In a randomized trial, subjects are assigned to treatment or control by the flip of a fair coin. In many nonrandomized or observational studies, subjects find their way to treatment or control in two steps, either or both of which may lead to biased comparisons. By a vague process, perhaps affected by proximity or sociodemographic issues, subjects find their way to institutions that provide treatment. Once at such an institution, a second process, perhaps thoughtful and deliberate, assigns individuals to treatment or control. In the current article, the institutions are hospitals, and the treatment under study is the use of general anesthesia alone versus some use of regional anesthesia during surgery. For a specific operation, the use of regional anesthesia may be typical in one hospital and atypical in another. A new matched design is proposed for studies of this sort, one that creates two types of nonoverlapping matched pairs. Using a new extension of optimal matching with fine balance, pairs of the first type exactly balance treatment assignment across institutions, so each institution appears in the treated group with the same frequency that it appears in the control group; hence, differences between institutions that affect everyone in the same way cannot bias this comparison. Pairs of the second type compare institutions that assign most subjects to treatment and other institutions that assign most subjects to control, so each institution is represented in the treated group if it typically assigns subjects to treatment or, alternatively, in the control group if it typically assigns subjects to control, and no institution appears in both groups. By and large, in the second type of matched pair, subjects became treated subjects or controls by choosing an institution, not by a thoughtful and deliberate process of selecting subjects for treatment within institutions. The design provides two evidence factors, that is, two tests of the null hypothesis of no treatment effect that are independent when the null hypothesis is true, where each factor is largely unaffected by certain unmeasured biases that could readily invalidate the other factor. The two factors permit separate and combined sensitivity analyses, where the magnitude of bias affecting the two factors may differ. The case of knee surgery in the study of regional versus general anesthesia is considered in detail.