Mark G. Vangel
Stefan D. Leigh
Statistical Engineering Division, CAML
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.
Toward this end, we have shown that the Mandel-Paule solution is equal to the maximum-likelihood estimate in the limit of large between-laboratory variability, and that this solution will nearly maximize the likelihood, provided that the MLEs of the within-laboratory variances are close to the sample variances. Also, the Mandel-Paule procedure can be derived in a natural way by approximating the likelihood equations, an approach which can be generalized.
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 very simple iteration which increases the likelihood with each step. By examining this surface, the MLE can be determined, along with all other stationary points. For example, the figure shows these calculations for an interlaboratory study on dietary fiber in apples. The large peak corresponds to the MLE. The secondary peak corresponds to downweighting an extreme outlying laboratory. If two laboratories are downweighted, then the remaining data can be pooled; this corresponds to the small peak on the boundary.
Future work will focus on confidence regions,
asymptotic theory, and extensions to the analysis
of interlaboratory studies which are represented
as two-way tables.
Figure 23: Relative Profile Likelihood for Apple Fiber Interlaboratory Study
Date created: 7/20/2001