Nonrandomized Comparative Clinical Studies - Proceedings of the International Conference on Nonrandomized Comparative Clinical Studies in Heidelberg, April 10 -11,1997

Modelling Response: Traditional approaches compared to the propensity score

E. Graf

• Abstract
• References

Abstract

Rosenbaum and Rubin (1983, 1984) proposed a method for reducing bias in observational studies designed to compare two treatments, Z=0 and Z=1. The data are stratified on a function of covariates X, called the propensity score. The propensity score is the conditional probability of receiving treatment Z=1, given X. Stratification on the propensity score yields an unbiased estimate for the average treatment effect, E(R₁ - R₀), where R_Z denotes the response to treatment Z.

A striking feature of this approach is that it assumes there are two responses (R₀, R₁) per patient, one to each treatment, of which only one can be observed depending on the treatment given. Conventionally the observed response R is modelled as a function of treatment Z and covariates X. From this the conclusion has wrongly been drawn that the two approaches are conceptionally different (Drake and Fisher 1995). The purpose of the talk is to show how they compare.

Without further assumptions, the approach based on (R₀, R₁) is a hyper-model to the one based on R in the following sense: given the joint distribution of (R₀, R₁, Z, X), the distribution of (R, Z, X) is completely specified, whereas for a given distribution of (R, Z, X), the distribution of (R₀, R₁, Z, X) can still vary. However in the hyper-model inference is only possible when an assumption called 'strongly ignorable treatment assignment' is made which concerns the relevance of the covariates to both treatment assignment and response. If it holds, the distribution of (R₀, R₁, Z, X) cannot vary for a given distribution of (R, Z, X), and thus direct comparisons of estimates resulting from the two approaches are meaningful. The hyper-model, allows a more elegant formal description of what is described informally as 'systematic differences between treatment groups'. For some special cases where a linear model holds, the standard estimate of the treatment effect adjusted for covariates can be shown to be superior to stratification on the propensity score.

References

[1]

Rosenbaum P.R., Rubin D.B. (1983): The Central Role of the Propensity Score in Observational Studies for Causal Effects. Biometrika 70, 41-55

[2]

Rosenbaum P.R., Rubin D.B. (1984): Reducing Bias in Observational Studies Using Subclassification on the Propensity Score. Journal of the American Statistical Association 79, 516-524

[3]

Drake C., Fisher L. (1995): Prognostic Models and the Propensity Score. International Journal of Epidemiology 24, 183-187