1. Targeted Maximum Likelihood Super Learning Application to assess effects in RCT,Observational Studies, and Genomics Mark van der Laan works.bepress.com/mark_van_der_laan Division of Biostatistics, University of California, Berkeley Workshop Brad Efron, December 2009

2. Outline Super Learning and Targeted Maximum Likelihood Learning Causal effect in observational studies Causal effect in RCTs Variable importance analysis in Genomics Multiple testing Case Control data

3. Motivation Avoid reliance on human art and parametric models Adapt the model fit to the data Target the fit to the parameter of interest Statistical Inference TMLE/SL Targeted Maximum Likelihood coupled with Super Learner methodology

4. 4 TMLE/SL Toolbox Targeted effects Effect of static or dynamic multiple time point treatments (e.g. on survival time) Direct and Indirect Effects Variable importance analysis in genomics Types of data Point treatment Longitudinal/Repeated Measures Censoring/Missingness/Time-dependent confounding. Case-Control Randomized clinical trials and observational data

5. Two-stage Methodology: SL/TMLE 1. Super Learning Works on a library of model fits Builds data-adaptive composite model by assigning weights Weights are optimized based on loss- function specific cross-validation to guarantee best overall fit 2. Targeted Maximum Likelihood Estimation Zooms in on one aspect of the model fit—the target Removes bias for the target.

6. Loss-Based Super Learning in Semiparametric Models Allows one to combine many data adaptive (e.g.) MLEs into one improved MLE. Grounded by oracle results for loss-function based cross-validation (vdL&D, 2003). Loss function needs to be bounded. Performs asymptotically as well as best (oracle) weighted combination, or achieves parametric rate of convergence.

7. Super Learner Flow Chart in Prediction

8. Super Learner Prediction

9. Targeted Maximum Likelihood Estimation Flow Chart Inputs Target feature map: Ψ( ) User Dataset The model is a set of possible probability distributions of the data Target Feature better estimates are closer to ψ(P TRUE ) Target feature values Initial P-estimator of the probability distribution of the data: P ˆ ˆ P TRUE P ˆ Ψ(P*) Ψ(P TRUE ) ˆ Ψ(P) Targeted feature estimator True value of the target feature Initial feature estimator Targeted P-estimator of the probability distribution of the data O(1), O(2), … O(n) Observations True probability distribution Model P* ˆ

10. Targeted Maximum Likelihood MLE/SL aims to do good job of estimating whole density Targeted MLE aims to do good job at parameter of interest General decrease in bias for parameter of Interest Fewer false positives Honest p-values, inference, multiple testing

11. (Iterative) Targeted MLE 1.Identify optimal strategy for “stretching” initial P –Small “stretch” -> maximum change in target 2.Given strategy, identify optimum amount of stretch by MLE 3.Apply optimal stretch to P using optimal stretching function -> 1 st -step targeted maximum likelihood estimator 4.Repeat until the incremental “stretch” is zero –Some important cases: 1 step to convergence 5.Final probability distribution solves efficient influence curve equation  (Iterative) T-MLE is double robust & locally efficient ^ ^

12. Example: Targeted MLE of Causal effect of point treatment on outcome Impact of Treatment on Disease

13. Likelihood of Point Treatment with Single Endpoint Outcome Draw baseline characteristics Draw treatment Draw missing indicator If not missing, draw outcome Counterfactual outcome distributions defined by intervening on treatment and enforcing no missingness Causal effects defined as user supplied function of these counterfactual distributions

14. TMLE for Average Causal Effect Observe predictors W, treatment A, missingness indicator Delta, and outcome Y: Target is additive causal effect: EY(1)-Y(0) Regress Y on treatment A and W and Delta=1 (e.g. Super Learning), and add clever covariate where –Then average regression over W for fixed treatment a: E n Y a Evaluate average effect: E n Y 1 -E n Y 0

15. TMLE is Collaborative Double Robust Suppose the initial fit minus true outcome regression is only a function of W through S Suppose the treatment mechanism adjusts correctly for a set of variables that includes S Then, the Targeted MLE is consistent. Thus the treatment mechanism only needs to adjust for covariates whose effect has not been captured by the initial fit yet. Formally,

16. TMLE/SL: more accurate information from less data Simulated Safety Analysis of Epogen (Amgen)

18. Example: Targeted MLE in RCT Impact of Treatment on Disease

19. The Gain in Relative Efficiency in RCT is function of Gain in R^2 relative to unadjusted estimator We observe (W,A,Y) on each unit A is randomized, P(A=1)=0.5 Suppose the target parameter is additive causal effect EY(1)-Y(0) The relative efficiency of the unadjusted estimator and a targeted MLE equals 1 minus the R-square of the regression 0.5 Q(1,W)+0.5 Q(0,W), where Q(A,W) is the regression of Y on A,W obtained with targeted MLE.

