Exact tests and confidence intervals for ratio of variance components in unbalanced two-and three-stage nested designs

Abstract

For the unbalanced two-stage nested design an exact size σ test for H0 : ρ≤ρ0 exists, where the variance component component in the first sampling stage σ2 e the error variance in the second stage. Using Wald's procudure one can construct and exact (1-σ) confidence interval for the ratio ρ of the variance components. For the three-stage nested design there are two interesting ratios of variance components where are the variance components in the first stage and second stage respectively, and is the error variance. The construction of exact size σ tests and (1-σ) confidence intervals for ρ2 is no problem using the same procedure as for the two-stage nested design. In the balanced three-stage nested design an exact size σ test is well known for H0: ρ ≥ ρ10 against H1:ρ ≥ ρ10 exists but depends also on the parameter ρ2. In the unbalanced three-stage nested design even the test for H1: ρ ≥ ρ10 exists but depends also on the parameter ρ2. In the unbalanced three-stage nested design even the test for H0:ρ1 = 0 ...