
3.4.5 Inference on a Common Mean in an Interlaboratory Study
Mark G. Vangel and Andrew L. Rukhin Data on a quantity measured by several laboratories often exhibits nonnegligible betweenlaboratory variability, as well as different withinlaboratory 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 oneway ANOVA and for twoway tables, and their use in interlaboratory studies.
For the following twostage hierarchical model
for oneway ANOVA
the joint posterior of and is proportional to the prior times the product of the densities of , where and are the sufficient statistics, and and Z are independent Studentt and standard normal random variables, respectively. This is implies that where is the density of . This result is intuitively appealing and computationally useful. We've investigated generalizations to twoway 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 betweenlaboratory 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 betweenlaboratory standard deviation
Date created: 7/20/2001 