Confidence regions for variance ratios in variance components model

Abstract

This study, using the method described by Hartley and Rao [10], derives confidence regions for the ratios of variances in random model. Each region is compared with a confidence region derived by Wald [22]. We derived confidence regions for the ratios of components of variance for the balanced one-way classification, balanced nested classification, balanced two-way classification mixed model without interaction, balanced two-way classification Model II without interaction and balanced incomplete block design. A full agreement was obtained between Hartley and Rao's [10] procedure, Wald's [22] procedure and the analysis of variance procedure for the one-way classification, nested classification and two-way classification mixed model without interaction. Wald's [22] confidence region was identical to the analysis of variance procedure for two-way classification Model II without interaction and the balanced incomplete block design. We then considered unbalanced one-way classification and unbalanced nested classification. The confidence interval for γ₁ in one-way classification with unequal numbers per subclass derived via Hartley and Rao's [10] procedure was identical to a confidence interval derived by Wald [19]. Three confidence regions for and γ₁ and γ₂ were derived based on the procedure outlined in Hartley and Rao [10]. Wald's [22] joint confidence region for γ₁ and γ₂ was derived by two methods then a confidence region for γ₂ was derived using Wald's [22] procedure.