  In observational studies, it is typically unrealistic to assume that
treatments are randomly assigned, even conditional on adjusting for all
observed covariates. Therefore, a sensitivity analysis is often needed to
examine how hidden biases due to unobserved covariates would affect inferences
on treatment effects. In matched observational studies where each treated unit
is matched to one or multiple untreated controls for observed covariates, the
Rosenbaum bounds sensitivity analysis is one of the most popular sensitivity
analysis models. In this paper, we show that in the presence of interactions
between observed and unobserved covariates, directly applying the Rosenbaum
bounds will almost inevitably exaggerate the report of sensitivity of causal
conclusions to hidden bias. We give sharper odds ratio bounds to fix this
deficiency. We illustrate our new method through studying the effect of
anger/hostility tendency on the risk of having heart problems.
