Estimation of a common mean and weighted means statistics

Measurements made by several laboratories may exhibit nonnegligible between-laboratory variability, as well as different within-laboratory variances. Also, the number of measurements made at each laboratory often differ. Questions of fundamental importance in the analysis of such data are how to form a best consensus mean, and what uncertainty to attach to this estimate. An estimation equation approach due to Mandel and Paule is often used at the National Institute of Standards and Technology (NIST), particularly when certifying standard reference materials. Primary goals of this article are to study the theoretical properties of this method, and to compare it with some alternative methods, in particular to the maximum Likelihood estimator (MLE). Toward this end, we show that the Mandel-Paule solution can be interpreted as a simplified version of the maximum likelihood method. A class of weighted means statistics is investigated for situations where the number of laboratories is large. This class includes a modified MLE and the Mandel-Paule procedure. Large-sample behavior of the distribution of these estimators is investigated. This study leads to a utilizable estimate of the variance of the Mandel-Paule statistic and to an approximate confidence interval for the common mean. It is shown that the Mandel-Paule estimator of the between-laboratory variance is inconsistent in this setting. The results of numerical comparison of mean squared errors of these estimators for a special distribution of within-laboratory variances are also reported.