Unbalanced ranked set sampling in cluster randomized studies

•The problem of imbalance due to use of URSS with CRDs/missing data is considered.•Nonparametric estimators & tests are developed to assess treatment effects.•Optimality, finite-sample & asymptotic properties of URSS estimators are studied.•For testing in RSS-embedded CRDs, we conservatively suggest to use BRSS over URSS.•The proposed methods work well when missing data causes imbalance.

We consider the use of unbalanced ranked set sampling (URSS) with cluster randomized designs (CRDs), and extend nonparametric estimators and testing methods, previously developed by Wang et al. (2016) for the use of balanced RSS (BRSS) with CRDs, to account for unbalanced stratified structures under different ranking schemes. We study the optimality, finite-sample and asymptotic properties of the URSS estimators, and numerically quantify and compare the relative efficiency of the URSS vs. BRSS estimators. We also study and compare the power of the URSS tests vs. their BRSS counterparts via simulation. Further, we investigate the application of the proposed methods to unbalanced data from BRSS-structured CRDs due to missing observations and illustrate it with an example using educational data. Finally, based on our results, we offer recommendations about when to use URSS/BRSS with CRDs.