Mark G. Vangel and Andrew L. Rukhin
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
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 Student-t and standard normal random variables, respectively. This is implies that
is the density of . 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
Date created: 7/20/2001