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Introduction to Propensity Score Weighting Weimiao Fan 10/10/2009 1

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Background Propensity Score Analysis (PSA) is used to adjust the confounding effect when studying the treatment effect in observational studies. Propensity score is the conditional probability of being in the treatment group given a set of covariates. There are four primary usages/methods of propensity scores (Posner& Ash ): 1.Random selection within strata. 2.Matching on the propensity score. 3.Regression adjustment. 4.Propensity score weighting. We are focusing on the weighting methods for this study. 2

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Propensity Score Weighting Weighted estimators (Lunceford, 2004; Robins, 1994) using propensity score can be used to adjust the sample selection bias. Weighting is based in the inverse of the propensity score. Such technique is referred to as ‘IPW’, which denotes ‘inverse probablity weighting’. Specificially, the weights are defined as: Wt=1/ps for treatment group Wc=1/(1-ps) for control group This means to give more weight to observations with lower propensity scores for treatment group; and give more weight to observations with higher propensity scores for control group. 3

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IPW Estimator Theoretically, the IPW estimator produces an unbiased estimate of the true treatment effect. Suppose Y is the outcome variable, T is the treatement indicator, p(x,T) is the propensity score, is the outcome for treated group. Similar conclusion is obtained for the control group. This suggests that IPW approach can be used to adjust the sample selection bias and obtain the true treatment effect. 4

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The IPW Estimators in Literature There are different methods to estimate the average causal effects (E(Yt)-E(Yc)) in the literature. 1.Rosenbaum and others (1998) proposed the following weighted estimators of the average causal effects. IPW2 is sometimes known as a ratio estimator in sampling literature, which normalizes the weights so that they add up to 1 in each treatment group. 2. Another estimator with ‘double robustness’ is created by Robins, Rotnitzky, and Zhao (1994). 5

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Weighting Within Strata and Proportional Weighting Within Strata One drawback of Inverse probability weighting approach is that it’s very sensitive to extreme values. For an observation with extremely small propensity score, the weight will be extremely large so that it will be very influential to the estimate. Posner and Ash proposed weighting within strata and proportional weighting within strata as alternative weighting methods. 6

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Application – in progress The IPW weighting method is applied to the birthweight data to determine the effect of smoking. Some preliminary results of the smoke effect are shown below ipw1= -0.6542768 ipw2= -0.5678549 While loess-based estimate using loess.psa gives an estimate of -0.5434 7

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References Model selection, confounder control, and marginal structural models: Review and new applications Joffe, MM AMERICAN STATISTICIAN Volume: 58 Issue: 4 Pages: 272-279 Published: NOV 2004 Model selection, confounder control, and marginal structural models: Review and new applications A comparison of propensity score methods: A case-study estimating the effectiveness of post-AMI statin use Austin PC, Mamdani MM STATISTICS IN MEDICINE Volume: 25 Issue: 12 Pages: 2084-2106 Published: JUN 30 2006A comparison of propensity score methods: A case-study estimating the effectiveness of post-AMI statin use A Generalization of Sampling Without Replacement From a Finite Universe D. G. Horvitz and D. J. Thompson Journal of the American Statistical Association, Vol. 47, No. 260 (Dec., 1952), pp. 663-685Journal of the American Statistical Association Estimation of Regression Coefficients When Some Regressors Are Not Always Observed James M. Robins, Andrea Rotnitzky and Lue Ping Zhao Journal of the American Statistical Association, Vol. 89, No. 427 (Sep., 1994), pp. 846-866Journal of the American Statistical Association Rosenbaum PR. Propensity Score. In Encyclopida of Biostatistics, Armitage P, Colton T (eds), vol. 5. Wiley: New York, 1998; 3551-3555 8

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