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3.3.6 Inference on a Common Mean in an Interlaboratory Study
Mark G. Vangel
Andrew Rukhin
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
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 |
boundary.