20. TMLE in Actual Phase IV RCT Study: RCT aims to evaluate safety based on mortality due to drug-to-drug interaction among patients with severe disease Data obtained with random sampling from original real RCT FDA dataset Goal: Estimate risk difference (RD) in survival at 28 days (0/1 outcome) between treated and placebo groups

21. TMLE in Phase IV RCT UnadjustedTMLE Estimate0.0340.043 p-value (RE)0.085 (1.000)0.009 (1.202) TMLE adjusts for small amount of empirical confounding (imbalance in AGE covariate) TMLE exploits the covariate information to gain in efficiency and thus power over unadjusted TMLE Results significant at 0.05

22. TMLE in RCT: Summary TMLE approach handles censoring and improves efficiency over standard approaches –Measure strong predictors of outcome Implications –Unbiased estimates with informative censoring –Improved power for clinical trials –Smaller sample sizes needed –Possible to employ earlier stopping rules –Less need for homogeneity in sample More representative sampling Expanded opportunities for subgroup analyses

23. Targeted MLE Analysis of Genomic Data Biomarker discovery, Impact of mutations on disease, or response to treatment

24. The Need for Experimentation Estimation of Variable Importance/Causal Effect requires assumption not needed for prediction “Experimental Treatment Assignment” (ETA) –Must be some variation in treatment variable A within every stratum of confounders W W must not perfectly predict/determine A g(a|W)>0 for all (a,W)

25. Biomarker Discovery: HIV Resistance Mutations Goal: Rank a set of genetic mutations based on their importance for determining an outcome –Mutations (A) in the HIV protease enzyme Measured by sequencing –Outcome (Y) = change in viral load 12 weeks after starting new regimen containing saquinavir How important is each mutation for viral resistance to this specific protease inhibitor drug? –Inform genotypic scoring systems

26. Stanford Drug Resistance Database All Treatment Change Episodes (TCEs) in the Stanford Drug Resistance Database –Patients drawn from 16 clinics in Northern CA 333 patients on saquinavir regimen Table 2: LPV Baseline Viral Load TCE (Change >= 1 Drug) Final Viral Load Change in Regimen <24 weeks 12 weeks

27. Parameter of Interest Need to control for a range of other covariates W –Include: past treatment history, baseline clinical characteristics, non-protease mutations, other drugs in regimen Parameter of Interest Variable Importance ψ = E[E(Y|A j =1,W)-E(Y|A j =0,W)] –For each protease mutation (indexed by j)

28. Parameter of Interest Assuming no unmeasured confounders (W sufficient to control for confounding)  Causal Effect is same as W-adjusted Variable Importance E(Y 1 )-E(Y 0 )=E[E(Y|A=1,W)-E(Y|A=0,W)]= ψ –Same advantages to T-MLE

29. Targeted Maximum Likelihood Estimation of the Adjusted Effect of HIV Mutation on Resistance to Lopinavir Stanford mutation score, http://hivdb.stanford.edu, accessed September, 1997http://hivdb.stanford.edu

30. Multiple Testing: Combining Targeted MLE with Type-I Error Control

31. Hypothesis Testing Ingredients  Data (X 1,…,X n ) Hypotheses Test Statistics Type I Error Null Distribution Marginal (p-values) or Joint distribution of the test statistics Rejection Region Adjusted p-values

32. Type I Error Rates FWER: Control the probability of at least one Type I error (V n ): P(V n > 0) ·  gFWER: Control the probability of at least k Type I errors (V n ): P(V n > k) ·  TPPFP: Control the proportion of Type I errors (V n ) to total rejections (R n ) at a user defined level q: P(V n /R n > q) ·  FDR: Control the expectation of the proportion of Type I errors to total rejections: E(V n /R n ) · 

