Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 1 EP 521 Spring 2007, Vol II, Part 9 (Under development) §10 Propensity Scores (balancing scores) §10.1 Potential outcomes, confounding, and conditional independence Problem: In randomized studies: when there are two groups, treated and control: We rely on randomization to “balance” the two groups. Balance is achieved on both observed and unobserved factors In observational studies: There are no assurances that the exposed and unexposed groups are similar, i.e, that there is balance on the covariates We thus face the problem of confounding Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 2 Stated more formally: Let 00,ABµµ be the outcomes in the groups A and B if both groups were untreated (or unexposed). We cannot observe both of these states. If group A is treated, we do not observe it as not treated. Assume that treatment =1x is applied to A only. Let 1Aµ = observed effect of applying treatment to A. Control treatment =0x . If B=control group, observed effect of applying the control to B =0Bµ. Goal is to measure treatment effect as10AAµ−µ. But 0Aµis not observed because it is the treatment group. 10ABµ−µ is observed. Confounding is present if 00ABµ≠µ. Then: Observed treatment effect (10ABµ−µ) ≠ the effect size of interest (10AAµ−µ).Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 3 Solution We want to ensure that comparison groups (treatment vs control, or exposed vs unexposed) are comparable because we cannot compare the group actually treated (actual outcome) with what would have happened if it had not been treated (potential outcome). What do we try to do when we adjust for confounding via regression? Let yT=α+β+ε, where T represents the treatment (treatment assignment). Key assumption in regression is that residuals, 'sε, are independent of T. State alternatively, if there is randomization, then T explains a systematic shift in y (the treatment effect) but there should be no residual ε’s that differ when 1T= than when 0.T= If residuals differ by group, then ˆβ will be biased estimate of treatment effect Goal of adjustment for confounding: Find variables Zthat are highly corrected with ε so that after adjusting for Z, T is no longer correlated with the revised residuals *ε. Then ˆ*β will no longer be biased estimate in the equation: 12***yTZ=α+β+β+ε Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 4 Propensity score methods (1) Find Z, a vector of covariates, that eliminates bias. Stated alternatively, we suspect that there are residual associations between Y and T. But attempt to subtract out of Y and out of T the residual association arising out of nonrandom treatment assignment. (2) Achieve this goal by focusing on modeling the treatment/exposure assignment (T) as a function of the covariates (Z), in an initial modeling exercise. The “propensity score model” (3) Facilitates making causal statements about effect T on Y in a subsequent “response model”Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 5 Assumptions of propensity scores (1) Treatment assignment (T=1 or 0) must be independent of the potential outcome conditional on the chosen set of covariates (Z). This assumptions suggests that we do not want to omit any important covariates in the propensity score model (2) SUTVA (stable unit treatment value assumption). – Potential outcome of a subject is independent of the treatment assignment (or exposure) of the other subjects, given the set of covariates, and the “treatment” does not vary from subject to subject. So, we want to make sure that subjects are not being assigned in groups to treatment/exposure unless we have controlled for the factors that make up those groups and we want to ensure that everyone assigned to treatment receives the same “treatment”. (3) Individuals with the same covariate pattern (Z) must be split, at least in part, between assignment to treatments (T=1 or 0) so that there will always be individuals with T=0 who can serve a comparators to similar persons with T=1. Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 6 §10.2 The shortcomings of controlling for confounding in analysis using regression (1) There might be many potential confounders (2) There are relatively few events (3) There might be an acceptable number of events per potential confounder, but there is only limited overlap in the distribution of the confounder by the treatment group when looking at each confounder. E.g. Weight of intercollegiate woman (gymnasts) and men (football players) How can one adjust for weight as a potential confounder? One question is whether weight is the only confounder in the analysis. This problem might occur in multivariable space – the lack of overlap might occur when considering more than one factor simultaneously. (4) Lack of overlap is called the “problem of common support” Group Min median Max Males 150 200 360 Females 85 100 125Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 7 (5) What is the correct functional form for the potential confounder? Linear Linear term + quadratic (21121xxβ+β ) Complex relationship (splines or piecewise model) Transformation of term (ln or sqrt) These issues might fall under the heading of “model misspecifiction”, and misspecification can lead to bias (6) As the number of covariates becomes large (or there are many levels to covariates), adjustment by regression becomes impossible. (The dimension of X, the set of variables) Vol II, Part 9. Propensity Scores. Copyright © 2006 Trustees of the University of Pennsylvania. Page 8 §10.3 Subclassification on propensity score - principles The following implementation of propensity scores is still probably the most common method. (1) Make groups of subjects that have comparable chances of being assigned to the treatment (exposure) group vs the control (unexposed) based on their characteristics So, we compare exposed people who have a 10% chance