Tutorial and spreadsheets for Bayesian evaluation of risks of false decisions on conformity of a multicomponent material or object due to measurement uncertainty

A tutorial and a user-friendly program for evaluating risks of false decisions in conformity assessment of a multicomponent material or object due to measurement uncertainty, based on a Bayesian approach, are presented. The developed program consists of two separate MS-Excel spreadsheets. It allows calculation of the consumer's and producer's risks concerning each component of the material whose concentration was tested ('particular risks') as well as concerning the material as a whole ('total risks'). According to the Bayesian framework, probability density functions of the actual/'true' component concentrations (prior pdfs) and likelihood functions (likelihoods) of the corresponding test results are used to model the knowledge about the material or object. Both cases of independent and correlated variables (the actual concentrations and the test results) are treated in the present work. Spreadsheets provide an estimate of the joint posterior pdf for the actual component concentrations as the normalized product of the multivariate prior pdf and the likelihood, starting from normal or log-normal prior pdfs and normal likelihoods, using Markov chain Monte Carlo (MCMC) simulations by the Metropolis-Hastings algorithm. The principles of Bayesian inference and MCMC are described for users with basic knowledge in statistics, necessary for correct formulation of a task and interpretation of the calculation results. The spreadsheet program was validated by comparison of the obtained results with analytical results calculated in the R programming environment. The developed program allows estimation of risks greater than 0.003% with standard deviations of such estimates spreading from 0.001% to 1.5%, depending on the risk value. Such estimation characteristics are satisfactory, taking into account known variability in measurement uncertainty associated with the test results of multicomponent materials.