33. Multivariate Normal Null Distribution Suppose null hypotheses involve testing of target parameters H_0: psi(j)<=0 We estimate target parameters with T- MLE, and use t-statistic for testing T-MLE as vector is asymptotically linear with known influence curve IC Valid joint null distribution for multiple testing is N(0,Sigma=E IC^2) Null distr can be inputted in any MTP (Dudoit, vdL, 2009, Springer)

34. GENERAL JOINT NULL DISTRIBUTION Let Q 0j be a marginal null distribution so that for j in set S 0 of true nulls Q 0j -1 Q nj (x)> x, for all x where Q nj is the j-th marginal distribution of the true distribution Q n (P) of the test statistic vector T n.

35. JOINT NULL DISTRUTION We propose as null distribution the distribution Q 0n of T n * (j)=Q 0j -1 Q nj (T n (j)), j=1,…,J This joint null distribution Q 0n (P) does indeed satisfy the wished multivariate asymptotic domination condition in (Dudoit, van der Laan, Pollard, 2004).

36. We estimate this null distribution Q 0n (P) with the bootstrap analogue: T n # (j)=Q 0j -1 Q nj # (T n # (j)) where # denotes the analogue based on bootstrap sample O 1 #,..,O n # of an approximation P n of the true distribution P. BOOTSTRAP BASED JOINT NULL DISTRIBUTION

37. Case-Control Weighted Targeted MLE Case-control weighting in targeted MLE successfully maps an estimation method designed for prospective sampling into a method for case-control sampling. This technique relies on knowledge of the true prevalence probability P(Y=1)=q 0 to eliminate the bias of the case-control sampling design. The procedure is double robust and locally efficient. It produces efficient estimators when its prospective sample counterpart is efficient.

38. Comparison to Existing Methodology Case-control weighted targeted MLE differs from other approaches as it can estimate any type of parameter, incorporates q 0, and is double robust and locally efficient.

39. Case-Control Weighted Targeted MLE We showed striking improvements in efficiency and bias in our case-control weighted method versus the IPTW estimator (Mansson 2007, Robins 1999), which does not utilize q 0. Our complete simulation results bolster our theoretical arguments that gains in efficiency and reductions in bias can be obtained by having known q 0 and using a targeted estimator. Table results for a sample of 500 cases and 1000 controls taken from a population of 120,000 where q0 = 0.035 Simulation Results

40. Closing Remarks True knowledge is embodied by semi or non- parametric models Semi-parametric models require fully automated state of the art machine learning (super learning) Targeted bias removal is essential and is achieved by targeted MLE Statistical inference is now sensible The machine learning algorithms are (super) efficient for the target parameters.

41. Closing Remarks (RC) Clinical Trials and Observational Studies can be analyzed with TMLE. TMLE outperforms current standards in analysis of clinical trials and observational studies, including double robust methods It is the only targeted method that is collaborative double robust, efficient, and naturally incorporates machine learning

42. Acknowledgements UC Berkeley –Oliver Bembom –Susan Gruber –Kelly Moore –Maya Petersen –Dan Rubin –Cathy Tuglus –Sherri Rose –Michael Rosenblum –Eric Polley –P.I. Ira Tager (Epi). Stanford Univ. –Robert Shafer Kaiser: Dr. Jeffrey Fessels.… FDA: Thamban Valappil, Greg Soon, Harvard: David Bangsberg, Victor DeGruttolas

43. References Oliver Bembom, Maya L. Petersen, Soo-Yon Rhee, W. Jeffrey Fessel, Sandra E. Sinisi, Robert W. Shafer, and Mark J. van der Laan, "Biomarker Discovery Using Targeted Maximum Likelihood Estimation: Application to the Treatment of Antiretroviral Resistant HIV Infection" (August 2007). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 221. http://www.bepress.com/ucbbiostat/paper221 http://www.bepress.com/ucbbiostat/paper221 Mark J. van der Laan and Susan Gruber, "Collaborative Double Robust Targeted Penalized Maximum Likelihood Estimation" (April 2009). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 246, http://www.bepress.com/ucbbiostat/paper246 http://www.bepress.com/ucbbiostat/paper246 Mark J. van der Laan, Eric C. Polley, and Alan E. Hubbard, "Super Learner" (July 2007). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 222. http://www.bepress.com/ucbbiostat/paper222 Mark J. van der Laan and Daniel Rubin, "Targeted Maximum Likelihood Learning" (October 2006). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 213. http://www.bepress.com/ucbbiostat/paper213 Oliver Bembom, Mark van der Laan (2008), A practical illustration of the importance of realistic individualized treatment rules in causal inference, Electronic Journal of Statistics. Mark J. van der Laan, "Statistical Inference for Variable Importance" (August 2005). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 188. http://www.bepress.com/ucbbiostat/paper188

