Confounding adjustment: Ideas in Action -a case study Xiaochun Li, Ph.D. Associate Professor Division of Biostatistics Indiana University School of Medicine

2 Description of the data set Quantity to be estimated Summary of baseline characteristics Approaches to data analyses Results Discussion Outline

3 Linder Center data described and analyzed in Kereiakes et al. (2000) 6 month follow-up data on 996 patients who  underwent an initial Percutaneous Coronary Intervention (PCI)  were treated with “usual care” alone or usual care plus a relatively expensive blood thinner (IIB/IIIA cascade blocker has10 variables  Y: 2 outcomes, mort6mo (efficacy) and cardcost (cost)  X: 1 treatment variable, and 7 baseline covariates, stent, height, female, diabetic, acutemi, ejecfrac and ves1proc Simulation Setup

4 Baseline characteristics Stentcoronary stent deployment femalepatient sex diabeticdiabetes mellitus acutemiacute myocardial infarction ves1procnumber of vessels involved in initial PCI heightIn centimeter ejecfracleft ejection fraction %

5 Simulation data set was based on the Linder Center data 17 copies of the clustered Lindner data, with fudge factors added to ejfract and hgt, and some clipping  same correlation among covariates, same clustering patterns Contains the values of 10 simulated variables for 10,325 hypothetical patients To simplify analyses, the data contain no missing values. Details and dataset available from Bob’s website The “LSIM10K” dataset

6 The population average treatment effect (ATE), i.e., E(Y 1 ) - E(Y 0 ) Y 1 and Y 0 are conterfactual outcomes In plain words: what if scenarios The expected response if treatment had been assigned to the entire study population minus the expected response if control had been assigned to the entire study population What do we want to estimate?

7 Baseline covariate balance assessment VariableC (Usual care alone) T (Usual care + Abciximab) P value stent63%69%<0.001 female33%34%0.36 diabetic23%19%<0.001 acutemi7%15%<0.001 ves1proc 1.4 ( ± 0.6)1.3 ( ± 0.6) <0.001 height (cm) ( ± 10)171.5 ( ± 10) <0.001 ejfract 53 ( ± 8)50 ( ± 10) <0.001

8 Visualizing overall imbalance C Deep blue = high values T

9 The following methods were applied to lsim10k Outcome regression adjustment (OR) Propensity score (PS) stratification Inverse-probability-treatment-weighted (IPTW) Doubly robust estimation Matching by  Mahalonobis distance  PS only Analytical Methods for confounding adjustment

A NALYSIS OF MORT 6 MO OR model for mort6mo : treatment indicator (trtm) main effect terms for all seven covariates quadratic terms for both height and ejfract Residual deviance: on degrees of freedom PS model: saturated model for the five categorical covariates (main effects and interaction terms up to fifth-order) main effects and quadratic terms for height and ejfract

Covariates Balance Evaluations based on PS Quintiles

1212 Stent

1313 Female

1414 Diabetic

1515 Acutemi

1616 Ves1proc

1717 Height strata 2 (0.95 cm) and 3 (-1.50cm)

1818 Height Existence of residual confounding after adjusting for PS quintiles The within-stratum between-group height difference means.d.p  Stratum 2:  Stratum 3:

1919 Ejfract strata 1 (0.81), 2 (-1.32) and 3 (-0.72)

2020 Existence of residual confounding after adjusting for PS quintiles The within-strata between-group height difference means.d.p-value  Stratum 1:  Stratum 2: e-5  Stratum 3: Ejfract

2121 Residual confounding within strata In PS stratification method, height and ejfract are further adjusted stratum specific  Treatment effect  Height, ejfract main effects and their quadratic terms PS Stratification

2 Results – mort6mo Methodu1u1 u0u0 △ SE Outcome Regression PS strat IPTW IPTW DR Match Mahalanobis PS NA Results of all methods are consistent, providing evidence of treatment effectiveness at preventing death at 6 months. True △=-0.036

A NALYSIS OF CARDCOST cardcost model: treatment indicator (trtm) main effect terms for all seven covariates quadratic terms for both height and ejfract PS MODEL : SAME AS BEFORE cardcost model of CA with PS stratification: stratum specific Treatment effect Height, ejfract main effects and their quadratic terms

2424 Model checking – OR Adjusted R-squared:

2525 Model checking – OR (log transformed) Adjusted R-squared:

2626 Results – cardcost Methodu1u1 u0u0 △ SE OR: original scale OR: Log transformed PS strat IPTW IPTW DR Match Mahalanobis PS NA

2727 IPTW 1 vs 2

2828 All methods give consistent results on the 2 outcomes All PS based results have similar variance except IPTW1 IPTWs depend on approx. correct PS model OR depends on approx. correct outcome model DR is a fortuitous combination of OR and IPTW: depends on one of models being right DR is a fortuitous combination of OR and IPTW: depends on one of models being right Nonparametric models of either models may be an alternative to parametric models Discussion

2929 Double Robustness MethodPSoutcome △ SE IPTW2wrongNA DR wrong right wrong right wrong wrong PS model: adjust for one covariate ‘acutemi’ only wrong OR model for card cost: adjust for the treatment indicator ‘trtm’ and the ‘acutemi’ covariate By “right”, we mean approximately.

3030 The majority applications in literature use a parametric logistic regression model that assume covariates are linear and additive on the log odds scale  May include selected interactions and polynomial terms Accurate PS estimation is impeded by  High dimensional covariates – which ones should we de- confound?  Unknown functional form – how do they relate to the treatment selection PS model misspecification can substantially bias the estimated treatment effect Nonparametric approach is flexible to accommodate nonlinear/non-additive relationship of covariates to treatment assignment, e.g., trees Propensity score estimation

3131 Nonparametric regression techniques Generalized Boosted Models (GBM) to estimate the propensity score function  Friedman, 2001; Madigan and Ridgeway, 2004; McCaffrey, Ridgeway, and Morral, 2004  R package: twang Regression tree model to predict cardcost  Ripley, 1996; Therneau and Atkinson, 1997  R package: rpart

3232 A multivariate nonparametric regression technique Sum of a large set of simple regression trees modelling log-odds  gbm finds mle of g(x)=log(p(x)/(1-p(x)), p(x)=P(T=1|x) Predict treatment assignment from a large number of pretreatment covariates – adaptively choose them Nonlinear No need to select variables Can model complex interactions Invariant to monotone transformations of x  E.g, same PS estimates whether use age, log(age) or age 2 Outperforms alternative methods in prediction error Generalized Boosted Models (GBM)

3 Results – cardcost nonparametric approach Methodu1u1 u0u0 △ SE DR: parametric models DR: Gbm + parametric model DR: Gbm + tree

3434 People try quintiles, deciles for propensity score stratification – need data driven approach (based on bias-variance tradeoff) for number of strata Model selection: PS model, and outcome model  Nonparametric estimation of models may be intuitive, but not clear about the properties of the causal estimates  Nonparametric caveat: still need to define a set of “confounders” based on knowledge of causal relationship among treatment, outcome and covariates rather than conditioning indiscriminatly on all covariates that have associations with treatment and outcome Future research