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Empirical Type 1 Error Rate and Power Comparisons of Normality Tests with R

Year 2018, Volume: 39 Issue: 3, 799 - 811, 30.09.2018

Ahmet Pekgör Murat Erişoğlu Aydın Karakoca Ülkü Erişoğlu

https://doi.org/10.17776/csj.426382

Abstract

Normality is one of the main presuppositions in
statistical tests. The multiplicity of the normality tests bring out another
problem of choosing the appropriate test for researchers. The free software R
which has a great popularity in the statistical analysis has 18 normality tests
in 4 different packages. In this study we compared performance of these
normality tests in terms of empirical type 1 error rate and power by Monte
Carlo simulation. As a result, regardless of the distribution of data (symetric
or asymmetric) the Shapiro-Francia test, also the Frosini B test performed
better than the other normality tests in terms of experimental type 1 error
rate. However the widely used Kolmogorov-Smirnov test showed worse performance
than other normality tests in terms of empirical type 1 error rate and power.

Keywords

Normality tests R Empirical type 1 error rate Power of test

References

  • [1] Pearson, K., X. on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 50-302 (1900) 157–175.
  • [2] Von Mises, R., Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik. Bull. Amer. Math. Soc 38 (1932) 169-170.
  • [3] Kolmogorov, A., Sulla determinazione empirica di una lgge di distribuzione, Inst. Ital. Attuari, Giorn. 4 (1933) 83–91.
  • [4] Geary, R. C., The ratio of the mean deviation to the standard deviation as a test of normality, Biometrika 27-3/4 (1935) 310–332.
  • [5] Anderson, T. W. and Darling, D. A., A test of goodness of fit, Journal of the American statistical association 49-268 (1954) 765–769.
  • [6] Kuiper, N. H., Tests concerning random points on a circle, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 63, (1960) 38–47.
  • [7] Shapiro, S. S. and Wilk, M. B., An analysis of variance test for normality (complete samples), Biometrika 52-3/4 (1965) 591–611.
  • [8] Lilliefors, H. W., On the kolmogorov-smirnov test for normality with mean and variance unknown, Journal of the American statistical Association 62-318 (1967) 399–402.
  • [9] Shapiro, S. S. and Francia, R., An approximate analysis of variance test for normality, Journal of the American Statistical Association 67-337 (1972) 215–216.
  • [10] Hegazy, Y. and Green, J., Some new goodness-of-fit tests using order statistics, Applied Statistics pp. (1975) 299–308.
  • [11] Weisberg, S. and Bingham, C., An approximate analysis of variance test for non-normality suitable for machine calculation, Technometrics 17-1 (1975) 133–134.
  • [12] Spiegelhalter, D., A test for normality against symmetric alternatives, Biometrika 64-2 (1977) 415–418.
  • [13] Frosini, B. V., A survey of a class of goodness-of-fit statistics, Universit`a degli studi. Facolt`a di scienze statistiche, demografiche ed attuariali (1978).
  • [14] Jarque, C. M. and Bera, A. K., A test for normality of observations and regression residuals, International Statistical Review/Revue Internationale de Statistique (1987) 163–172.
  • [15] Urzua, C. M., On the correct use of omnibus tests for normality, Economics Letters 3-54 (1997) 301.
  • [16] Esteban, M., Castellanos, M., Morales, D. and Vajda, I., Monte carlo comparison of four normality tests using different entropy estimates, Communications in Statistics-Simulation and computation 30-4 (2001) 761–785.
  • [17] Yazici, B. and Yolacan, S., A comparison of various tests of normality, Journal of Statistical Computation and Simulation 77-2 (2007) 175–183.
  • [18] Yap, B. W. and Sim, C. H.,Comparisons of various types of normality tests, Journal of Statistical Computation and Simulation 81-12 (2011) 2141–2155.
  • [19] Noughabi, H. A. and Arghami, N. R. ,Monte carlo comparison of seven normality tests, Journal of Statistical Computation and Simulation 81-8 (2011) 965–972.
  • [20] Royston, J., An extension of shapiro and wilk’s w test for normality to large samples, Applied statistics pp. (1982) 115–124.
  • [21] Razali, N. M., & Wah, Y. B.. Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. Journal of statistical modeling and analytics, 2-1(2011) 21-33.

