The problem of estimating the intraclass correlation coefficient rho under balanced cluster sampling in a finite population is well known and solutions are available. This paper deals with the estimation of rho if the values of a study variable are only known for a simple random subsample of respondents from the original sample, e.g through non-response. We propose an estimator of rho which is similar to the well known ANOVA estimator in certain scenarios but explicitly incorporates non-response in the data through a subsample process.
This estimator is derived using a fundamental formula for the difference of the variance of the Horvitz-Thompson estimator and the variance of the Hansen-Hurwitz estimator for the original sample and the subsample. We compare our estimator with three commonly used estimators on the basis of a Monte Carlo simulation study.