  In matched observational studies with binary treatments, the Rosenbaum bounds
framework is arguably the most widely used sensitivity analysis framework for
assessing sensitivity to unobserved covariates. Unlike the binary treatment
case, although widely needed in practice, sensitivity analysis for matched
observational studies with treatment doses (i.e., non-binary treatments such as
ordinal treatments or continuous treatments) still lacks solid foundations and
valid methodologies. We fill in this blank by establishing theoretical
foundations and novel methodologies under a generalized Rosenbaum bounds
sensitivity analysis framework. First, we present a criterion for assessing the
validity of sensitivity analysis in matched observational studies with
treatment doses and use that criterion to justify the necessity of
incorporating the treatment dose information into sensitivity analysis through
generalized Rosenbaum sensitivity bounds. We also generalize Rosenbaum's
classic sensitivity parameter $\Gamma$ to the non-binary treatment case and
prove its sufficiency. Second, we study the asymptotic properties of
sensitivity analysis by generalizing Rosenbaum's classic design sensitivity and
Bahadur efficiency for testing Fisher's sharp null to the non-binary treatment
case and deriving novel formulas for them. Our theoretical results showed the
importance of appropriately incorporating the treatment dose into a test, which
is an intrinsic distinction with the binary treatment case. Third, for testing
Neyman's weak null (i.e., null sample average treatment effect), we propose the
first valid sensitivity analysis procedure for matching with treatment dose
through generalizing an existing optimization-based sensitivity analysis for
the binary treatment case, built on the generalized Rosenbaum sensitivity
bounds and large-scale mixed integer programming.
