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The clustered observational study (COS) design is the observational counterpart to the clustered randomized trial. COSs are common in both education and health services research. In education, treatments may be given to all students within some schools but withheld from all students in other schools. In health studies, treatments may be applied to clusters such as hospitals or groups of patients treated by the same physician. In this paper, we study the identification of causal effects in COS designs. We focus on the prospect of differential selection of units to clusters, which occurs when the units’ cluster selections depend on the clusters’ treatment assignments. Extant work on COSs has made an implicit assumption that rules out the presence of differential selection. We derive the identification results for designs with differential selection and that contexts with differential cluster selection require different adjustment sets than standard designs. We outline estimators for designs with and without differential selection. Using a series of simulations, we outline the magnitude of the bias that can occur with differential selection. We then present 2 empirical applications focusing on the likelihood of differential selection. 

Summary Nonparametric covariate adjustment is considered for log-rank-type tests of the treatment effect with right-censored time-to-event data from clinical trials applying covariate-adaptive randomization. Our proposed covariate-adjusted log-rank test has a simple explicit formula and a guaranteed efficiency gain over the unadjusted test. We also show that our proposed test achieves universal applicability in the sense that the same formula of test can be universally applied to simple randomization and all commonly used covariate-adaptive randomization schemes such as the stratified permuted block and the Pocock–Simon minimization, which is not a property enjoyed by the unadjusted log-rank test. Our method is supported by novel asymptotic theory and empirical results for Type-I error and power of tests.