Total risk of a false decision on conformity of an alloy due to measurement uncertainty and correlation of test results

Total risk (probability) of a false decision on conformity of an alloy due to measurement uncertainty and correlation of test results is quantified. As an example, a dataset of test results of a PtRh alloy is studied when contents of four components of the alloy composition are under control. There are specification limits for contents of 1) Pt and 2) Rh; 3) three precious impurities - Au, Ir and Pd, and 4) eight impurities, both precious Au, Ir, Pd, and non-precious Fe, Pb, Si, Sn, Zn. Test results of 100 batches of the alloy produced at the same plant, obtained by X-ray fluorescence and optical atomic emission spectrometry methods at the plant laboratory, were in the dataset. The Pt content was tested based on the mass balance. Measurement uncertainties of the test results are estimated summarizing data of validation reports of the measurement procedures for different elements/analytes. These test results are correlated because of the natural chemical origin of the raw materials used in the alloy production and mass balance constraints. Correlations between test results for two pairs of the components (Pt vs. Rh, and the three vs. the eight impurities) were strong. To assess the correlation effects on the total risk, the study was performed for two scenarios considering 1) correlated test results for all four components, and 2) practically uncorrelated test results for two components only - Rh and the eight impurities. A multiparameter Bayesian approach was applied for total risk evaluation, where the observed correlations are taken into account within the experimental correlation matrix. This matrix influenced all subsequent multivariate calculation results. It was shown that simplification of the testing by reducing the number of components under control leads to a significant increase of the probability of a false decision on conformity of an alloy batch randomly drawn from a statistical population of such batches. Core of the developed R code, used for the risk calculations, is presented.