clustering is unsupervised • Two tuning parameters • Cautionary note on simulation studies 24/25
  • 52. Limitations • There must be a high-dimensional signature of the exposure • Clustering is unsupervised • Two tuning parameters • Cautionary note on simulation studies • Need more samples . . . Got data? 24/25
  • 53. acknowledgements • Dr. Celia Greenwood • Dr. Blanchette and Dr. Yang • Dr. Luigi Bouchard, André Anne Houde • Dr. Steele, Dr. Kramer, Dr. Abrahamowicz • Maxime Turgeon, Kevin McGregor, Lauren Mokry, Dr. Forest • Greg Voisin, Dr. Forgetta, Dr. Klein • Mothers and children from the study 25/25