Nonetheless, a number of papers have shown that propensity score methods can be extended to the multiple treatment case with three or more conditions of interest (e.g., treatment A, treatment B, and control ). Most studies that use propensity scores to control for imbalances compare just two treatment groups of interest (e.g., treatment and control).

This feature of propensity score adjustments is valuable because it eliminates the potential for the choice of model specification for pretreatment variables to be influenced by its impact on the estimated treatment effect. Lastly, propensity score adjustments can be implemented using only the pretreatment covariates and treatment assignments of study participants without any use of the outcomes. Fourth, propensity score methods avoid extrapolating beyond the observed data unlike parametric regression modeling for outcomes which extrapolate whenever the treatment and control groups are disparate on pretreatment variables. This can help avoid bias from misspecification of that model. Third, propensity score methods do not require modeling the mean for the outcome. Second, propensity score methods derive from a formal model for causal inference, the potential outcomes framework, so that causal questions can be well-defined and explicitly specified and not conflated with the modeling approach as they are with traditional regression approaches.

This characteristic of propensity scores is particularly advantageous over standard adjustment methods when there exists a potentially large number of pretreatment covariates. First, by summarizing all pretreatment variables to a single score, propensity scores are an important dimension reduction tool for evaluating treatment effects. Propensity score techniques are advantageous compared with regression-based, covariate-adjustment techniquesвЂ“ which correct for imbalances between groups on pretreatment covariates by controlling for them in regression models for the outcomesвЂ“ for at least five reasons. Propensity scores have been used to match, stratify (subclassify), or weight the samples from the treatment and control groups so that the distributions (or features of the distributions such as the means) of observed pretreatment characteristics are similar across the treatment and control groups, thereby reducing or eliminating confounding. The use of propensity scores to control for pretreatment imbalances on observed variables in non-randomized or observational studies examining the causal effects of treatments or interventions has become widespread over the past decade.

A case study examining the effects of three treatment programs for adolescent substance abuse demonstrates the methods. Tools for assessing balance and overlap of pretreatment variables among treatment groups in the context of multiple treatments are also provided.

We present a detailed plan for using GBM to estimate propensity scores and using those scores to estimate weights and causal effects. We define the causal quantities that may be of interest to studies of multiple treatments and derive weighted estimators of those quantities. The goals of this paper are two-fold: (1) to provide step-by-step guidance for researchers who want to implement propensity score weighting for multiple treatments and (2) to propose the use of generalized boosted models (GBM) for estimation of the necessary propensity score weights. However, there is not such guidance for analyses of three or more treatments. There are tools (e.g., the twang package in R) and guidance for implementing this method with two treatments. For settings with two conditions of interest such as a treatment and a control, inverse probability of treatment weighted (IPTW) estimation with propensity scores estimated via boosted models has been shown in simulation studies to yield causal effect estimates with desirable properties.

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