  Subclassification and matching are often used in empirical studies to adjust
for observed covariates; however, they are largely restricted to relatively
simple study designs with a binary treatment and less developed for designs
with a continuous exposure. Matching with exposure doses is particularly useful
in instrumental variable designs and in understanding the dose-response
relationships. In this article, we propose two criteria for optimal
subclassification based on subclass homogeneity in the context of having a
continuous exposure dose, and propose an efficient polynomial-time algorithm
that is guaranteed to find an optimal subclassification with respect to one
criterion and serves as a 2-approximation algorithm for the other criterion. We
discuss how to incorporate dose and use appropriate penalties to control the
number of subclasses in the design. Via extensive simulations, we
systematically compare our proposed design to optimal non-bipartite pair
matching, and demonstrate that combining our proposed subclassification scheme
with regression adjustment helps reduce model dependence for parametric causal
inference with a continuous dose. We apply the new design and associated
randomization-based inferential procedure to study the effect of
transesophageal echocardiography (TEE) monitoring during coronary artery bypass
graft (CABG) surgery on patients' post-surgery clinical outcomes using Medicare
and Medicaid claims data, and find evidence that TEE monitoring lowers
patients' all-cause $30$-day mortality rate.