44. References Oliver Bembom, Maya L. Petersen, Soo-Yon Rhee, W. Jeffrey Fessel, Sandra E. Sinisi, Robert W. Shafer, and Mark J. van der Laan, "Biomarker Discovery Using Targeted Maximum Likelihood Estimation: Application to the Treatment of Antiretroviral Resistant HIV Infection" (August 2007). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 221. http://www.bepress.com/ucbbiostat/paper221 Mark J. van der Laan, Eric C. Polley, and Alan E. Hubbard, "Super Learner" (July 2007). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 222. http://www.bepress.com/ucbbiostat/paper222 Mark J. van der Laan and Daniel Rubin, "Targeted Maximum Likelihood Learning" (October 2006). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 213. http://www.bepress.com/ucbbiostat/paper213 Yue Wang, Maya L. Petersen, David Bangsberg, and Mark J. van der Laan, "Diagnosing Bias in the Inverse Probability of Treatment Weighted Estimator Resulting from Violation of Experimental Treatment Assignment" (September 2006). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 211. http://www.bepress.com/ucbbiostat/paper211 Oliver Bembom, Mark van der Laan (2008), A practical illustration of the importance of realistic individualized treatment rules in causal inference, Electronic Journal of Statistics. Mark J. van der Laan, "Statistical Inference for Variable Importance" (August 2005). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 188. http://www.bepress.com/ucbbiostat/paper188

45. Collaborative T-MLE: Building the Propensity Score Based on Outcome Data Initial outcome regression based on super learning Construct rich set of one dimensional dimension reductions of W, that will be used as main terms below Select main terms in propensity score using forward selection based on emp. fit (e.g, loglik) of T-MLE If no main term increases emp. fit of TMLE, then carry out T-MLE update to update initial outcome regression Proceed to generate a sequence of T-MLE’s using increasingly nonparametric treatment mechanisms Select the wished T-MLE with cross-validation

46. The Likelihood for Right Censored Survival Data It starts with the marginal probability distribution of the baseline covariates. Then follows the treatment mechanism. Then it follows with a product over time points t At each time point t, one writes down likelihood of censoring at time t, death at time t, and it stops at first event Counterfactual survival distributions are obtained by intervening on treatment, and censoring. This then defines the causal effects of interest as parameter of likelihood.

47. TMLE with Survival Outcome Suppose one observes baseline covariates, treatment, and one observes subject up till end of follow up or death: One wishes to estimate causal effect of treatment A on survival T Targeted MLE uses covariate information to adjust for confounding, informative drop out and to gain efficiency

48. TMLE with Survival Outcome Target ψ 1 ( t 0 )= Pr(T 1 > t 0 ) and ψ 0 ( t 0 )= Pr(T 0 > t 0 ) – thereby target treatment effect, e.g., 1) Difference: Pr(T 1 > t 0 ) - Pr(T 0 > t 0 ), 2) Log RH: Obtain initial conditional hazard fit (e.g. super learner for discrete survival) and add two time-dependent covariates –Iterate until convergence, then use updated conditional hazard from final step, and average corresponding conditional survival over W for fixed treatments 0 and 1

49. TMLE analogue to log rank test The parameter, corresponds with Cox ph parameter, and thus log rank parameter Targeted MLE targeting this parameter is double robust

50. TMLE in RCT with Survival Outcome Difference at Fixed End Point % BiasPower95% Coverage Relative Efficiency KM<1%0.790.951.00 TMLE<1%0.910.951.44 % BiasPower95% Coverage Relative Efficiency KM13%0.880.921.00 TMLE<1%0.920.951.50 Independent Censoring Informative Censoring  TMLE: gain in power over KM  TMLE: unbiased

