Mark G. Vangel
Stefan D. Leigh
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. An estimation-equation approach due to Mandel and Paule is often used at NIST, particularly when certifying standard reference materials. However, the theoretical properties of this procedure were not well understood. Primary goals of the present research are to study the properties of this widely-used method, and to compare it with competitors, in particular to maximum-likelihood.
We have shown that the Mandel-Paule solution is equivalent to an approximate REML method, where the within-laboratory variances are estimated by the usual sample variances, instead of their restricted MLEs. Similarly, a trivial modification of Mandel-Paule can be shown to be an excellent approximation to maximum-likelihood. A very simple approximate variance for the Mandel-Paule mean estimate has been found. In numerical examples, this approximate variance agrees closely with delta-method and observed Fisher information results.
In addition, a reparametrization of the likelihood has been found which enables the entire profile-likelihood surface, in the plane of the consensus mean and between-laboratory standard deviation, to be calculated efficiently and reliably. This calculation is performed by a simple iteration which increases the likelihood with each step. By examining this surface, the MLE can be determined, along with all other stationary points. This also facilitates straightforward Bayesian computation, using a non-informative prior and numerical integration.
In the figure, the joint marginal posterior distribution for the mean and between-laboratory variance is displayed for data from an interlaboratory study in which 28 laboratories measured arsenic in NIST oyster tissue SRM 1566a. Estimates of the mean and between-laboratory standard deviations are as follows:
Figure 24: A Bayesian Analysis of Interlaboratory Data on Arsenic in SRM 1566a (Oyster Tissue)
Date created: 7/20/2001