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Jeff Leek's JHU Job Talk from 2009 on surrogate variable analysis.

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- 1. A General Framework for Multiple Testing Dependence Jeffrey Leek Johns Hopkins University School of Medicine
- 2. High-dimensional multiple hypothesis testing is common. Problem: Dependence between tests can result in incorrect statistical and scientific results. A solution: Define and address multiple testing dependence at the level of the data – not the P-values. Big Picture Ideas
- 3. High-Dimensional Multiple Testing Is Common Spatial EpidemiologyBrain Imaging Molecular Biology
- 4. 4 Inflammation and the Host Response to Injury mRNA Expression ~50,000 genes Clinical Data >150 clinical variables Patient 1 Patient 2 Patient 166…. MOF measures severity of injury
- 5. Data at Initial Time Point Multiple Organ Failure
- 6. Simple Analysis 1. Fit the model to the data, xi, for gene i: xi = ai + biMOF + ei 2. Calculate P-values for testing the hypotheses: H0: bi = 0 vs. H1: bi ≠ 0 3
- 7. Four “Replicated” Studies Phase 1 Phase 3 Phase 2 Phase 4 P-value P-value P-value P-value Frequency Frequency Frequency Frequency
- 8. • Data for test i: • “Primary variable(s)”: • Model: • Hypothesis test i: € xi = xi1,xi2,…,xin( ) € Y = y1,y2,…,yn( ) € xij = ai + biksk y j( ) k=1 d ∑ + eij H0i :bi ∈ Ω0 H1i :bi ∈ Ω1 {m hypothesis tests, n observations per test} Start With The Whole Data
- 9. = + X = B S(Y) + E observations tests Underlying Model
- 10. A Simple Simulated Example Independent E Dependent E Genes Genes Arrays Arrays
- 11. Null P-Value Distributions Independent E Dependent E Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency P-value P-value P-value P-value P-value P-value P-value P-value
- 12. Null P-Value Distributions |ρ| = 0.40 |ρ| = 0.31 |ρ| = 0.10 |ρ| = 0.00Correlation Independent E Dependent E Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency P-value P-value P-value P-value P-value P-value P-value P-value
- 13. Null Distribution Behavior Dependent E Independent E
- 14. False Discovery Rate Estimates Independent E Dependent E
- 15. Ranking Estimates Independent E Dependent E
- 16. Data X Fit Model X= BS + E Obtain and R € ˆB Calculate P-values Form P-value Threshold When To Address Dependence? Form Test-Statistics and Null Distribution
- 17. Data X Fit Model X= BS + E Obtain and R € ˆB Calculate P-values Form P-value Threshold When To Address Dependence? Form Test-Statistics and Null Distribution Existing Approaches Empirical null approaches modify the null distribution at the test-statistic level Dependence adjustments conservatively modify the P-value threshold
- 18. Examples of Existing Approaches • Empirical Null – Devlin and Roeder Biometrics (1999) – Efron JASA (2004) – Schwartzman AOAS (2008) • Error Rate Adjustments – Benjamini and Yekutieli Annals of Statistics (2001) – Romano, Shaikh, and Wolf Test (2001) – Dudoit, Gilbert, van der Laan Biometrical Journal (2008)
- 19. Data X Fit Model X= BS + E Obtain and R € ˆB Calculate P-values Form P-value Threshold When To Address Dependence? Form Test-Statistics and Null Distribution Our Approach Fit the model: X = BS + ΓG + U where G is a valid dependence kernel
- 20. Dependence and bias are no longer present at any of these steps; standard methods can be used. Data X Fit Model X= BS + E Obtain and R € ˆB Calculate P-values Form P-value Threshold When To Address Dependence? Form Test-Statistics and Null Distribution Our Approach Fit the model: X = BS + ΓG + U where G is a valid dependence kernel
- 21. New Dependence Definitions Definition – Data X are population-level multiple testing dependent if: Definition - Data X are estimation-level multiple testing dependent if: Leek and Storey (2008)
- 22. Structure in E Array MOF1Genes Signal + Dependent Noise Dependent Noise Independent Noise
- 23. = + X = B S + E observations tests data random variation primary variables Decomposing E
- 24. = + X = B S + H + U tests + independent variation observations data primary variables dependent variation Decomposing E
- 25. = + X = B S + Γ G + U tests + independent variation observations data primary variables dependence kernel Decomposing E H
- 26. Decomposing E Theorem Let the data be distributed according to the model: Suppose that for each ei there is no Borel measurable function, g, such that ei =g(ei,…,ei-1,ei+1,…,em) almost surely. Then there exist matrices Γ(m×r), G(r×n) (r ≤ n) and U(m×n) such that: where the rows of U are independent and ui ≠ 0 and ui=hi(ei) for a non-random Borel measurable function hi. Leek and Storey (2008)
- 27. Dependence Kernel Leek and Storey (2008) Definition – Dependence Kernel An r ×n matrix G forms a dependence kernel for the data X, if the following equality holds: X = BS + E = BS + ΓG + U where the rows of U are independent.
