A Multivariate Extension of McNemar’s Test Based on Permutations

Grzegorz Kończak

doi:10.18778/0208-6018.349.06

Authors

Grzegorz Kończak University of Economics in Katowice, College of Management, Department of Statistics Econometrics and Mathematics http://orcid.org/0000-0002-4696-8215

DOI:

https://doi.org/10.18778/0208-6018.349.06

Keywords:

permutation test, McNemar’s test, multivariate test

Abstract

The purpose of this publication is to propose a permutation test to detect the departure from symmetry in multidimensional contingency tables. The proposal is a multivariate extension of McNemar’s test. McNemar’s test could be applied to 2 × 2 contingency tables. The proposal may be also treated as a modification of Cochran’s Q test which is used for testing dependency for multivariate binary data. The form of the test statistics that allows us to detect the departure from counts symmetry in multidimensional contingency tables is presented in the article. The permutation method of observations was used to estimate the empirical distribution of the test statistics. The considerations were supplemented with examples of the use of a multivariate test for simulated and real data. The application of the proposed test allows us to detect the asymmetrical distribution of counts in multivariate contingency tables.

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References

Agresti A., Klingenberg B. (2005), Multivariate tests comparing binomial probabilities, with application to safety studies for drugs, “Applied of Statistics”, vol. 54, pp. 691–816. DOI: https://doi.org/10.1111/j.1467-9876.2005.05437.x

Bowker A. H. (1948), A test for symmetry in contingency tables, “Journal of American Statistical Association”, vol. 43, pp. 572–574. DOI: https://doi.org/10.1080/01621459.1948.10483284

Diagnoza społeczna: zintegrowana baza danych, www.diagnoza.com [accessed: 23.11.2019].

Donald S., Shahren A. Z.A. (2018), Cochran’s Q with Pairwise McNemar for Dichotomous Multiple Responses Data: a Practical Approach, “International Journal of Engineering & Technology”, vol. 7, no. 3(18), pp. 4–6, https://www.sciencepubco.com/index.php/ijet/article/view/16662 [accessed: 23.11.2019]. DOI: https://doi.org/10.14419/ijet.v7i3.18.16662

Fay M. (2011), Exact McNemar’s Test and Matching Confidence Intervals, https://www.researchgate.net/publication/267448346_Exact_McNemar’s_Test_and_Matching_Confidence_Intervals [accessed: 23.11.2019].

Feuer E. J., Kessler L. G. (1989), Test Statistic and Sample Size for a Two‑Sample McNemar Test, “Biometrics”, vol. 45, no. 2, pp. 629–636. DOI: https://doi.org/10.2307/2531505

Klingenberg B., Agresti A. (2006), Multivariate Extensions of McNemar’s Test, “Biometrics”, vol. 62, pp. 921–928. DOI: https://doi.org/10.1111/j.1541-0420.2006.00525.x

Kończak G. (2016), Testy permutacyjne. Teoria i zastosowania, Uniwersytet Ekonomiczny w Katowicach, Katowice.

Maxwell A. E. (1970), Comparing the classification of subjects by two independent judges, “British Journal of Psychiatry”, vol. 116(535), pp. 651–655, http://doi.org/10.1192/bjp.116.535.651 DOI: https://doi.org/10.1192/bjp.116.535.651

McNemar Q. (1947), Note on the sampling error of the difference between two correlated proportions in percentages, “Psychometrica”, vol. 12, pp. 153–157. DOI: https://doi.org/10.1007/BF02295996

Oden A., Wedel H. (1975), Arguments for Fisher’s Permutation Test, “The Annals of Statistics”, vol. 3, no. 2, pp. 518–520. DOI: https://doi.org/10.1214/aos/1176343082

Oyeka I. C.A. (2012), Modified McNemar Test, “International Journal of Statistics in Medical Research”, vol. 1, pp. 73–78. DOI: https://doi.org/10.6000/1929-6029.2012.01.01.07

Pesarin F. (2001), Multivariate Permutation Tests with Applications in Biostatistics, John Wiley & Sons, Chichester.

Sheskin D. J. (2011), Handbook of Parametric and Nonparametric Statistical Procedures, Chapman & Hall/CRC, Boca Raton.

Stuart A. A. (1955), A test for homogeneity of the marginal distributions in a two‑way classification, “Biometrika”, vol. 42(3–4), pp. 412–416, http://doi.org/10.1093/biomet/42.3–4.412 DOI: https://doi.org/10.1093/biomet/42.3-4.412

Westfall P. H., Troendle J. F., Pennello G. (2010), Multiple McNemar Tests, “Biometrics”, vol. 66, no. 4, pp. 1185–1191. DOI: https://doi.org/10.1111/j.1541-0420.2010.01408.x