1-to-n Propensity Matching Methods

Introduction: The goal of the propensity matching method (PMM) is to efficiently estimate treatment effects within a non-randomized controlled study, by reducing bias and confounds that arise within a non-experimental design. In doing so, causality can be inferred. There are many approaches one can utilize to achieve PMM however. The overarching goal of this paper is to discuss the relative benefits of a 1-to-n matching as compared to 1-to-1 matching using either greedy, non-greedy, or optimal matching. The 1-to-n approach can dramatically increase sample size and increase the precision of an estimate of treatment effects. Using data from a state-wide scale-up of a tiered intervention framework in schools, called Positive Behavior Intervention and Supports (PBIS; Horner & Sugai, 2002), this paper will provide the optimal number, n, when using the 1-to-n matching. Based on data that suggests there are substantial gains in bias reduction, when using up to seven matches, but a little loss in efficiency when multiple controls are matched to each treated participant (Gu and Rosenbaum, 1993; Haviland et al., 2007). Moreover, extant literature focuses mainly on cross-sectional data when utilizing such matching; this study will be extended to structured (i.e., longitudinal and multilevel) data.

Methods: In this study, the optimal PMM will be considered, with the goal of minimizing the total distance utilizing one of three weight functions in Rosenbaum (2002) and improving mean differences of variables after matching (Ho, Imai, King, & Stuart, 2011). Instead of considering an arbitrary n, the optimal number, n, will be bounded by minimum of 7 and 2(1-p)/p (Hansen, 2004) where p is the proportion of treatment samples over control samples. To take into account missing data, multiple imputation (m=5) was applied. R package, MatchIt, was used. The data used are 659 elementary schools across the state of Maryland. Schools implementing and not implementing PBIS were matched, to determine the state-wide scale-up effect of PBIS.

Results: In data, there are 54 schools implementing PBIS and 604 control schools in year 2006. Among 16 variables (15 covariates and 1 propensity score distance), 11 variables indicate positive improvement with 1-to-4 matching while average number of variables improved via PSM is six variables. That is, a final matched sample of 270 (54+216) is recommended for post-PSM analysis.

Conclusions: This finding demonstrates that 1-to-4 matching with small treatment sample is best for bias reduction. Although the optimal n may vary from study to study, considering a 1-to-n propensity matching approach may be necessary and be especially recommended for small treatment and large control group samples.