Elsevier

Talanta

Volume 174, 1 November 2017, Pages 789-796
Talanta

Risk of false decision on conformity of a multicomponent material when test results of the components’ content are correlated

https://doi.org/10.1016/j.talanta.2017.06.073Get rights and content

Highlights

  • Correlation among test results of a multicomponent material complicates its conformity assessment.

  • A statistical procedure for evaluation of a total risk of false decision on the drug conformity is developed.

  • A case of correlated test results for four components under control in a drug is studied.

  • Strong correlation may lead either to decreasing or increasing of the total risk, depending on the actual test results.

Abstract

The probability of a false decision on conformity of a multicomponent material due to measurement uncertainty is discussed when test results are correlated. Specification limits of the components’ content of such a material generate a multivariate specification interval/domain. When true values of components’ content and corresponding test results are modelled by multivariate distributions (e.g. by multivariate normal distributions), a total global risk of a false decision on the material conformity can be evaluated based on calculation of integrals of their joint probability density function. No transformation of the raw data is required for that. A total specific risk can be evaluated as the joint posterior cumulative function of true values of a specific batch or lot lying outside the multivariate specification domain, when the vector of test results, obtained for the lot, is inside this domain. It was shown, using a case study of four components under control in a drug, that the correlation influence on the risk value is not easily predictable. To assess this influence, the evaluated total risk values were compared with those calculated for independent test results and also with those assuming much stronger correlation than that observed. While the observed statistically significant correlation did not lead to a visible difference in the total risk values in comparison to the independent test results, the stronger correlation among the variables caused either the total risk decreasing or its increasing, depending on the actual values of the test results.

Introduction

Risk of false decision on conformity of a multicomponent material due to measurement uncertainty was recently discussed in the position paper of the IUPAC task group [1]. There are several kinds of such risk. The probability of accepting a batch or lot of the material, when it should have been rejected, is named ‘consumer's risk’, whereas the probability of falsely rejecting the lot is the ‘producer's risk’. For a specified lot, they are referred to as the ‘specific consumer's risk’ and the ‘specific producer's risk’, Rci*, for i-th particular component of the material under control, i = 1, 2, …, n, respectively. The risks of incorrect conformity assessment of a lot randomly drawn from a statistical population of such lots are the ‘global consumer's risk’ and the ‘global producer's risk’, Rci, for i-th particular component, since they characterize the material production globally [2]. Even if conformity assessment for each i-th component of a material is successful (i.e. the particular specific Rci* or global Rci risks are small enough), the total probability of a false decision concerning the material as a whole (the total specific Rtotal* or global Rtotal risks) might still be significant. A scheme summarizing the terminology used here is shown in Fig. 1.

A model of the total risk for the case of independent quantities has been formulated on the basis of the law of total probability [3]. Using this model, the total risk can be evaluated as a combination of the particular risks of conformity assessment of the material components. For a more complicated task, i.e. for an increased number of components of the material under control, the total risk increases. Examples for three and four components (n = 3 and 4, respectively) are given below in the Appendix A. General expressions for evaluating the total global consumer's risk for any number n of the material components are also provided [1]. The counterpart models for the total producer's risk are easily obtainable as well.

However, the problem is that the assumption of independence of true values of each component content ci from other(s) and independence of corresponding measurement/test results cim is not always acceptable. Correlation of true values may be caused by stoichiometry of native compounds, or by technological conditions in production of materials, etc. In their turn, test results may be correlated because of correlation of true values, and/or due to systematic effects in the measurement/test process, common for two or more analytes.

The task of evaluating the total risks for correlated quantities is detailed in the present paper, based on a case study of test results of NyQuil tablets. This cold/flu medication contains four active components: 1) acetaminophen (APAP) as a pain reliever and fever reducer; 2) dextromethorphan hydrobromide (DEX) as a cough suppressant; 3) doxylamine succinate (DOX) as an antihistamine and hypnotic; and 4) phenylephrine hydrochloride (PE) as a nasal decongestant [4]. However, there are publications which have claimed that the last component (PE) is no more effective than placebo [5]. Therefore, the case study is performed for both scenarios: when particular risks of conformity assessment of four and three only (without PE) components contribute to the total risks. To assess influence of the correlation of the test results on the evaluated total risk values, they are compared with those calculated for independent test results by formulas (A.1), (A.2), (A.3), (A.4) shown in the Appendix A, and also with the values obtained supposing much stronger correlation than that observed.

Section snippets

Specification and acceptance limits

The assay test lower and upper specification limits, lsli and usli, for the product release for each active component i = 1, 2, 3, 4 are 95.0–105.0 % of the labeled amount li, respectively. The labeled amounts are the following: l1 = 325 mg for APAP, l2 = 10 mg for DEX, l3 = 6.25 mg for DOX, and l4 = 5 mg for PE, per tablet (a tablet mass is of 775 mg on average). The acceptance limits of test results coincide with the specification limits in this study.

Sample preparation

A sample of the tablets is weighted, dissolved

Global distributions of the components’ content values

A total of N = 105 lots of the medication produced and released at the same factory during a year were tested in the same laboratory belonging to the factory. Histograms of the test results cim are shown in Fig. 2 for: a) APAP, i = 1; b) DEX, i = 2; c) DOX, i = 3; and d) PE, i = 4. Mean, mi, and standard deviation, si, values of the test results are presented in Table 1.

Note that the si values are smaller than the target measurement uncertainty ui = (urel/100 %) cim = 0.028 cim, % of labeled

Conclusions

Correlation among test results of a multicomponent material complicates its conformity assessment. When true values of components’ content and corresponding test results are modelled by multivariate distributions, for example normal pdfs (prior and likelihood, respectively), a total global risk of a false decision on the material conformity can be evaluated based on calculation of integrals of their joint pdf. No transformation of the raw data is required for that.

A total specific consumer's

Acknowledgement

This research was supported in part by the International Union of Pure and Applied Chemistry (IUPAC Project 2016–007–1–500).

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