Estimation of variance components in unbalanced nested designs

Motohiro Yamasaki, Yoshikazu Ojima, Seiichi Yasui

Research output: Contribution to journalArticlepeer-review


To evaluate the precision of measurement method, nested designs are often used for estimating variance components. Variance components are the element of dispersion by a factor conducting dispersion of measurement result. So, it is the useful evaluation of measurement method for us to estimate variance components. In general, its estimator is derived from the sums of squares by ANOVA, and it is defined as the ordinary method in this paper. But it is possible to conduct the alternative sums of squares by ANOVA in the unbalanced design. Therefore, in two-stage nested design, that is one-way ANOVA model, the estimator for variance components by using the sums of squares that conducted overall mean as a simple mean of mean of each level is proposed. It is defined as the proposed method in this paper. Additionally, the expectations and variances of estimators are derived by using the positive semi-definite quadratic form in our study. And, the performance of estimators obtained from two methods is evaluated by their variances. As a result, the performance of the proposed method was superior to that of the ordinary in many conditions.

Original languageEnglish
Pages (from-to)469-482
Number of pages14
JournalJournal of Quality
Issue number6
Publication statusPublished - 1 Dec 2010


  • Unbalanced nested design
  • Unbiased estimators
  • Variance components


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