Evaluating Additional Observations of the Same Units to Estimate Misclassification Probabilities in Measurement System Analysis

Abstract

Measurement System Analysis evaluates the accuracy and precision of measurement processes; in the literature, part variability and measurement error are typically assumed to follow normal distributions, and we adopt this convention. We derive closed‐form formulas for Types I and II misclassification probabilities using univariate and bivariate normal cumulative distribution functions, avoiding numerical integration and enabling efficient computation (e.g., in R). Building on these results, we derive explicit maximum likelihood estimates of misclassification probabilities for both the classical approach based on measurements from different parts and the repeated‐measurement approach using multiple measurements on the same part at different times. A Monte Carlo study shows that incorporating repeated measurements reduces bias and mean squared error. A brief numerical example with simulated data demonstrates practical implementation.

Keywords

• {Monte Carlo simulation
• maximum likelihood estimation
• measurement system analysis
• misclassification probabilities
• quality control

References

1. R. K. Burdick, G. A. Larsen, Confidence intervals on measures of variability in R&R studies, Journal of Quality Technology 29 (3) (1997) 261–273.
2. K. K. Dolezal, R. K. Burdick, N. J. Birch, Analysis of a two-factor R&R study with fixed operators, Journal of Quality Technology 30 (2) (1998) 163–170.
3. R. K. Burdick, A. E. Allen, G. A. Larsen, Comparing variability of two measurement processes using R&R studies, Journal of Quality Technology 34 (1) (2002) 97–105.
4. R. K. Burdick, C. M. Borror, D. C. Montgomery, A review of methods for measurement systems capability analysis, Journal of Quality Technology 35 (4) (2003) 342–354.
5. B. P. Weaver, M. S. Hamada, S. B. Vardeman, A. G. Wilson, A Bayesian approach to the analysis of gauge R&R data, Quality Engineering 24 (4) (2012) 486–500.
6. J. Quiroz, R. Baumgartner, Interval Estimations for Variance Components: A Review and Implementations, Taylor & Francis, 2019.
7. D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons, 2019.
8. A. Shirodkar, S. Rane, Evaluation of coordinate measuring machine using gage repeatability & reproducibility, International Journal of System Assurance Engineering and Management 12 (1) (2021) 84–90.

9. W. O. S. Soares, R. S. Peruchi, R. A. V. Silva, P. Rotella Junior, Gage R&R studies in measurement system analysis: A systematic literature review, Quality Engineering 34 (3) (2022) 382–403.
10. J. Plura, D. Vykydal, F. Tošenovský, P. Klaput, Graphical tools for increasing the effectiveness of gage repeatability and reproducibility analysis, Processes 11 (1) (2022) 1.
11. M. Croquelois, C. Ferraris, M. Achibi, F. Thiebaut, Process capability indices with student mixture models applied to aircraft engines GD&T, Journal of Manufacturing Science and Engineering 144 (10) (2022) 101008.
12. M. Saha, Application of a new process capability index to electronic industries, Communications in Statistics: Case Studies, Data Analysis and Applications 8 (4) (2022) 574–587.
13. M. Saha, S. Dey, Uses of a new asymmetric loss-based process capability index in the electronic industries, Communications in Statistics: Case Studies, Data Analysis and Applications 9 (2) (2023) 135–151.
14. M. Saha, A. Devi, P. Pareek, Applications of process capability indices for suppliers selection problems using generalized confidence interval, Communications in Statistics: Case Studies, Data Analysis and Applications 9 (3) (2023) 270–286.
15. K. Karakaya, Inference on process capability index Spmk for a new lifetime distribution, Soft Computing 28 (19) (2024) 10929–10941.
16. K. Karakaya, İ. Kınacı, Y. Akdoğan, B. Saraçoğlu, C. Kuş, Statistical inference on process capability index cpyk for inverse Rayleigh distribution under progressive censoring, Pakistan Journal of Statistics and Operation Research 20 (1) (2024) 37–47.
17. K. Karakaya, A general novel process capability index for normal and non-normal measurements, Ain Shams Engineering Journal 15 (6) (2024) 102753.
18. W. A. Hamdi, Y. Akdoğan, T. Erbayram, M. Alqawba, A. Z. Afify, Statistical inference of the generalized process capability index for the discrete Lindley distribution, Scientific Reports 15 (1) (2025) 6776.
19. D. C. Montgomery, Introduction to statistical quality control, John Wiley & Sons, 2020.
20. R. G. Easterling, M. E. Johnson, T. R. Bement, C. J. Nachtsheim, Statistical tolerancing based on consumer’s risk considerations, Journal of Quality Technology 23 (1) (1991) 1–11.
21. N. Doganaksoy, Assessment of impact of measurement variability in the presence of multiple sources of product variability, Quality Engineering 13 (1) (2001) 83–89.
22. G. A. Larsen, Measurement system analysis in a production environment with multiple test parameters, Quality Engineering 16 (2) (2003) 297–306.
23. R. K. Burdick, Y.-J. Park, D. C. Montgomery, C. M. Borror, Confidence intervals for misclassification rates in a gauge R&R study, Journal of Quality Technology 37 (4) (2005) 294–303.
24. F.-K. Wang, S. W. Yang, Applying bootstrap method to the types I–II errors in the measurement system, Quality and Reliability Engineering International 24 (1) (2008) 83–97.
25. D. Cocchi, M. Scagliarini, A robust approach for assessing misclassification rates under the two-component measurement error model, Applied Stochastic Models in Business and Industry 26 (4) (2010) 389–400.
26. N. Doganaksoy, A simplified formulation of likelihood ratio confidence intervals using a novel property, Technometrics 63 (1) (2021) 127–135.
27. N. L. Johnson, S. Kotz, N. Balakrishnan, Continuous multivariate distributions, Vol. 7, Wiley, New York, 1972.