Combining propensity score matching with additional adjustments for prognostic covariates

Propensity score matching refers to a class of multivariate methods used in comparative studies to construct treated and matched control samples that have similar distributions on many covariates. This matching is the observational study analog of randomization in ideal experiments, but is far less complete as it can only balance the distribution of observed covariates, whereas randomization balances the distribution of all covariates, both observed and unobserved. An important feature of propensity score matching is that it can be easily combined with model-based regression adjustments or with matching on a subset of special prognostic covariates or combinations of prognostic covariates that have been identified as being especially predictive of the outcome variables. We extend earlier results by developing approximations for the distributions of covariates in matched samples created with linear propensity score methods for the practically important situation where matching uses both the estimated linear propensity scores and a set of special prognostic covariates. Such matching;on a subset of special prognostic covariates is an observational study analog of blocking in a randomized experiment. An example combining propensity score matching with Mahalanobis metric matching and regression adjustment is presented that demonstrates the flexibility of these methods for designing an observational study that effectively reduces both bias due to many observed covariates and bias and variability due to a more limited subset of covariates. Of particular importance, the general approach, which includes propensity score matching, was distinctly superior to methods that focus only on a subset of the prognostically most important covariates, even if those covariates account for most of the variation in the outcome variables. Also of importance, analyses based on matched samples were superior to those based on the full unmatched samples, even when regression adjustment was included.