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

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.

诸多工业产品、食品及环境样本均需依据规范或立法文件中设定的容限限值或区间，对各类化学参数的检测值开展合规核验。在部分场景中，由于物料获取方式、化学约束条件，或是检测采用了共享分析操作与分析效应的多分析物检测流程（multi-analyte procedures），不同参数的测量值之间可能存在相关性。在此类情形下，若仍基于各测量值对应的测量不确定度（measurement uncertainty），对被测样品的每个测量值分别单独设定合格判定准则，则需通过构建多元判定准则来处理该多元问题。此类多元准则被设定为控制错误合格判定的最大总风险，该总风险是将每个测量值与其对应限值比对后，由被测样品被拒收或接收的各项特定风险构成的复杂函数。目前已开发出多款计算工具，可在测量值独立或存在相关性的场景下，估算被测样品因被错误判定为符合或不符合容限限值时，针对各被测组分的总特定风险。但此类工具需针对每项检测单独应用，以验证总特定风险是否处于可接受范围。本研究开发了一款可设定多元合格限值的工具，该工具适用于存在相关性的测量场景，且可基于预设的总特定风险进行参考计算。借助该多元合格限值，可通过将测量值与合格限值直接比对，快速完成被测样品的合格性判定。该合格限值可通过一款用户友好的迭代式工具估算得到，该工具已集成于MS-Excel电子表格中，并可在补充材料（Supplementary Material）中获取。该工具已成功应用于各类合格性判定场景。通过对比基于信息性先验信息与非信息性先验信息的合格限值，可批判性梳理贝叶斯（Bayesian）与频率学派（frequentist）合格性评估方法的优势与现存问题。