In a nonrandomized or observational study, propensity scores may be used to balance observed covariates and trajectory groups may be used to control baseline or pretreatment measures of outcome. The trajectory groups also aid in characterizing classes of subjects for whom no good matches are available and to define substantively interesting groups between which treatment effects may vary. These and related methods are illustrated using data from a Montreal-based study. The effects on subsequent violence of gang joining at age 14 are studied while controlling for measured characteristics of boys prior to age 14. The boys are divided into trajectory groups based on violence from ages 11 to 13. Within trajectory group, joiners are optimally matched to a variable number of controls using propensity scores, Mahalanobis distances, and a combinatorial optimization algorithm. Use of variable ratio matching results in greater efficiency than pair matching and also greater bias reduction than matching at a fixed ratio. The possible impact of failing to adjust for an important but unmeasured covariate is examined using sensitivity analysis.
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