  Motivated by an observational study of the effect of hospital ward versus
intensive care unit admission on severe sepsis mortality, we develop methods to
address two common problems in observational studies: (1) when there is a lack
of covariate overlap between the treated and control groups, how to define an
interpretable study population wherein inference can be conducted without
extrapolating with respect to important variables; and (2) how to use
randomization inference to form confidence intervals for the average treatment
effect with binary outcomes. Our solution to problem (1) incorporates existing
suggestions in the literature while yielding a study population that is easily
understood in terms of the covariates themselves, and can be solved using an
efficient branch-and-bound algorithm. We address problem (2) by solving a
linear integer program to utilize the worst case variance of the average
treatment effect among values for unobserved potential outcomes that are
compatible with the null hypothesis. Our analysis finds no evidence for a
difference between the sixty day mortality rates if all individuals were
admitted to the ICU and if all patients were admitted to the hospital ward
among less severely ill patients and among patients with cryptic septic shock.
We implement our methodology in R, providing scripts in the supplementary
material.