R ile Normallik Testlerinin Deneysel 1. Tip Hata Oranı ve Güç Karşılaştırması

Year 2018, Volume: 39 Issue: 3, 799 - 811, 30.09.2018

Ahmet Pekgör Murat Erişoğlu Aydın Karakoca Ülkü Erişoğlu

https://doi.org/10.17776/csj.426382

Abstract

Normallik istatistiksel testlerde ana
varsayımlardan biridir. Normallik testlerinin çokluğu, araştırmacılar için
uygun normallik testini seçme konusunda başka bir problem ortaya çıkarmaktadır.
İstatistiksel analizlerde sıklıkla kullanılan ücretsiz bir yazılım olan R’da 4
farklı pakette 18 normallik testi bulunmaktadır. Bu çalışmada, normallik
testlerinin performansını ampirik 1. tip hata oranı ve güç açısından Monte
Carlo simülasyonu ile karşılaştırılmıştır. Çalışma sonucunda verinin dağılımı (simetrik veya asimetrik) ne olursa olsun
Shapiro-Francia testi, ayrıca Frosini B testi de deneysel tip 1 hata oranı
açısından diğer normallik testlerinden daha iyi performans göstermiştir.Ayrıca
yaygın olarak kullanılan Kolmogorov-Smirnov testi ampirik tip 1 hata oranı ve
testin gücü açısından diğer normallik testlerinden daha kötü performans
göstermiştir.

Keywords

Normallik testi R Deneysel 1. tip hata oranı Testin gücü

References

  • [1] Pearson, K., X. on the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 50-302 (1900) 157–175.
  • [2] Von Mises, R., Wahrscheinlichkeitsrechnung und ihre Anwendung in der Statistik und theoretischen Physik. Bull. Amer. Math. Soc 38 (1932) 169-170.
  • [3] Kolmogorov, A., Sulla determinazione empirica di una lgge di distribuzione, Inst. Ital. Attuari, Giorn. 4 (1933) 83–91.
  • [4] Geary, R. C., The ratio of the mean deviation to the standard deviation as a test of normality, Biometrika 27-3/4 (1935) 310–332.
  • [5] Anderson, T. W. and Darling, D. A., A test of goodness of fit, Journal of the American statistical association 49-268 (1954) 765–769.
  • [6] Kuiper, N. H., Tests concerning random points on a circle, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 63, (1960) 38–47.
  • [7] Shapiro, S. S. and Wilk, M. B., An analysis of variance test for normality (complete samples), Biometrika 52-3/4 (1965) 591–611.
  • [8] Lilliefors, H. W., On the kolmogorov-smirnov test for normality with mean and variance unknown, Journal of the American statistical Association 62-318 (1967) 399–402.
  • [9] Shapiro, S. S. and Francia, R., An approximate analysis of variance test for normality, Journal of the American Statistical Association 67-337 (1972) 215–216.
  • [10] Hegazy, Y. and Green, J., Some new goodness-of-fit tests using order statistics, Applied Statistics pp. (1975) 299–308.
  • [11] Weisberg, S. and Bingham, C., An approximate analysis of variance test for non-normality suitable for machine calculation, Technometrics 17-1 (1975) 133–134.
  • [12] Spiegelhalter, D., A test for normality against symmetric alternatives, Biometrika 64-2 (1977) 415–418.
  • [13] Frosini, B. V., A survey of a class of goodness-of-fit statistics, Universit`a degli studi. Facolt`a di scienze statistiche, demografiche ed attuariali (1978).
  • [14] Jarque, C. M. and Bera, A. K., A test for normality of observations and regression residuals, International Statistical Review/Revue Internationale de Statistique (1987) 163–172.
  • [15] Urzua, C. M., On the correct use of omnibus tests for normality, Economics Letters 3-54 (1997) 301.
  • [16] Esteban, M., Castellanos, M., Morales, D. and Vajda, I., Monte carlo comparison of four normality tests using different entropy estimates, Communications in Statistics-Simulation and computation 30-4 (2001) 761–785.
  • [17] Yazici, B. and Yolacan, S., A comparison of various tests of normality, Journal of Statistical Computation and Simulation 77-2 (2007) 175–183.
  • [18] Yap, B. W. and Sim, C. H.,Comparisons of various types of normality tests, Journal of Statistical Computation and Simulation 81-12 (2011) 2141–2155.
  • [19] Noughabi, H. A. and Arghami, N. R. ,Monte carlo comparison of seven normality tests, Journal of Statistical Computation and Simulation 81-8 (2011) 965–972.
  • [20] Royston, J., An extension of shapiro and wilk’s w test for normality to large samples, Applied statistics pp. (1982) 115–124.
  • [21] Razali, N. M., & Wah, Y. B.. Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. Journal of statistical modeling and analytics, 2-1(2011) 21-33.
There are 21 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

Ahmet Pekgör

Murat Erişoğlu

Aydın Karakoca 0000-0001-6503-3872

Ülkü Erişoğlu

Publication Date September 30, 2018
Submission Date May 23, 2018
Acceptance Date September 3, 2018
Published in Issue Year 2018Volume: 39 Issue: 3

Cite

APA Pekgör, A., Erişoğlu, M., Karakoca, A., Erişoğlu, Ü. (2018). Empirical Type 1 Error Rate and Power Comparisons of Normality Tests with R. Cumhuriyet Science Journal, 39(3), 799-811. https://doi.org/10.17776/csj.426382