Mark G. Vangel
Andrew L. Rukhin
Statistical Engineering Division, ITL
Data on a quantity measured by several laboratories often exhibits non-negligible between-laboratory variability, as well as different within-laboratory variances. Also, the number of measurements made at each laboratory can differ. A question of fundamental importance in the analysis of such data is how to best estimate a consensus mean, and what uncertainty to attach to this estimate.
We have been engaged
in a detailed investigation of this problem, and
its generalizations and applications. Recent
results include a representation of the posterior
distribution of the common mean under a Bayesian
hierarchical model with `noninformative' prior
distributions as a product of symmetric `generalized
is the common mean, is the between-lab standard deviation, xi is the sample mean, ti is the sample standard deviation of this mean, and is the degrees of freedom for this standard deviation.
This posterior distribution can lead to approximate confidence regions for and for situations where exact frequentist results are not available. The figure illustrates the results of a small simulation study which compares frequentist intervals with non-informative-prior Bayesian intervals for a special case.
Figure 29: Results of a small simulation comparing Bayesian equal-tailed probability intervals with corresponding confidence intervals on for a one-way random-effects ANOVA model with 5 groups of 5, a true mean of 0, standard deviation 1, and intraclass correlation of 1/2. The Bayesian intervals are always slightly larger, which is to be expected since these intervals do not require the assumption that the within-laboratory variances are equal.
Date created: 7/20/2001