Propensity Score Methods for Causal Inference John Pura BIOS790 October 2, 2015 John PuraBIOS790 Propensity Score Methods for Causal Inference Causal inference Philosophical problem, statistical solution Important in various disciplines (e.g. Koch’s postulates, Bradford Hill criteria, Granger causality) Good reference on history of causal inference: Paul Holland “Statistics and Causal Inference” JASA, 1986 John PuraBIOS790 Propensity Score Methods for Causal Inference What can we estimate? Potential Outcomes Framework (Rubin’s Causal Model) Notation: Z (1=treated, 0=control), baseline covariates X =(X1,...,Xp), outcome Y potential outcomes Y0, Y1 We observe (Z, Y , X)foranindividual Y = ZY (1)+(1 Z)Y (0) ≠ Causal effect of treatment: Y (1) Y (0) ≠ Average causal effect: ∆ = E[Y (1) Y (0)] ≠ John PuraBIOS790 Propensity Score Methods for Causal Inference What can we estimate? Average causal effect ACE or ATE = E[Y (1) Y (0)]) All ≠ ACE or ATT = E[Y (1) Y (0) Z = 1]) Exp ≠ | ACE or ATU = E[Y (1) Y (0) Z = 0]) Un ≠ | Estimand and statistical methods depends on the study goal/question John PuraBIOS790 Propensity Score Methods for Causal Inference Assumptions 1. ZprecedesY 2. Stable Unit Treatment Value Assumption (SUTVA) non-interference no variation in treatment 3. Strongly Ignorable Treatment Assigment (SITA) 0 < P(Z = 1 X) < 1 (this is the propensity score) | (Y (0), Y (1)) Z X (very strong assumption) ‹ | no unobserved confounders John PuraBIOS790 Propensity Score Methods for Causal Inference Randomized Controlled Trials vs. Observational Studies RCTs Treatment effects on outcome considered as causal Z is determined for each participant at random, (Y (0), Y (1)) Z ‹ E[(Y Z = 1) (Y Z = 0)] is unbiased estimate of ∆ | ≠ | ATT = ATE Observational Study Z is not controlled, (Y (0), Y (1)) Z ”‹ E(Y Z = 1)=E(Y (1) Z = 1) = E(Y (1)). Cannot obtain | | ” unbiased estimate by direct comparison. But... John PuraBIOS790 Propensity Score Methods for Causal Inference Potential Solution In observational studies, assuming SITA assumption is met then treatment assignment, Z, among individuals with particular X is essentially random and independent of potential outcomes Rosenbaum and Rubin (1983) - conditioning on the propensity score (PS) we can identify E(Y (0)) and E(Y (1)) from the observed data (Z, Y , X)andultimatelyestimate∆. John PuraBIOS790 Propensity Score Methods for Causal Inference Propensity Score Austin, 2011: “The propensity score is a balancing score: conditional on the propensity score, the distribution of observed baseline covariates will be similar between treated and untreated subjects” This is a large sample property Unknown in practice, but can be estimated from the data, given some assumptions on e(X) (e.g. parametric regression model, CRTs) . Mathematically: e(X)=P(Z = 1 X).R&Rshowedthat | X Z e(X) and in addition to the SITA assumption, ‹ | (Y (0), Y (1)) Z e(X). ‹ | For theoretical properties see R&R (1983) and Lunceford and Davidian (2004) John PuraBIOS790 Propensity Score Methods for Causal Inference The Propensity Score Model Goal: Covariate balance Popular method for estimating PS is logistic regression, though others exist (e.g. tree-based methods, random forests, neural networks, etc.) Regress logit[P(Z = 1 X)] on X and obtain predicted | probabilities (ˆe(X)) R&R (1984) and Austin 2011 describe an iterative approach: 1. Specify an initial model to estimate ˆe(X) 2. Perform diagnostics to assess covariate balance for each treatment 3. Modify PS by adding covariates, interactions, or using non-linear terms 4. Important: Each step should not be motivated by statistical significance but by objective John PuraBIOS790 Propensity Score Methods for Causal Inference The Propensity Score Model Goal: Covariate balance What covariates do we include? Selection driven by subject-matter knowledge Only baseline variables Include all confounders and possible non-linear transformations (e.g. interactions). Overfitting generally not an issue (unless treatment is uncommon) Always include variables that affect the outcome even if they don’t affect treatment assignment (Brookhart et al. (2006)) John PuraBIOS790 Propensity Score Methods for Causal Inference Diagnostics How do we know