Assessing balance in measured baseline covariates when using many-to-one matching on the propensity-score

The propensity score is defined to be a subject's probability of treatment selection, conditional on observed baseline covariates. Conditional on the propensity score, treated and untreated subjects have similar distributions of observed baseline covariates. Propensity-score matching is a commonly used propensity score method for estimating the effects of treatment on outcomes. Balance diagnostics have been previously described for use when 1:1 matching on the propensity score is employed. We illustrate that these methods can be misleading when many-to-one matching on the propensity score is employed. We then propose modifications of these methods that involve weighting each untreated subject by the inverse of the number of untreated subjects in the matched set. We describe both quantitative and qualitative methods to assess the balance in baseline covariates between treated and untreated subjects in a sample obtained by many-to-one matching on the propensity score. The quantitative method uses the weighted standardized difference. The qualitative methods employ graphical methods to compare the distribution of continuous baseline covariates between treated and untreated subjects in the weighted sample. We illustrate our methods using a large sample of patients discharged from hospital with a diagnosis of a heart attack (acute myocardial infarction). The exposure was receipt of a prescription for a statin at hospital discharge.