Bayesian Causal Inference in High Dimensional Data Settings Jacob - PowerPoint PPT Presentation

Bayesian Causal Inference in High Dimensional Data Settings Jacob Spertus and Sharon-Lise Normand Harvard Medical School & School of Public Health Funded by R01-GM111339 and U01-FDA004493 Thanks to Organizer: Mariel Finucane , Mathematica Policy Research June 2017 sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 1 / 14

Introduction OUTLINE Thanks Motivating problem What has been done What we add to literature Revisit motivating problem Concluding remarks sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 2 / 14

Introduction Motivating problem DRUG ELUTING (DES)VERSUS BARE METAL (BMS) CORONARY STENTS MASSACHUSETTS, 2011 DES (approved 2003) and BMS Stent Type (approved in 1990s) frequently Characteristic BMS DES implanted to keep treated arteries Outcomes, % clear & supported 1 Year Mortality 10.2 3.3 DES improves target-vessel 1 Year TVR 9.0 6.5 revascularization (TVR) more than Confounders BMS Age, yrs 66.4 63.7 DES associated with late stent STEMI, % 35.7 18.2 thrombosis (death) Cardiomyopathy or LVSD, % 11.1 8.4 Have 9000 patients and 500 Emergent, % 38.3 20.3 confounders Shock, % 3.8 0.8 Do DES cause fewer revascularizations STEMI = ST-elevated myocardial infarction; LVSD = left ventricular systolic dysfunction compared to BMS ? Spertus and Normand, 2017 (under review) sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 3 / 14

Introduction Motivating problem CONSIDERATIONS Observational setting - no randomization Combination of clinical and claims data Clinical data obtained by trained data managers in each hospital Approximately 131 confounders from clinical data (demographics, pre-existing conditions, presentation severity, procedural characteristics) Approximately 500 confounders in claims data (Present on Admission codes) Sparsity : considered claims diagnoses having � 10 coded as present Goals: avoid strong parametric specifications, adhere to ignorable treatment assignment assumption, and adopt a design-based approach Final dataset: 8718 patients and 495 potential confounders sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 4 / 14

What has been done PROPENSITY SCORES With high-dimensional data Schneeweiss et al. (2009) proposed algorithm in high-dimensional settting But no accounting for uncertainty in variable selection & outcome is used to inform propensity score model Bayesian ? McCandless et al. 2009: uses propensity score as a latent variable and jointly models the latent score and the outcome model . Kaplan and Chen, 2012: 2-step approach (propensity score model then a parametric outcome model) Saarela et al., 2015: marginal model specification coupled with inverse probability of treatment weighting 2-step procedure † Zigler and Dominici, 2014: include the propensity score as a linear predictor in the outcome model & use Bayesian model averaging † Only Bayesian method proposed in the high-dimensional setting sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 5 / 14

What we add APPROACH WANT: ∆ = E( Y DES) - E( Y BMS) Two-step approach 1 Step 1: estimate propensity score model via regularization 2 Step 2: assume a binomial likelihood for binary outcome, weighted to generate a pseudo-population Not Bayesian per se Incorporates uncertainty from propensity score estimation Addresses the large k problem Simple diagnostic tools to assess balancing properties Maintains separation between treatment and outcome Outcome model does not assume a parametric function of treatment sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 6 / 14

What we add Models MODELS ( T Binary Treatment, Y Binary Outcome, X Confounders, k Large) Step 1: Treatment Model T i ∼ Bern ( π ( X i )) � k � � logit − 1 π ( X i ) = β 0 + β j X ij j = 1 π ( X i ) = propensity score Priors required for β j Typically centered at 0 Horseshoe prior (Carvahlo et al., 2010) N ( 0, λ 2 j τ 2 ) β j ∼ Cauchy + ( 0, 1 ) λ j , τ ∼ Mimics Bayesian Model Averaging (with heavy-tailed discrete mixtures) sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 7 / 14

