MEASUREMENT SCIENCE REVIEW, Volume 1, Number 1, 2001 ON STATISTICAL MODELS FOR CONSENSUS VALUES Viktor Witkovsk´ y and Gejza Wimmer Institute of Measurement Science and Mathematical Institute Slovak Academy of Sciences D´ ubravsk´ a cesta 9 842 19 Bratislava, Slovakia E-mail: umerwitk@savba.sk Abstract We consider the problem of measurements made by several laboratories (or methods) on virtually the same object of interest. In general, the number of measurements made at each laboratory may differ. Moreover, the laboratories may exhibit the between-laboratory variability caused by the systematic error due to each laboratory, as well as different within-laboratory variances caused by different within-laboratory precision of the used measurement method. In this paper we try to describe statistical models and methods that are appropriate for derivation of the consensus mean of the unknown (measured) value as well as related problems concerning the statistical inference on the unknown value. 1. Introduction We consider that the measurements on virtually the same object of interest are made by k ≥ 2 laboratories. The ith laboratory repeats its measurements n i times, n i ≥ 2. The laboratories may exhibit the between laboratory variability, as well as different within-laboratory variances (heteroscedasticity). In this paper we will assume that the measurements follow normal distribution. The results of a typical interlaboratory study (given in aggregated form) are presented in Table 1: In [2] the data on Selenium in non-fat milk powder were reported. The measurements are based on four independent measurement methods. Table 1. Selenium in non-fat milk powder Method n i ¯ y i (mean) s 2 i (variance) A 8 105.00 85.711 B 12 109.75 20.748 C 14 109.50 2.729 D 8 113.25 33.640 As pointed in [7]: A question of fundamental importance in the analysis of such data is how to form the best consensus mean, and what uncertainty to attach to this estimate. This fundamental question is followed by a series of other questions regarding the statistical inference on the unknown common mean. The problem, although not new in statistical literature, see e.g. [1, 4], is not completely solved. Many questions remain still open and unsolved. The problem of interlaboratory comparisons is of particular interest for applications that are looking for harmonization of industrial and scientific practice. The general problem covers many aspects of which the major ones are: (1) the choice of the appropriate model, (2) the choice of associated statistical methods and (3) the identification and verification of the model. 33