the PS model has been adequately specified? Assess standardized differences of each covariate between treatment groups (very useful) Assess PS distributions by treatment (need common support condition) Compare distributions of the covariates between treatments Varies with PS method Difficult in practice with high dimensional data Assess the sensitivity of study conclusions to the SITA assumption. John PuraBIOS790 Propensity Score Methods for Causal Inference Methods utilizing PS Matching Stratification Inverse PS weighting Covariate adjustment by PS PS methods allow for estimation of the marginal treatment effect. The first three separate the design of the study from the analysis of the study. Can do subsequent regression adjustment to eliminate residual imbalance in prognostically important covariates after first three PS methods John PuraBIOS790 Propensity Score Methods for Causal Inference Matching Simple formulation for ATT For each treated subject, select single untreated subject (without replacement) with same value of ˆe(X) or its logit (R&R, 1985) Take difference of outcomes for the matched pair and average over all matched pairs Calculating ATE and ATU require slightly different sampling, possibly with replacement Advantage: Eliminates large proportion of systematic differences in baseline characteristics between treated and untreated subjects Disadvantage: Inexact matching may lead to bias. Unmatched individuals are discarded, leading to loss in statistical power. Discarding individuals may also alter our estimand (Hill, 2008) John PuraBIOS790 Propensity Score Methods for Causal Inference Stratification Easily estimate ATT: Create quantiles (e.g. quintiles) of the PS values, thereby dividing the subjects into equal-sized strata Within each stratum estimate treatment effect Calculate weighted average of within-strata estimates of treatment effect. Weight of each stratum is simply the percent of the quantile Estimating ATE and ATU require weighting by fraction of treated or untreated individuals, respectively, per stratum Advantage: Easy to construct and estimate causal effects. Disadvantage: Small number of strata may result in residual confounding within the strata, resulting in bias. ATT estimates largely biased (compared to weighting) John PuraBIOS790 Propensity Score Methods for Causal Inference Inverse weighting Weighted linear regression of outcome on treatment where Z 1 Z w = + ≠ w1 w2 For ATE, w1 = e(X), w2 = 1 e(X);forATT,w1 = 1, e(X) ≠1 e(X) ≠ w2 = 1 e(X) ;ForATU,w1 = e(X) , w2 = 1. (Morgan & Todd, 2008)≠ Advantage: Uses all available data; Can deal with more complex non-linear link functions (e.g. odds ratio); generally less biased than stratification (Lunceford & Davidian, 2004) Disadvantage: An individual with PS close to 0 or 1 will have unstable weights, leading to potentially spurious treatment effects with high variance and wide CIs. John PuraBIOS790 Propensity Score Methods for Causal Inference Covariate adjustment using PS Fit model: E(Y Z, X)=– + —Z + “f (e(X)) (may include | interaction of Z and e(X)) Can obtain ATE, ATT, and ATU by evaluating — at different values of ˆe(X) Advantage: Allows for flexible relationship between PS and outcome (e.g. use of splines for PS) Disadvantage: Sensitive to whether PS has been accurately estimated. Analyst may be tempted to work toward desired or anticipated result, given that outcome is in sight. John PuraBIOS790 Propensity Score Methods for Causal Inference Final Thoughts PS methods can be done without reference to outcome - i.e. separate study design from analysis Balance of covariates can be easily checked PS methods more robust to model misspecification compared to traditional outcome regression (all we care about is balance) Measures a different quantity, namely, the marginal/population treatment effect (vs. conditional/individual treatment effect in traditional regression) Important to distinguish the two in relation to study goals Omitted variable bias affects internal validity of both approaches similarly Strategy so far is to balance covariates. Another idea is to find an “instrument”" S that is randomly assigned and affects Y only through Z John PuraBIOS790 Propensity Score Methods for Causal Inference.