  Matching is one of the most widely used study designs for adjusting for
measured confounders in observational studies. However, unmeasured confounding
may exist and cannot be removed by matching. Therefore, a sensitivity analysis
is typically needed to assess a causal conclusion's sensitivity to unmeasured
confounding. Sensitivity analysis frameworks for binary exposures have been
well-established for various matching designs and are commonly used in various
studies. However, unlike the binary exposure case, there still lacks valid and
general sensitivity analysis methods for continuous exposures, except in some
special cases such as pair matching. To fill this gap in the binary outcome
case, we develop a sensitivity analysis framework for general matching designs
with continuous exposures and binary outcomes. First, we use probabilistic
lattice theory to show our sensitivity analysis approach is
finite-population-exact under Fisher's sharp null. Second, we prove a novel
design sensitivity formula as a powerful tool for asymptotically evaluating the
performance of our sensitivity analysis approach. Third, to allow effect
heterogeneity with binary outcomes, we introduce a framework for conducting
asymptotically exact inference and sensitivity analysis on generalized
attributable effects with binary outcomes via mixed-integer programming.
Fourth, for the continuous outcomes case, we show that conducting an
asymptotically exact sensitivity analysis in matched observational studies when
both the exposures and outcomes are continuous is generally NP-hard, except in
some special cases such as pair matching. As a real data application, we apply
our new methods to study the effect of early-life lead exposure on juvenile
delinquency. We also develop a publicly available R package for implementation
of the methods in this work.
