A tutorial on propensity score estimation for multiple treatments using generalized boosted models

Abstract

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. For settings with two conditions of interest such as a treatment and a control, inverse probability of treatment weighted estimation with propensity scores estimated via boosted models has been shown in simulation studies to yield causal effect estimates with desirable properties. There are tools (e.g., the twang package in R) and guidance for implementing this method with two treatments. However, there is not such guidance for analyses of three or more treatments. The goals of this paper are twofold: (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. We define the causal quantities that may be of interest to studies of multiple treatments and derive weighted estimators of those quantities. We present a detailed plan for using GBM to estimate propensity scores and using those scores to estimate weights and causal effects. We also provide tools for assessing balance and overlap of pretreatment variables among treatment groups in the context of multiple treatments. A case study examining the effects of three treatment programs for adolescent substance abuse demonstrates the methods.

Keywords: GBM; causal effects; causal modeling; inverse probability of treatment weighting; twang.

Figure 1

Effect size plots for assessing the balance of pretreatment variables on youth like those receiving Community care for estimating pairwise ATT effects for Community Treatment.

Figure 2

Effect size plots for assessing the balance of pretreatment variables on youth like those receiving MET/CBT-5 for estimating pairwise ATT effects for MET/CBT-5.

Figure 3

Effect size plots for assessing the balance of pretreatment variables on youth like those receiving SCY for estimating pairwise ATT effects for SCY.

Figure 4

Overlap Assessment. Each panel presents box plots by treatment group of the estimated propensity scores for one of the treatments, pr(T[t] = 1 | X) for ever youth in the sample. The top panel present Community (t = 1), the middle panel presents MET/CBT-5 (t = 2), and the bottom panel presents SCY (t = 3).