Optimal allocation of subjects in a cluster randomized trial with fixed number of clusters when the ICCs or costs are heterogeneous over clusters

The intra-cluster correlation coefficient (ICC) plays an important role while designing the cluster randomized trials (CRTs). Often optimal CRTs are designed assuming that the magnitude of the ICC is constant across the clusters. However, this assumption is hardly satisfied. In some applications, the precise information about the cluster specific correlation is known in advance. In this article, we propose an optimal design with non-constant ICC across the clusters. Also in many situations, the cost of sampling of an observation from a particular cluster may differ from that of some other cluster. An optimal design in those scenarios is also obtained assuming unequal costs of sampling from different clusters. The theoretical findings are supplemented by thorough numerical examples.