Abstract
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 in the recent literature. Following the definition of propensity function and its parameterizations (called the propensity parameter in this paper) proposed by Imai and van Dyk, we explore sufficient conditions for selecting propensity parameters to control for confounding for continuous treatments in the context of regression-based adjustment in linear models. Typically, investigators make parametric assumptions about the form of the dose-response function for a continuous treatment. Such assumptions often allow the analyst to use only a subset of the propensity parameters to control confounding. When the treatment is the only predictor in the structural, that is, causal model, it is sufficient to adjust only for the propensity parameters that characterize the expectation of the treatment variable or its functional form. When the structural model includes selected baseline covariates other than the treatment variable, those baseline covariates, in addition to the propensity parameters, 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.