Interval estimation of variance components is studied for the unbalanced one-way random effects model. An easily calculated function, $W$, of the harmonic mean of the class sizes and of the sample variance of the class means is found to be important. The exact distribution of $W$ is found and is shown to be excellently approximated by a chi-square distribution. The random variable $W$ is used to construct interval estimates for (i) the between classes variance component and (ii) the ratio of the variance components and thus for the intraclass correlation and heritability. For most one-way unbalanced designs use of these approximate interval estimators will work very well.
"Interval Estimation for the Unbalanced Case of the One-Way Random Effects Model." Ann. Statist. 6 (3) 582 - 587, May, 1978. https://doi.org/10.1214/aos/1176344202