51. TMLE in RCT with survival outcome: Log rank analogue Independent Censoring Informative Censoring % Bias Power95% Coverage Relative Efficiency Log rank<2%0.130.951.00 TMLE (correct λ)<1%0.220.951.48 TMLE (mis-spec λ)<1%0.190.951.24  TMLE: gain in power over log rank % BiasPower95% Coverage Relative Efficiency Log rank32%0.20*0.931.00 TMLE (correct λ, correct G)<1%0.180.951.44 TMLE (mis-spec λ, correct G)<1%0.150.951.24  TMLE: unbiased

52. Tshepo Study The Tshepo Study is an open-label, randomized, 3x2x2 factorial design study conducted at Princess Marina Hospital in Gaborone, Botswana to evaluate the efficacy, tolerability, and development of drug resistance of six different first-line cART regimens Analysis Team: Mark van der Laan, Kelly Moore, Ori Stitelman, Victor De Gruttola Tshepo Study: C. William Wester, Ann Muir Thomas, Hermann Bussmann, Sikhulile Moyo, Joseph Makhema, Tendani Gaolathe, Vladimir Novitsky 2, Max Essex, Victor deGruttola, and Richard G. Marlink

53. Causal Effect of treatment NVP/EFV on time till death, time till virologic failure/dealth/treatment discontinuation, among other endpoints. Effect modification by baseline CD4 and Gender Causal effect modification by baseline CD4 Tshepo Study

54. Simulation Study Generated 500 data sets of 500 observations each per scenario. Varied How Well Hazard Was Specified (Well/Badly) Varied Level of ETA Violation (Low/Med/High) Observed performance of: IPCW, G-comp, Double Robust Augm IPCW, TMLE, C-TMLE and TMLE excluding the variable that causes the eta violation in the treatment mechanism.

55. MSE And Relative Efficiency of Estimating P(T A=1 >t 0 ) Low ETAMedum ETAHigh ETA Method Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ IPCW 0.00086 (1.1) 0.00086 (1.1) 0.00117 (1.2) 0.00117 (1.2) 0.00698 (1.9) 0.00698 (1.9) G-comp 0.00071 (0.9) 0.00086 (1.1) 0.00073 (0.7) 0.00161 (1.6) 0.00071 (0.2) 0.00231 (0.6) DR-IPCW 0.00085 (1.1) 0.00088 (1.1) 0.00106 (1.1) 0.00116 (1.2) 0.00455 (1.3) 0.00819 (2.3) TMLE 0.00085 (1.1) 0.00088 (1.1) 0.00105 (1.1) 0.00111 (1.1) 0.00347 (1.0) 0.00375 (1.0) TMLE w/o ETA 0.00084 (1.0) 0.00088 (1.1) 0.00083 (0.8) 0.00087 (1.2) 0.00079 (0.2) 0.00085 (0.2) c-TMLE 0.00085 (1.1) 0.00104 (1.3) 0.00085 (0.9) 0.00122 (1.2) 0.00080 (0.2) 0.00101 (0.3)

56. Percent Of Time Influence Curve Based 95 Percent CI Includes Truth Low ETAMedum ETAHigh ETA Method Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ IPCW 98.4% 97.2% 89.8% Double Robust 94.2%95.4%95.0%96.4%94.8%95.8% TMLE 94.0%95.2%95.4%95.6%84.8%87.0% TMLE w/o ETA 94.2%94.8%94.4%95.6%94.6%95.0% c-TMLE 94.6%93.6%96.0%95.0%94.8%93.2%

57. Properties of c-TMLE Algorithm Low ETAMedum ETAHigh ETA Method Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ Well Specified λ Badly Specified λ Mean Moves 2.79.12.17.32.97.1 % No Moves 68.8%0.2%74.2%6.4%62.6%1.6% % ETA Variable 27.8%60.4%1.2%6.0%0.0%0.4%

58. Case-Control Weighted Targeted MLE Nested case-control studies are quite common, and can be particularly beneficial in biomarker studies. We simulated a nested case- control study within a cohort study examining the impact of higher physical activity on death in an aging population. With known q 0, we obtained similar point estimates in the case-control analysis. Full cohort analysis sample size was 2066. Case-control analysis was repeated in 100 simulations with 269 cases and 538 randomly selected controls. “% Rej” indicates average percent rejected tests among the 100 simulations. Data Analysis

59. Description of Simulation –100 subjects each with one random X (say a SNP’s) uniform over 0, 1 or 2. –For each subject, 100 binary Y’s, (Y 1,...Y 100 ) generated from a model such that: first 95 are independent of X Last 5 are associated with X All Y’s correlated using random effects model –100 hypotheses of interest where the null is the independence of X and Y i. –Test statistic is Pearson’s  2 test where the null distribution is  2 with 2 df. –In this case, Y 0 is the outcome if, counter to fact, the subject had received A=0. –Want to contrast the rate of miscarriage in groups defined by V,R,A if among these women, one removed decaffeinated coffee during pregnancy.

