Propensity Parameters for Causal Inference

Propensity scores are widely used to control for confounding when estimating the effect of a binary treatment in observational studies. They have been generalized to ordinal and continuous treatments (Joffe and Rosenbaum 1999; Imbens 2000; Imai and van Dyk 2004) in recent literature. Following the definition of propensity function and its parameterizations (called the propensity parameter) in Imai and van Dyk 2004 (Imai and van Dyk 2004), we explore sufficient conditions for selecting propensity parameters to control for confounding for an arbitrary treatment, with special attention to continuous treatments. Typically, investigators make parametric assumptions about the form of the dose-response function for a continuous treatment and how it varies with baseline covariates. Such assumptions often allow control for only a subset of the propensity parameter; we require adjustment only for the propensity parameters that characterize the expectation of the treatment variable or its functional form when the treatment is the only predictor in the structural model. When the structural model includes selected baseline covariates other than the treatment variable, adjustment for the propensity parameters for the treatment alone is insufficient, and those covariates must also be adjusted in the model. We demonstrate these points with an example estimating the dose-response relationship for the effect of erythropoietin on hematocrit level in patients with end stage renal disease.