Naihua Duan UCLA and RAND May 2000 Selection bias in treatment assignment/delivery

1. Research Designs, Statistical Strategies for Dealing with Selection Bias in Treatment Delivery, and Limitations • Naihua Duan • UCLA and RAND • May 2000 • Selection bias in treatment assignment/delivery • Research designs • Mitigating for overt selection bias • Dealing with hidden selection bias • Discussions NIDA/NIMH Substance Abuse Conference

2. Selection Bias in Treatment Delivery In naturalistic settings: Pre-treatment health  treatment delivered Pre-treatment health  outcome Treated group dissimilar from untreated group Direct comparison of treated vs. untreated results in biased estimate for treatment effect Need to mitigate selection bias in order to assess treatment effect more appropriately NIDA/NIMH Substance Abuse Conference

3. Selection Bias in Treatment Delivery: Typology Overt selection bias Treatment related to covariates T  X Given covariates, treatment independent of outcome T  Y | X (ignorability) Like a stratified randomized experiment Hidden selection bias Given covariates, treatment still related to outcome T  Y | X Rosenbaum (1995) Observational Studies, Springer-Verlag NIDA/NIMH Substance Abuse Conference

4. Research Designs • Ideal randomized clinical trial (RCT)Imperfect RCT with noncomplianceRandomized encouragement design (RED)Observational studies • Settings: controlled vs. naturalistic • Treatment assignment/delivery: mandated vs. choice • Treated vs. untreated groups: balance vs. imbalance • Research questions: efficacy vs. adoption, program effect, and efficacy • Analytic strength: interval validity vs. external validity NIDA/NIMH Substance Abuse Conference

5. Randomized Clinical Trial Intensive efforts made to mandate assignment NIDA/NIMH Substance Abuse Conference

6. Randomized Encouragement Design Encouragement: training, providing information, case management, reducing barriers (child care, transportation, flexible hours, reducing co-payment…), decorate waiting room,... NIDA/NIMH Substance Abuse Conference

7. Randomized Encouragement Design: Features Analogous to marketing experiment Encouragement  higher adoption rate?  better overall outcomes?  better outcomes for new users? Naturalistic, incorporate user preferences, facilitate choice Broader participation, external validity, dissemination Zelen (1979 NEJM, 1990 Stat. in Medicine: randomized consent design), Holland (1988) in Clogg CC, ed. Sociological Methodology, Hirano et al. (2000, Biostatistics), Wells et al. (2000, JAMA), Duan et al. (2000, manuscript) NIDA/NIMH Substance Abuse Conference

8. Mitigating Overt Selection Bias Assume overt selection bias: T  X Assume no hidden selection bias: T  Y | X Covariate adjustment through ANCOVA Stratification (through propensity score method) Matching (through propensity score method) NIDA/NIMH Substance Abuse Conference

9. Covariate Adjustment Y =  + T b + X g (+ T X d) + e Extrapolation can be risky when imbalance is substantial Y T = 1 T = 0 X: Pre-Tx health NIDA/NIMH Substance Abuse Conference

10. Limitations for Covariate Adjustment Extrapolation can be risky when imbalance is substantial Compare apples and oranges, rely on model to adjust Careful model diagnosis is essential Multivariate imbalance might be more problematic Why so popular? Ease of push-botton analysis Almost always gives an answer Could be a bad answer! NIDA/NIMH Substance Abuse Conference

11. Stratification When Covariate Is Univariate Stratify, then compare by stratum Compare apples and apples, oranges and oranges Y T = 1 T = 0 X: Pre-Tx health NIDA/NIMH Substance Abuse Conference

12. Stratification: Procedure Stratify, then compare treated vs. untreated by stratum Two-sample comparison within each stratum ANCOVA within each stratum Assess interactions across strata Synthesize treatment effects across strata Weighted average Overall intervention effect on treated Overall intervention effect on untreated Overall intervention effect on entire pool Can be specified as ANCOVA with interactions Nonparametric regression of Y on X, stratified by T NIDA/NIMH Substance Abuse Conference

13. Covariate Adjustment, Nonparametric Version OK for low dimension X Curse of dimensionality for high dimension X Y T = 1 T = 0 X: Pre-Tx health NIDA/NIMH Substance Abuse Conference

