Setting Multivariate and Correlated Acceptance Limits for Assessing the Conformity of Items

Abstract

Many industrial products, foodstuffs and environmental samples are checked for values of different chemical parameters against tolerance limits or intervals defined in a specification or legislation. In some cases, the measured values of the different parameters are correlated due to how materials are obtained, chemical constraints and/or due to the simple fact that determinations are performed by multi-analyte procedures that share analytical operations and effects. In these cases, instead of defining an acceptance criterion for each measured value on the tested item separately based on the respective measurement uncertainty, the multivariate problem should be addressed by defining multivariate criteria. These multivariate criteria are set for a maximum total risk of wrong conformity decisions that is a complex function of all particular risks of the item being rejected or accepted by comparing each measured value with its respective limit. Computational tools have been developed to estimate the total specific risk of an item being wrongly considered to conform or not to conform with tolerance limits for various components when the measured values are independent or correlated. However, these tools must be applied for each test to check if the total specific risk is acceptable. This work presents a tool for setting multivariate acceptance limits applicable to correlated measurements and referenced to a defined total specific risk. The acceptance limits allow the decision about conformity of an item based on the simple comparison of the measured values with the acceptance limits. The acceptance limits are estimated by a user-friendly and iterative tool implemented in a MS-Excel spreadsheet and available in the Supplementary Material. This tool is successfully applied to various conformity problems. Acceptance limits based on informative and non-informative prior information are compared for a critical review of the merits and problems associated with Bayesian or frequentist conformity assessments.

Keywords:

• Multivariate conformity
• acceptance limits
• tolerance limits
• measurement uncertainty
• risk assessment
• Markov Chain Monte Carlo
• Bayesian probability

Acknowledgements

The authors thank Prof Ilya Kuselman for useful comments and discussions. The authors gratefully acknowledge to Fundação para a Ciência e Tecnologia (Projects SFRH/BPD/110186/2015, UIDB/00100/2020 and UIDP/00100/2020) for funding the work.

Conflict of interest

The authors declare that they have no conflicts of interest.