Estimating exposure effects by modelling the expectation of exposure conditional on confounders

In order to estimate the causal effects of one or more exposures or treatments on an outcome of interest, one has to account for the effect of "confounding factors" which both covary with the exposures or treatments and are independent predictors of the outcome. In this paper we present regression methods which, in contrast to standard methods, adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders. In the special case of a univariate dichotomous exposure or treatment, this conditional expectation is identical to what Rosenbaum and Rubin have called the propensity score. They have also proposed methods to estimate causal effects by modelling the propensity score. Our methods generalize those of Rosenbaum and Rubin in several ways. First, our approach straightforwardly allows for multivariate exposures or treatments, each of which may be continuous, ordinal, or discrete. Second, even in the case of a single dichotomous exposure, our approach does not require subclassification or matching on the propensity score so that the potential for "residual confounding," i.e., bias, due to incomplete matching is avoided. Third, our approach allows a rather general formalization of the idea that it is better to use the "estimated propensity score" than the true propensity score even when the true score is known. The additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model. To illustrate our methods, we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2,713 adult white males. We compare the results with those obtained using standard methods.