14. Stratification: Features Why not used as widely as ANCOVA? Does not always give an answer Provides warning where imbalance is too severe Not a push-button operation, but not difficult How to stratify? Clinical judgement Usually not critical; sensitivity analysis recommended Cochran-Rubin-Rosenbaum recommend 5 strata How to stratify with multi-dimensional covariates? Curse of dimensionality Use propensity score method to reduce dimensionality NIDA/NIMH Substance Abuse Conference

15. Propensity Score Method Assumeovert selection bias, no hidden selection bias T  Y | X p = p(X) = P(T = 1 | X) is the propensity score Example: logit(p(X)) = a + X b p(X) is a balancing score (most parsimonious) T  X | p(X) Given p(X), treatment independent of outcome T  Y | p(X) Need only stratify by propensity score Other dimensions of X can be neglected in assessing treatment effect NIDA/NIMH Substance Abuse Conference

16. Propensity Score Method: Procedure Estimate p(X) = P(T = 1 | X) Logistic regression of T on X Stratify sample (X, T, and Y) by estimated p(X) or Xb Sort out apples and oranges Analyze each stratum, compare treated vs. untreated Two sample comparison within stratum ANCOVA within stratum Assess interactions across strata Synthesize treatment effects across strata Weighted average... NIDA/NIMH Substance Abuse Conference

17. Propensity Score Method: Stratification for Y Stratify, then compare by stratum Compare apples and apples, oranges and oranges Y T = 0 T = 1 Xb NIDA/NIMH Substance Abuse Conference

18. Propensity Score Method: Model Specification Specification of propensity score model Lean towards over-fitting vs. under-fitting? Model diagnosis: are the covariates balanced across treatment groups within each stratum? Stratify by propensity score and key covariates (one or two)? Model misspecification less serious than ANCOVA? Only rank of estimated propensity score is used Stratification not sensitive to minor perturbations in model Limited empirical evidence (Drake 1993 Biometrics, Dehejia and Wahba 1999 JASA) NIDA/NIMH Substance Abuse Conference

19. Propensity Score Method: Options Stratification Matching (case-control) Curse of dimensionality relevant, less critical Mahalonobis distance matching Match on propensity score (+ a few key covariates?) Design stage vs. analysis stage Primary vs. secondary data collection ANCOVA: regress Y on T and propensity score (+ a few key covariates? + interactions?) Nonparametric regression? Stratified by T? NIDA/NIMH Substance Abuse Conference

20. Dimension Reduction Fundamental challenge in ANCOVA Valid assessment of treatment effect can be obtained using nonparametric regression of Y on X, stratified by T Curse of dimensionality No obvious way to reduce dimensionality? Propensity score method is an elegant way to reduce dimensionality Alternative dimension reduction methods? Slicing regression (Duan and Li 1991 Annals of Statistics, Li 1991 JASA): use inverse regression of X on Y... NIDA/NIMH Substance Abuse Conference

21. Propensity Score Method: References Rosenbaum and Rubin (1983 Biometrika, 1984 JASA) Lavori, Dawson, and Mueller (1994 Stat. in Medicine) Rosenbaum (1995) Observational Studies, Springer-Verlag Rubin (1997) Annals of Internal Medicine D’Agastino (1998 Stat. in Medicine) Normand et al. (2000 manuscript) Hirano et al. (2000 manuscript) NIDA/NIMH Substance Abuse Conference

22. Dealing with Hidden Selection Bias T  Y | X Very challenging problem, no easy solutions Given X, how does treatment depend on outcome? Overt selection bias can be made to look like stratified randomized experiment Hidden selection bias cannot be made to… Rosenbaum-Rubin’s sensitivity analysis Instrumental variable analysis a la Rubin Causal Model Selection modeling NIDA/NIMH Substance Abuse Conference

23. Rosenbaum’s Sensitivity Analysis: General Principle How robust is the observed treatment effect against hidden selection bias? Analogous to pattern mixture model for missing data Formulate a family of plausible models for hidden selection bias (from mild to severe) Assess treatment effect under each model Determine how much hidden selection bias wipes out treatment effect Is this much hidden selection bias realistic? Specificity analysis NIDA/NIMH Substance Abuse Conference

