Limits for a Variance Component with an Exact Confidence Coefficient

Abstract

This paper deals with confidence interval estimation and hypothesis testing for components of variance, in analysis-of-variance situations embraced by the Model II of Eisenhart [1], including also the so-called nested classifications. Several authors have treated the problem of setting confidence limits for variance components, and several approximate methods have been proposed. Four approximate methods are described in Anderson and Bancroft [2] and briefly in Crump [3], and references to original sources and extensive bibliographies are given in both [2] and [3]. In [4], Green gives more refined approximations which are, however, not presently in a form for practical use. Huitson [5], Welch [6] and Cochran [7] discuss related problems involving linear combinations of variances, and offer approximate methods for these problems. The many references on approximate tests and confidence limits in variance component problems emphasize the absence of exact methods. The present paper points out that, using a randomization device, exact confidence limits and tests for a variance component become available in a simple way. These exact confidence limits will usually but not always define a single confidence interval; in the exceptional case (having small probability in practice) the exact limits may define an interval with a gap in it. Numerical illustrations are given, together with comparisons with results using some of the available approximate methods. Also, asymptotic power comparisons between the exact test and two approximate tests are discussed. There are at least three notions of confidence that can be associated with the statement: "$a(x) \leqq \theta \leqq b(x)$ is a $100(1 - \alpha){\tt\%}$ confidence interval" for the parameter $\theta$ with possible nuisance parameters $\eta$, based on observations $x$. They are \begin{equation*}\begin{align*}\mathrm{(a)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \equiv 1 - \alpha\quad \text{for all} \quad \theta, \eta \\ \mathrm{(b)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \geqq 1 - \alpha\quad \text{for all} \quad \theta, \eta \quad \text{with equality for some} \quad \theta, \eta. \\ \mathrm{(c)} \quad \operatorname{Pr}\{a(x) &\leqq \theta \leqq b(x)\} \geqq 1 - \alpha \quad \text{for all} \quad \theta, \eta. \end{align*}\end{equation*} The phrase ``exact confidence" we shall interpret in the sense of (a) above. So far as the author is aware, for a variance component no confidence limits satisfying either of the notions (a) or (b) have previously been constructed. An interval satisfying (c) has been constructed in [8], using a two-stage sampling procedure. In the present approach, the mathematical difficulties ordinarily caused in variance component analysis by the presence of nuisance parameters are circumvented by a process of randomization. The resulting confidence limits depend not only on the mean squares of the analysis of variance table, but also on auxiliary observations on a random variable with known normal distribution. A consequence is that two statisticians confronted with the same analysis of variance table will in general construct different confidence limits. While practically this may afford some discomfiture, it remains true nevertheless that these exact confidence limits meet the ordinary claim (a) as to probability of containing the true variance component. In the examples tried, the limits are plausible and they are not difficult to compute. Moreover, the agreement between numerical results using the method proposed herein and the usual approximate methods may serve to increase one's faith in the approximate methods in small samples.