- 28. Fitting S & G Results In Independent Tests Leek and Storey (2008) Theorem Let G be any valid dependence kernel for the data X. Suppose that the model: is fit by least squares resulting in residuals: if the rowspace jointly spanned by S and G has dimension less than n, then the ri and the are jointly independent given S and G and: € ˆbi
- 29. = + X = B S + Γ G + U tests + independent variation observations data primary variables dependence kernel A “Blessing” of Dimensionality
- 30. Iteratively Reweighted Surrogate Variable Analysis 1. Estimate the row dimension, , of G. 2. Form an initial estimate equal to the first right singular vectors of R = X - S. 3. Estimate . 4. Weight the ith row of X by and set to be the first right singular vectors of the weighted matrix. ˆG(b+1) € ˆr € ˆB Iterate for b=0,…,B: € ˆG0 ˆr € X = BS + ΓG + U € xi = biS + γiG + ui Whole data: Test i data: € ˆr
- 31. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 32. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 33. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 34. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 35. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 36. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 37. An Example of the IRW-SVA Algorithm The Data True GEstimate of GPr(G & !S)
- 38. Iteratively Re-weighted Surrogate Variable Analysis 1. Estimate the row dimension, , of G. 2. Form an initial estimate equal to the first right singular vectors of R = X - S. 3. Estimate . 4. Weight the ith row of X by and set to be the first right singular vectors of the weighted matrix. ˆG(b+1) € ˆr € ˆB € ˆG0 ˆr € X = BS + ΓG + U € xi = biS + γiG + ui Whole data: Test i data: € ˆr Iterate for b=0,…,B:
- 39. 1. Buja and Eyuboglu (1992) proposed a permutation approach. 2. Patterson, Price, and Reich (2006) proposed a sequential testing strategy based on Tracey- Widom theory. 3. Leek (in preparation) proposes an eigenvalue estimator that is consistent in the number of tests. Estimating The Row Dimension of G
- 40. 1. Assume the data follow X = BS + ΓG + U, where G and S have row dimensions r and d, r + d < n. 2. Calculate the singular values s1,…, sn of X and choose b, such that r+d < b. 3. Calculate the eigenvalues, λ1,…, λn of where P = I - S(STS)-1ST and R = XP. 4. Set ˆr = 1 λj > m−1/ 3 ( ) j=1 n ∑ € € 1 m RT R − sb 2 P[ ] Estimating The Row Dimension of G
- 41. Theorem As , is a consistent estimate of the row dimension of G, provided that: (1) uij are independent (2) E[uij]=0 (3) (4) (5) ΓTΓ is positive definite with unique eigenvalues € m → ∞ € E[uij 2 ] = σi 2 < M1 € E[uij 4 ] < M2 € lim m→∞ 1 m Leek (In Prep.) € ˆr = 1 λj > m−1/ 3 ( ) j=1 n ∑ Estimating The Row Dimension of G
- 42. Iteratively Re-weighted Surrogate Variable Analysis 1. Estimate the row dimension, , of G. 2. Form an initial estimate equal to the first right singular vectors of R = X - S. 3. Estimate . 4. Weight the ith row of X by and set to be the first right singular vectors of the weighted matrix. ˆG(b+1) € ˆr € ˆB € ˆG0 ˆr € X = BS + ΓG + U € xi = biS + γiG + ui Whole data: Test i data: € ˆr Iterate for b=0,…,B:
- 43. Break The Estimation Into Two Components
- 44. 1. Form F-statistics F1,…,Fm for testing the hypotheses: 2. Bootstrap from the conditional null model to obtain null- statistics , k =1,…K. 3. From Bayes’ Theorem: where and . Estimating the Probability Weights € F1 0k ,...,Fm 0k € Fi 0k ~ g0 € Fi ~ π0g0 + (1− π0)g1
- 45. 1. Form F-statistics F1,…,Fm for testing the hypotheses: 2. Bootstrap from the conditional null model to obtain null- statistics , k =1,…K. 3. From Bayes’ Theorem: 4. Estimate the ratio of the densities with a non-parametric logistic regression where Fi are “successes” and Fi 0k are “failures” (Anderson and Blair 1982). where and . . Estimating the Probability Weights € F1 0k ,...,Fm 0k € Fi 0k ~ g0 € Fi ~ π0g0 + (1− π0)g1
- 46. 1. Form F-statistics F1,…,Fm for testing the hypotheses: 2. Bootstrap from the conditional null model to obtain null- statistics , k =1,…K. 3. From Bayes’ Theorem: 4. Estimate the ratio of the densities with a non-parametric logistic regression where Fi are “successes” and Fi 0k are “failures” (Anderson and Blair 1982). 5. Estimate π0 according to Storey (2002). where and . Estimating the Probability Weights € F1 0k ,...,Fm 0k € Fi 0k ~ g0 € Fi ~ π0g0 + (1− π0)g1
- 47. Estimating the Probability Weights Estimate of posterior probability bi ≠ 0.