What we add Models MODELS ( T Binary Treatment, Y Binary Outcome, X Confounders, k Large) Step 1: Treatment Model Step 2: Outcome Model T i ∼ Bern ( π ( X i )) � k � � Y T | n T , p T , π ( X ) Binomial ( n T , p T ) ∼ logit − 1 π ( X i ) = β 0 + β j X ij p T | α T 0 , α T 1 ∼ Beta ( α T 0 , α T 1 ) j = 1 π ( X i ) = propensity score n T = number of subjects receiving treatment T Priors required for β j p T = probability of outcome under Typically centered at 0 treatment T Horseshoe prior (Carvahlo et al., 2010) Independent samples in treatment groups N ( 0, λ 2 j τ 2 ) β j ∼ A-posteriori Cauchy + ( 0, 1 ) λ j , τ ∼ p T | Y , π ( X ) ∼ Beta ( a T , b T ) Mimics Bayesian Model Averaging (with heavy-tailed discrete mixtures) sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 8 / 14

What we add Models MODEL ( T Binary Treatment, Y Binary Outcome, X Confounders, k Large) Posterior Distribution: p T | Y , π ( X ) ∼ Beta ( a T , b T ) � n � n � � T i Y i T i ( 1 − Y i ) � � a 1 = α 11 + γ 1 b 1 = α 10 + γ 1 π i π i i = 1 i = 1 � �� � � �� � weight weight � n � n � � ( 1 − T i ) Y i ( 1 − T i )( 1 − Y i ) � � a 0 = α 00 + γ 0 b 0 = α 01 + γ 0 1 − π i 1 − π i i = 1 i = 1 � �� � � �� � weight weight Renormalization terms: � n � n i = 1 ( 1 − T i ) i = 1 T i ; γ 0 = γ 1 = � n � n i = 1 T i /π i i = 1 ( 1 − T i ) / ( 1 − π i ) sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 9 / 14

What we add Operating characteristics DOES UNCERTAINTY IN STEP 1 MATTER? 500 simulations, n = 1000, k= 100 (18 β j � = 0), P ( Y i = 1 ) ≈ 0.10 BART = Bayesian Additive Regression Trees 95% CI Coverage ˆ Bias Width 95% CI ∆ Integrated Propensity Score Student- t 3 ( 0, 3 2 ) -.011 .220 95.2% Horseshoe Priors .016 .110 93.0% BART .011 .123 96.8% Mean Propensity Score Student- t 3 ( 0, 3 2 ) -.001 .095 79.2% Horseshoe Priors .018 .092 86.0% BART .015 .093 87.2% Other Methods Naive Estimate .030 .092 73.0% IPW -.001 .151 92.8% TMLE .006 .075 81.4% Bottom Line: useful to integrate over the propensity score distribution for large k sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 10 / 14

What we add Causal estimates for BMS vs DES WEIGHTED STANDARDIZED MEAN DIFFERENCES IN CONFOUNDERS sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 11 / 14

Frequentist Bayesian What we add Causal estimates for BMS vs DES 1-YEAR TARGET VESSEL REVASCULARIZATION Favours DES Favours BMS BART HShoe t(0,9) TMLE IPW Naive −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 1−Year Revascularization,% sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 12 / 14

Conclusions SUMMARY OF OUR APPROACH Advantages Not fully Bayesian Propensity score is simple, widely used, and familiar Easy to assess balancing properties Fixing propensity score at posterior mean: Good coverage for small k Underestimates variance for large k Some bias Maintains separation between treatment and outcome models Honestly reflects uncertainty sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 13 / 14

Conclusions SUMMARY OF OUR APPROACH Disadvantages Advantages Not fully Bayesian Not fully Bayesian Robustness? Propensity score is simple, widely used, and familiar Binomial likelihood Easy to assess balancing properties Balancing property may be lost when incorporating uncertainty Fixing propensity score at posterior mean: Thank you! Good coverage for small k Underestimates variance for large k Some bias Maintains separation between treatment and outcome models Honestly reflects uncertainty sharon@hcp.med.harvard.edu (HMS) AcademyHealth 2017 June 2017 14 / 14