  This paper studies inference in two-stage randomized experiments under
covariate-adaptive randomization. In the initial stage of this experimental
design, clusters (e.g., households, schools, or graph partitions) are
stratified and randomly assigned to control or treatment groups based on
cluster-level covariates. Subsequently, an independent second-stage design is
carried out, wherein units within each treated cluster are further stratified
and randomly assigned to either control or treatment groups, based on
individual-level covariates. Under the homogeneous partial interference
assumption, I establish conditions under which the proposed
difference-in-"average of averages" estimators are consistent and
asymptotically normal for the corresponding average primary and spillover
effects and develop consistent estimators of their asymptotic variances.
Combining these results establishes the asymptotic validity of tests based on
these estimators. My findings suggest that ignoring covariate information in
the design stage can result in efficiency loss, and commonly used inference
methods that ignore or improperly use covariate information can lead to either
conservative or invalid inference. Finally, I apply these results to studying
optimal use of covariate information under covariate-adaptive randomization in
large samples, and demonstrate that a specific generalized matched-pair design
achieves minimum asymptotic variance for each proposed estimator. The practical
relevance of the theoretical results is illustrated through a simulation study
and an empirical application.