24. Unobserved Confounder Model logit(p(Xi)) = a + Xib + Uig 0  Ui  1 g > 0: maximum impact of unobserved hidden bias G = exp(g) is the upper bound between p(Xi)’s | X Example: 2 x 2 table (analyzed with Fisher’s exact test) Worst case scenario for hidden bias: Unobserved health is a perfect predictor of survival Healthy patients are more likely to receive treatment Ui = 1 for all survivors; = 0 for all deceaseds Null distribution is a tilted hypergeometric distribution Given g, derive P-value under tilted hypergeometric distribution NIDA/NIMH Substance Abuse Conference

25. Rosenbaum’s Sensitivity Analysis: Limitations Does not give THE answer (should we expect one?) Rosenbaum’s sensitivity analysis is based on permutation test (tilted by hidden selection bias) Permutation test is the foundation for randomized trials, but rarely used: heavy computation burden Used more in recent years, e.g., COMMIT Special software required for tilted permutation test Programming logic not difficult Very heavy computation burden Inertia for users to stay with familiar packages NIDA/NIMH Substance Abuse Conference

26. Instrumental Variable (IV) Analysis for RED,a la Rubin Causal Model Encouragement intervention serves as instrumental variable Assume binary intervention (I = 0, 1) binary treatment (T = 0, 1) T(0) T(1) Category 0 0 Never takers 0 1 Compliers (new users) 1 0 Defiers (assumed to be absent) 1 1 Always takers Very likely different beyond observed characteristics NIDA/NIMH Substance Abuse Conference

27. IV Analysis: Observed Compliance Status I = 0: Untreated: C or N Treated: A or D I = 1: Untreated: N or D Treated: C or A Randomized encouragement design Compliance status distributed similarly across intervention groups %(C) = %(treated | I = 1) - %(treated | I = 0) = %(untreated | I = 0) - %(untreated | I = 1) NIDA/NIMH Substance Abuse Conference

28. IV Analysis: Intervention Effect by Subgroups Key assumption: Effect of encouragement intervention mediated entirely through treatment (exclusion restriction) Always takers and never takers: no treatment variation  no intervention effect [exclusion restriction]  cannot assess treatment effect Intervention effect manifested entirely through compliers NIDA/NIMH Substance Abuse Conference

29. Complier Average Causal Effect Treatment “Efficacy” on compliers: CACE = Program effect / Incremental adoption rate Program effect = intent-to-treat effect for encouragement intervention on outcome Incremental adoption rate = intent-to-treat effect for encouragement intervention on adoption Distribute intervention effect on outcome over compliers NIDA/NIMH Substance Abuse Conference

30. IV Analysis: External Validity Treatment effect estimable only for compliers (new users) Intrinsic limitation of design (RED or imperfect RCT) Should we be concerned about treatment effect for always takers and never takers? Yes for efficacy trials, less so for RED Never taker might never adopt treatment voluntarily Mandate vs. choice Universal dissemination vs. practical dissemination Always takers more critical; absent for new treatments Presence of defier likely to cancel some intervention effect IV estimate is conservative for true CACE NIDA/NIMH Substance Abuse Conference

31. IV Analysis: Discussions Exclusion restriction needs to be entertained carefully Likelihood and Bayesian methods available under weaker assumptions Non-randomized encouragement design (observational studies with instrumental variables) Example: McClellan et al. JAMA 1994, distance to alternative types of hospitals IV analysis usually deflates precision substantially Bias-variance trade-off? Combine propensity score analysis with IV analysis? NIDA/NIMH Substance Abuse Conference

32. IV Analysis: References Sommer and Zeger (1991 Stat. in Medicine) Angrist, Imbens, and Rubin (1996 JASA) Imbens and Rubin (1997 Annals of Statistics) Little and Yau (1998 Psych Methods) Hirano, Imbens, Rubin, and Zhou (2000 Biostatistics) Wells, et al. (2000, manuscript) NIDA/NIMH Substance Abuse Conference

33. Discussions Formulate research questions Treatment effect for whom? Adoption? Careful design usually more effective than analytic solutions Matching to avoid severe imbalance Promising methods for mitigating overt selection bias Careful modeling warranted Propensity score method worth exploring Nonparametric regression worth exploring Hidden selection bias very challenging Rosenbaum’s sensitivity analysis warranted IV analysis and selection model require careful assessment NIDA/NIMH Substance Abuse Conference