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3.4.5 Inference on a Common Mean in an Interlaboratory Study

Mark G. Vangel and 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. Some recent work has focused on hierarchical models for one-way ANOVA and for two-way tables, and their use in interlaboratory studies.

For the following two-stage hierarchical model for one-way ANOVA

$\displaystyle p(x_{ij} \vert \theta_i, \sigma_i^2)$ = $\displaystyle {\rm N\,}(\theta_i, \sigma_i^2)$  
$\displaystyle p(\sigma_i)$ $\textstyle \propto$ $\displaystyle 1/\sigma_i$  
$\displaystyle p(\theta_i \vert \mu, \sigma^2)$ = $\displaystyle {\rm N\,}(\mu,\sigma^2)$  
$\displaystyle p(\mu)$ $\textstyle \propto$ 1  
$\displaystyle p(\sigma)$ $\textstyle \propto$ 1.  

the joint posterior of $\mu$ and $\sigma$ is proportional to the prior $p(\sigma)$ times the product of the densities of $U_i, i=1,\dots,k$, where

\begin{displaymath}U_i = \sqrt{\frac{s_i}{n_i}} T_{\nu_i}
+\bar{x}_i +\sigma Z,

$\{\bar{x}_i\}$ and $\{s_i\}$ are the sufficient statistics, and $T_{\nu_i}$ and Z are independent Student-t and standard normal random variables, respectively. This is implies that

\begin{displaymath}p(\mu,\sigma\vert \{x_{ij}\}) \propto
p(\sigma) \prod_{i=1}^{...
...x}_i-\mu}{s_i/\sqrt{n_i}};\frac{2 n_i\sigma^2}{s_i^2} \right),


\begin{displaymath}f_{\nu}\left(x; \gamma\right) \equiv
...1+\frac{x^2}{\gamma y
+\nu}\right]}} {\sqrt{\gamma y +\nu}} dy

is the density of $T_{\nu}+Z\sqrt{\gamma/2}$. This result is intuitively appealing and computationally useful. We've investigated generalizations to two-way mixed models, but the numerical integrations required become much more difficult. The figure illustrates posterior calculations for interlaboratory study data on arsenic in oyster tissue.


Figure 28: Marginal posterior densities for the consensus mean and between-laboratory standard deviation for an interlaboratory study on arsenic in oyster tissue. Calculations were done by numerical quadrature, using the hierarchical model given above, with a uniform prior on the between-laboratory standard deviation

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Date created: 7/20/2001
Last updated: 7/20/2001
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