  One central goal of design of observational studies is to embed
non-experimental data into an approximate randomized controlled trial using
statistical matching. Despite empirical researchers' best intention and effort
to create high-quality matched samples, residual imbalance due to observed
covariates not being well matched often persists. Although statistical tests
have been developed to test the randomization assumption and its implications,
few provide a means to quantify the level of residual confounding due to
observed covariates not being well matched in matched samples. In this article,
we develop two generic classes of exact statistical tests for a biased
randomization assumption. One important by-product of our testing framework is
a quantity called residual sensitivity value (RSV), which provides a means to
quantify the level of residual confounding due to imperfect matching of
observed covariates in a matched sample. We advocate taking into account RSV in
the downstream primary analysis. The proposed methodology is illustrated by
re-examining a famous observational study concerning the effect of right heart
catheterization (RHC) in the initial care of critically ill patients. Code
implementing the method can be found in the supplementary materials.
