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Subclassification and matching are often used in empirical studies to adjustfor observed covariates; however, they are largely restricted to relativelysimple study designs with a binary treatment and less developed for designswith a continuous exposure. Matching with exposure doses is particularly usefulin instrumental variable designs and in understanding the dose-responserelationships. In this article, we propose two criteria for optimalsubclassification based on subclass homogeneity in the context of having acontinuous exposure dose, and propose an efficient polynomial-time algorithmthat is guaranteed to find an optimal subclassification with respect to onecriterion and serves as a 2-approximation algorithm for the other criterion. Wediscuss how to incorporate dose and use appropriate penalties to control thenumber of subclasses in the design. Via extensive simulations, wesystematically compare our proposed design to optimal non-bipartite pairmatching, and demonstrate that combining our proposed subclassification schemewith regression adjustment helps reduce model dependence for parametric causalinference with a continuous dose. We apply the new design and associatedrandomization-based inferential procedure to study the effect oftransesophageal echocardiography (TEE) monitoring during coronary artery bypassgraft (CABG) surgery on patients' post-surgery clinical outcomes using Medicareand Medicaid claims data, and find evidence that TEE monitoring lowerspatients' all-cause $30$-day mortality rate.

Statistical matching and subclassification with a continuous dose: characterization, algorithm, and application to a health outcomes study