3.4.5 Inference on a Common Mean in an Interlaboratory Study

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 t-densities':

$$\pi(\mu,\sigma\vert\{x_{ij}\}) \propto \pi(\sigma) \prod_{i=1... ..._i}\left[ \frac{x_i-\mu}{t_i};\frac{2\sigma^2}{t_i^2} \right],$$

where

$$f_{\nu}\left(x; \theta\right) = \frac{1}{\Gamma_{\nu/2}\sqrt{... ...\frac{x^2}{\theta y +\nu}\right]}} {\sqrt{\theta y +\nu}} dy,$$

$\mu$ is the common mean, $\sigma$ is the between-lab standard deviation, x_i is the sample mean, t_i is the sample standard deviation of this mean, and $\nu_i$ is the degrees of freedom for this standard deviation.

This posterior distribution can lead to approximate confidence regions for $\mu$ and $\sigma$ 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 $\mu$ 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
Last updated: 7/20/2001
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