- 48. SVA-Adjusted Analysis 1. Estimate G with IRW-SVA 2. Fit 3. Test the hypotheses € H0i :bi ∈ Ω0 H1i :bi ∈ Ω1
- 49. A Simple Simulated Example Independent E Dependent E Genes Genes Arrays Arrays
- 50. Null Distribution Behavior Dependent E Independent E Dependent E + IRW-SVA
- 51. False Discovery Rate Estimates Independent E Dependent E Dependent E + IRW-SVA True False Discovery Rate True False Discovery Rate True False Discovery Rate Q-value Q-value Q-value
- 52. Ranking Estimates Independent E Dependent E Dependent E + IRW-SVA Ranking by True Signal to Noise Ranking by True Signal to Noise Ranking by True Signal to Noise AverageRankingbyT-Statistic AverageRankingbyT-Statistic AverageRankingbyT-Statistic
- 53. 53 Inflammation and the Host Response to Injury mRNA Expression ~50,000 genes Clinical Data >150 clinical variables Patient 1 Patient 2 Patient 166…. MOF1 measures severity of injury
- 54. Phase 1 Phase 2 Phase 3 Phase 4 Four “Replicated” Studies FrequencyFrequency P-value P-value P-value P-value P-value P-value P-value P-value Frequency Frequency Frequency Frequency Frequency Frequency Frequency
- 55. Functional Enrichment Across Phases Number of phases in which a significant pathway appears Percentoftotalsignificantpathways 1 of 4 2 of 4 3 of 4 4 of 4 Unadjusted IRW-SVAAdjusted
- 56. • High-dimensional hypothesis testing is common. • Dependence between tests can result in incorrect statistical and scientific inference. • We can define and address dependence at the level of the model using the dependence kernel. • IRW-SVA can be used to improve inference in high-dimensional multiple hypothesis testing. Summary
- 57. Future Work • Multiple Testing – Develop dependence kernel estimates for spatial data – Develop diagnostic tests for multiple testing procedures • High-Dimensional Asymptotics – Extend methods for asymptotic SVD to binary data • Feature Selection for High-Dimensional Classifiers – Extensions of top-scoring pairs (TSP) to survival data – Theoretical connections to LDA and SVM – Embedding TSP in a logic regression framework
- 58. Thank You
- 59. 1. Calculate the residuals R = X - S. 2. Calculate the singular values of R, d1,…,dn. 3. Permute each row of R individually to get R0. 4. Take the SVD of the residuals R* = R0 - S to obtain null singular values . 5. Compare di to for k=1,…,K to calculate a P- value for the ith right singular vector. Estimating The Row Dimension of G € ˆB € ˆB0 € di0 k € di0 k For k =1,…,K do steps 3-4: Buja and Eyuboglu (1992)
- 60. Why Does This Work? Leek and Storey (2007), Leek and Storey (2008) Useful Fact: X = BS + E = BS + ΓG + U = BS + ΛH + U if G and H have the same column space.
- 61. • References: Benjamini Y and Hochberg Y. (1995), “Controlling the false discovery rate – a practical and powerful approach to multiple testing.” JRSSB, 57: 289-300. De Castro MC, Monte-Mor RL, Sawyer DO, and Singer, BH. (2005), “Malaria risk on the amazon frontier.” PNAS, 103: 2452-2457. Delin B and Roeder K. (1999), “Genomic control for association studies.” Biometrics, 55: 997-1004. Efron B. (2004) “Large-scale simultaneous hypothesis testing: The choice of a null hypothesis.” JASA, 99: 96-104. Leek JT and Storey JD. (2008) “A general framework for multiple testing dependence.” Proceedings of the National Academy of Sciences , 105: 18718-18723. Leek JT and Storey JD. (2007) “Capturing heterogeneity in gene expression studies by ‘Surrogate Variable Analysis’.” PLoS Genetics, 3: e161. Taylor JE and Worsley KJ. (2007) “Detecting sparse signals in random fields, with applications to brain mapping.” JASA, 102: 913-928. Thank You
- 62. 1. Perform each hypothesis test individually. 2. Obtain the test-statistic for each test. 3. Compare distribution of test-statistics to the theoretical null distribution. 4. Adjust theoretical null so that it matches the observed statistics in a low signal region. Empirical Null
- 63. Theoretical Null Efron (2004)
- 64. Theoretical Null Empirical Null Efron (2004)
- 65. Empirical Null Results in Incorrect Null Distribution Dep. Kernel
- 66. • Observed statistics or observed P-values come from mixture distribution: π0g0 + π1g1 • Dependence distorts g0 … can go either way: • Must use full data set to capture dependence With Confounding Empirical Null is Ill-Posed

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