Multinomial Extension of Propensity Score Trimming Methods: A Simulation Study

Abstract

Crump et al. (Biometrika. 2009;96(1):187-199), Stürmer et al. (Am J Epidemiol. 2010;172(7):843-854), and Walker et al. (Comp Eff Res. 2013;2013(3):11-20) proposed propensity score (PS) trimming methods as a means to improve efficiency (Crump) or reduce confounding (Stürmer and Walker). We generalized the trimming definitions by considering multinomial PSs, one for each treatment, and proved that these proposed definitions reduce to the original binary definitions when we have only 2 treatment groups. We then examined the performance of the proposed multinomial trimming methods in the setting of 3 treatment groups, in which subjects with extreme PSs more likely had unmeasured confounders. Inverse probability of treatment weights, matching weights, and overlap weights were used to control for measured confounders. All 3 methods reduced bias regardless of the weighting methods in most scenarios. Multinomial Stürmer and Walker trimming were more successful in bias reduction when the 3 treatment groups had very different sizes (10:10:80). Variance reduction, seen in all methods with inverse probability of treatment weights but not with matching weights or overlap weights, was more successful with multinomial Crump and Stürmer trimming. In conclusion, our proposed definitions of multinomial PS trimming methods were beneficial within our simulation settings that focused on the influence of unmeasured confounders.

Keywords: multinomial treatment; propensity score; propensity score trimming; propensity score weighting.

Figure 1.

Visual explanation of 3 existing 2-group trimming methods, using simulated data. A) Crump method (4), 50% treated; B) Stürmer method (5), 50% treated; C) Walker method (6), 50% treated; D) Crump method, 25% treated; E) Stürmer method, 25% treated; F) Walker method, 25% treated; G) Crump method, 10% treated; H) Stürmer method, 10% treated; I) Walker method, 10% treated. The hypothetical propensity-score distribution densities were generated from beta distributions. The dotted line represents the propensity score density in the untreated group, whereas the solid line represents the propensity score density in the treated group. In each panel, the gray region represents the retention region that applies to both treated and untreated groups. Individuals outside the retention region are removed regardless of their treatment status. Crump trimming is the same regardless of the prevalence, whereas the other 2 methods adapt to skewed propensity score distributions due to less frequent treatment. See Web Figures 1 and 2 for further examples.

Figure 2.

Simulated samples size after trimming at different thresholds, using simulated data. The scales for the thresholds were the propensity score scale for the Crump method (4) (A), quantiles of propensity score for the Stürmer method (5) (B), and the preference score scale for the Walker method (6) (C). The vertical broken hairlines are at the tentative thresholds used for the empirical data illustration. The solid line with circles represents the 33:33:33 treatment prevalence. The dotted line with triangles represents the 10:45:45 treatment prevalence. The broken line with squares represents the 10:10:80 treatment prevalence. The original sample size was n = 6,000 in all prevalence scenarios. Both Stürmer and Walker methods trimmed similarly regardless of treatment prevalence given that they accommodated skewed PS distributions. Crump trimming, on the other hand, trimmed differently at the same trimming threshold across treatment prevalence scenarios.