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 against H₁:ρ₁>0 depends also on ρ₂. Also an exact (1-σ) confidence interval for ρ₁ irrespective of ρ₂ is not possible. Many approximate solutions were proposed in the past. One way to overcome this difficulty is to construct simultaneous confidence proposed, which produces exact tests and confidence intervals for ρ₁ given a certain value of ρ₂. The experimenter who is interested in a test or confidence interval for ρ₁ decides beforehand in which ranvge of ρ₂ he is interested in his experiment to make decisions about ρ₁. Exact size σ tests and (1-σ) confidence intervals for ρ₁ given a value of ρ₂ are presented here.

Keywords:

• projections on vector spaces
• exact distribution of ratio of SS
• Wald's procedure