60. Description of Simulation, cont. –Simulated data 1000 times –Performed the following MTP’s to control FWER at 5%. Bonferroni Null centered, re-scaled bootstrap (NCRB) – based on 5000 bootstraps Quantile-Function Based Null Distribution (QFBND) –Results NCRB anti-conservative (inaccurate) Bonferroni very conservative (actual FWER is 0.005) QFBND is both accurate (FWER 0.04) and powerful (10 times the power of Bonferroni).

61. Empirical Bayes/Resampling TPPFP Method We also devised another resampling based multiple testing procedure, controlling the proportion of false positives to total rejections. This procedure involves: –Randomly sampling a guessed set of true null hypotheses: H 0 (j)~Bernoulli (Pr(H 0 (j)=1|T j )=p 0 f 0 (T j )/f(T j ) ) based on the Empirical Bayes model: T j |H 0 =1 ~f 0 T j ~f p 0 =P(H 0 (j)=1) (p0=1 conservative) –Our joint bootstrap null distribution of test statistics.

62. Emp. BayesTPPFP Method 1.Grab a column from the null distribution of length M. 2.Draw a length M binary vector corresponding to S 0n. 3.For a vector of c values calculate: 4.Repeat 1. and 2. 10,000 times and average over iterations. 5.Choose the c value where P(r n (c) > q)· .

63. Examples/Simulations

64. Bacterial Microarray Example Airborne bacterial levels in specific cities over a span of several weeks are being collected and compared. A specific Affymetrics array was constructed to quantify the actual bacterial levels in these air samples. We will be comparing the average (over 17 weeks) strain-specific intensity in San Antonio versus Austin, Texas.

65. 420 Airborne Bacterial Levels 17 time points San Antonio vs Austin Procedure  Number Rejected BonferroniFWE  = 0.05 5  = 0.10 6 AugmentationTPPFP  = 0.05 11  = 0.10 14 E.Bayes/BootstrapTPPFP  = 0.05 13  = 0.10 21

66. CGH Arrays and Tumors in Mice 11 Comparative genomic hybridization (CGH) arrays from cancer tumors of 11 mice. DNA from test cells is directly compared to DNA from normal cells using bacterial artificial chromosomes (BACs), which are small DNA fragments placed on an array. With CGH: –differentially labeled test [tumor] and reference [healthy] DNA are co-hybridized to the array. –Fluorescence ratios on each spot of the array are calculated. –The location of each BAC in the genome is known and thus the ratios can be compiled into a genome-wide copy number profile

67. Plot of Adjusted p-values for 3 procedures vs. Rank of BAC (ranked by magnitude of T-statistic)

68. Acknowledgements: UC Berkeley –Susan Gruber –Ori Stittelman –Kelly Moore –Maya Petersen –Dan Rubin –Cathy Tuglus –Sherri Rose –Michael Rosenblum –Eric Polley -P.I. Ira Tager (Epi). Stanford: Dr. Robert Shafer Kaiser: Dr. Jeffrey Fessels Harvard: Dr. David Bangsberg UCSF: Dr. Steve Deeks FDA: Thamban Vallappil, Greg Soon

69. TargetDiscovery demo

70. Targeted Maximum Likelihood in Semiparametric Regression Implementation just involves adding a covariate h(A,W) to the regression model –When m(A,W|  ) is linear Requires estimating E(A|W) –E.g. Expected value of A given confounders W Robust: Estimate is consistent if either –E(A|W) is estimated consistently –E(Y|A,W) is estimated consistently

71. Model-based Variable Importance - Measure of Importance: When the variable of interest (A) is continuous - Given Observed Data: O=(A,W,Y)~P o W*={possible biomarkers, demographics, etc..} A=W* j (current biomarker of interest) W=W* -j 

