This proposal is motivated by our development of the beta-approach that has worked successfully for many variance components models, but it has not been implemented in any software package. Earlier results indicate that the beta distribution might be an approximate (or exact) distribution for intraclass correlation coefficients (ICCs) in general. However, the simulations have not been exhaustive and a more theoretical justification via finite sampling asymptotical results is needed. We will study various scenarios of sample size limits (asymptotics) due to the possible structures of the data to justify the beta-approach using theory of mathematical statistics. Additionally, we will create a publicly available R package. The theoretical results will help us to understand when the calculations are reliable, so that the software can provide warnings if these settings are not satisfied.

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