Abstract: The propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates. Previous theoretical arguments have shown that subclassification on the propensity score will balance all observed covariates. Subclassification on an estimated propensity score is illustrated, using observational data on treatments for coronary artery disease. Five subclasses defined by the estimated propensity score are constructed that balance 74 covariates, and thereby provide estimates of treatment effects using direct adjustment. These subclasses are applied within subpopulations, and model-based adjustments are then used to provide estimates of treatment effects within these subpopulations. Two appendixes address theoretical issues related to the application: the effectiveness of subclassification on the propensity score in removing bias, and balancing properties of propensity scores with incomplete data.

INTRODUCTION: SUBCLASSIFICATION AND THE PROPENSITY SCORE

Adjustment by Subclassification in Observational Studies

In observational studies for causal effects, treatments are assigned to experimental units without the benefits of randomization. As a result, treatment groups may differ systematically with respect to relevant characteristics and, therefore, may not be directly comparable. One commonly used method of controlling for systematic differences involves grouping units into subclasses based on observed characteristics, and then directly comparing only treated and control units who fall in the same subclass. Obviously such a procedure can only control the bias due to imbalances in observed covariates.

Cochran (1968a) presents an example in which the mortality rates of cigarette smokers, cigar/pipe smokers, and nonsmokers are compared after subclassification on the covariate age.