72. Evaluation of Biomarker methods: simulation No appreciable difference in ranking by importance measure or p-value –plot above is with respect to ranked importance measures List Length for linear regression and randomForest increase with increasing correlation, Variable Importance w/LARS stays near minimum (10) through ρ=0.6, with only small decreases in power Linear regression list length is 2X Variable Importance list length at ρ=0.4 and 4X at ρ=0.6 RandomForest (RF2) list length is consistently short than linear regression but still is 50% than Variable Importance list length at ρ=0.4, and twice as long at ρ=0.6 Variable importance coupled with LARS estimates true causal effect and outperforms both linear regression and randomForest Minimal List length to obtain all 10 “true” variables

73. Example: Biomarker Discovery HIV resistance mutations Accounting for ETA violations

74. The Need for Experimentation Estimation of Variable Importance/Causal Effect requires assumption not needed for prediction “Experimental Treatment Assignment” (ETA) –Must be some variation in treatment variable A within every stratum of confounders W W must not perfectly predict/determine A g(a|W)>0 for all (a,W)

75. Stanford Drug Resistance Database All Treatment Change Episodes (TCEs) in the Stanford Drug Resistance Database –Patients drawn from 16 clinics in Northern CA 333 patients on saquinavir regimen Table 2: LPV Baseline Viral Load TCE (Change >= 1 Drug) Final Viral Load Change in Regimen <24 weeks 12 weeks

76. Parameter of Interest Need to control for a range of other covariates W –Include: past treatment history, baseline clinical characteristics, non-protease mutations, other drugs in regimen Parameter of Interest: Variable Importance ψ = E[E(Y|A j =1,W)-E(Y|A j =0,W)] –For each protease mutation (indexed by j)

77. Unadjusted estimates

78. Delta-adjusted T-MLE

79. Realistic Causal Effect of Physical Activity Level Given elderly population (SONOMA), we wish to establish effect of activity on 5 year mortality. A realistic individualized treatment rule indexed by activity level a is defined as, given individuals covariates W, realistic activity level closest to assigned level a: Causal relative risk:E(Y(d(a))/E(Y(d(0))

80. Realistic Rules Indexed by 5 Activity Levels

81. Sparse-Data(ETA)-Bias for Different Causal Effects

82. Estimates of Realistic/Intention to Treat/Static Causal Relative Risks

84. Evaluation of Biomarker Discovery Methods > Univariate Linear Regression Importance measure: Coefficient value with associated p-value Measures marginal association > RandomForest (Breiman 2001) Importance measures (no p-values) RF1: variable’s influence on error rate RF2: mean improvement in node splits due to variable > Variable Importance with TMLE based on LARS Importance measure: causal effect Formal inference, p-values provided LARS used to fit initial E[Y|A,W] estimate W={marginally significant covariates}  All p-values are FDR adjusted > Test methods ability to determine “true” variables under increasing correlation conditions Ranking by measure and p-value Minimal list necessary to get all “true”? > Variables Block Diagonal correlation structure: 10 independent sets of 10 Multivariate normal distribution Constant ρ, variance=1 ρ={0,0.1,0.2,0.3,…,0.9} > Outcome Main effect linear model 10 “true” biomarkers, one variable from each set of 10 Equal coefficients Noise term with mean=0 sigma=10 –“realistic noise” Methods Simulation Study

85. THEOREM Consider any generalized linear regression model from the normal, binomial, Poisson, Gamma, Inverse Gaussian, with canonical link function, in which the linear part contains the treatment variable as main term and contains intercept. Let r be cts diff. Then the MLE of r(EY(0),EY(1)) is asymptotically consistent and locally efficient.

86. Median number of false positives

87. Patients Power TargetImpact advantage CrudeANCOVATargetImpactCrudeANCOVA Viral Load below 50 copies/mL 50014.6%x32.3%2.4x 100031.7%x60.4%2.6x 300070.1%x97.0%2.2x Change in CD4 counts 50013.1%97.5%99.4%27.51.4 100023.9%100% 31.21.5 300056.3%100% 30.51.5 Average number of patients needed in a GST: 2,593 with a crude analysis 1,980 with TargetImpact $15M or 23% savings (given the typical cost of $25K per patient)