Inter-laboratory comparisons.

A number of alloy bars were manufactured to some very precise specifications. Certain scientific institutions then performed chemical analyses and made several measurements of the content of some chemical elements of interest. The measurements made on each of the alloy bars can be considered a set of repeated measurements. Modelling techniques for repeated measurements are now well established. Many of these techniques are based on the multivariate normal distribution with some specified mean and covariance structure. Modelling of the covariance structure is necessary so that efficient and meaningful inferences may be made about the mean structure. For the example of repeated measurements made on an alloy bar, the set of measurements is assumed to follow a multivariate normal distribution with a mean mu and a covariance structure Sigma. The choice of mu and Sigma is explored. Experiments which produce sets of repeated measurements can quite often result in a large amount of data being collected. This means that the use of statistical techniques to fit the model to the data can become computationally demanding. The use of maximum likelihood estimation is considered. Several aspects of constructing computationally efficient algorithms to maximise the likelihood function of the data are addressed. When the proposed model has been fitted to the data the suitability of the model and its assumptions are investigated. A score test is constructed to assess the correctness of the proposed covariance structure. Normal plots of the standardised residuals are used to assess other possible defects in the model, such as an incorrect assumption of normally distributed data. The work which has been carried out was motivated specifically by experiments where the set of repeated measurements came from a chemical analysis of an alloy material. It is the percentage content of a number of chemical elements which is of interest and the choice of statistical models was made with this in mind. However, it is demonstrated how the statistical techniques and models for the analysis of the chemical data may be used to analyse repeated measurements which arise from other kinds of experiments