INVESTIGATION OF SOME UNIVARIATE NORMALITY TESTS IN TERMS OF TYPE-I ERRORS AND TEST POWER

In this study, Shapiro-Wilk, Kolmogorov-Smirnov, Skewness, Kurtosis, Lilliefors, Jargue-Bera and D'Agostino -Pearson tests, which are univariate normality tests, were compared in point of type-I error and power performances. For comparisons, samples were created in various distributions and sample volumes by simulation technique, and the probability of type-I error was taken as 0.05 in comparisons. Thus, it is aimed to determine the best test to check whether the normality condition is met in univariate data. As a result of the comparison, it was determined that the Jargue-Bera test gave better results than the other tests in point of type-I error probability. In addition, when the normality tests examined in all distributions were taken into account and compared, it was concluded that the Shapiro-Wilk gives better results than other tests in general for normal and non-normal distributions, and that D'Agostino -Pearson, Skewness and Jargue-Bera tests were also stronger than the other tests. In addition, it was determined that the increase in sample sizes increased power of the test. In conclusion, it can be said that in addition to the distribution pattern, type-I error probability and sample size are also very important factors for test power.

Keywords:

Univariate normality tests Type-I error, Power of test,

PDF

___

[1] Noughabi, H. A., and Arghami, N. R. (2009). Monte carlo comparison of seven normality tests. Journal of Statistical Computation and Simulation, 81, 965-972.
[2] Adefisoye, J., Golam Kibria, B., and George, F. (2016). Performances of several univariate tests of normality: An empirical study. Journal of Biometrics & Biostatistics, 7, 1-8.
[3] Özer, A. (2007). Comparison of normality tests. Unpublished Master's Thesis, University of Ankara, Ankara.
[4] Dufour, J. M., Farhat, A., Gardiol, L., and Khalaf, L. (1998). Simulation-based finite sample normality tests in linear regressions. Econometrics Journal, 1, 154-173.
[5] Banerjee, A., Chitnis, U. B., Jadhav, S. L., Bhawalkar, J. S., and Chaudhury, S. (2009). Hypothesis testing, type I and type II errors. Industrial Psychiatry Journal, 18, 127-131.
[6] Eygü, H. (2020). Çözümlü güncel örneklerle olasılık ve istatistik. Ankara: Nobel Yayınları. ss.251-252.
[7] Elsayir, H. A. (2018). Factors determining the power of a statistical test for the difference between means and proportions. American Journal of Mathematics and Statistics, 8, 171-178.
[8] Ugar, H. (1999). Functions of random variables and applications to probability distributions. Unpublished Master's Thesis, University of Istanbul, Istanbul.
[9] Gordon, S. (2006). The normal distribution. Produced by UPS, Sydney.
[10] Lancaster, H. O., and Seneta, E. (2005). Chi-square distribution, p. armitage and t. Colton in Encyclopedia of Biostatistics, Wiley, Chichester.
[11] Karagöz, Y. (2003). Demonstration of relationship between exponential and chi-square distributions with random numbers produced by simulation. C.Ü. İktisadi ve İdari Bilimler Dergisi, 4(1), 197-209 .
[12] Dündar, D. (1987). Hipotez testi ve ki-kare ile bir uygulama. İstanbul Üniversitesi İktisat Fakültesi Mecmuası, 45, 196-212.
[13] Wackerly, D. D., Mendenhall, W., and Scheaffer, R. L. (2008). Mathematical statistics with applications. Cengage Learning, USA.
[14] Koepf, W., and Masjed-Jamei, M. (2006). A generalization of student's t-distribution from the viewpoint of special functions. Integral Transforms and Special Functions, 17, 863-875.
[15] Ramachandran, K. M., and Tsokos, C. P. (2009). Mathematical statistics with applications. Academic Press, USA.
[16] Saraçoğlu, Ö. (2002). Düşük akım hidrolojisi ve akdeniz bölgesi’nde uygulanması. Unpublished Master's Thesis, University of Istanbul, Istanbul.
[17] Cain, M. K., Zhang, Z., and Yuan, K. (2016). Univariate and multivariate skewness and kurtosis for measuring nonnormality: Prevalence, influence and estimation. Behavior Research Methods, 49, 1716-1735.
[18] Nosakhare, U. H., and Bright, A. F. (2017). Statistical analysis of strength of w/s test of normality against non-normal distribution using monte carlo simulation. American Journal of Theoretical and Applied Statistics, 6, 62-65.
[19] Shapiro, S. S., and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples). Biometrica, 52, 591-611.
[20] Sheskin, D. J. (2000). Handbook of parametric and nonparametric statistical procedures. Chapman & Hall/CRC, USA.
[21] D'Agostino, R. B., and Pearson, E.,S. (1973). Tests for departure from normality. empirical results for the distributions of b 2 and √b 1. Biometria, 60, 613-622.
[22] D'Agostino, R. B. (1986). In tests of the normal distribution, rb d'agostino and ma stephens, goodness of fit techniques. Marcel Dekker, Inc, New York.
[23] Zar, J.H. (1999). Biostatistical analysis. Prentice Hall, New Jersey.
[24] Dong, L. B., and Giles, D. E. A. (2004). An empirical likelihood ratio test for normality. Communications in Statistics-Simulation and Computation, 36, 197-215.
[25] Razali, N. M., and Yap, B. W. (2010). Power comparisons of some selected normality tests. Proceedings of the Regional Conference on Statistical Sciences, July 2010, Selangor, Malaysia. 126-138.
[26] Abdi, H., and Molin, P. (2007). Lilliefors test of normality, N.J. Salkind, Encyclopedia of measurement and statistics. SAGE Publications, Inc, California.
[27] Soest, J. V. (1967). Some experimental results concerning tests of normality. Statistica Neerlandica, 21, 91-97.
[28] Lilliefors, H. W. (1967). On the kolmogorov-smirnov test for normality with mean and variance unknown. Journal of the American Statistical Association, 62, 399-402.
[29] Dagnelie, P. (1968). A propos de l'emploi du test de Kolmogorov-Smirnov comme test de normalité. Biométrie-Praximétrie, 9, 3-13.
[30] Genceli, M. (2007). Kolmogorov-smirnov, lilliefors and shaphiro-wilk tests for normality. Sigma Journal of Engineering and Natural Sciences, 25, 306-328.
[31] Akçadağ, H. İ. (2013). Tek değişkenli ve çok değişkenli bazı normallik testlerinin karşılaştırılması, Yayımlanmamış Doktora Tezi, Selçuk Üniversitesi Fen Bilimleri Enstitüsü, Konya.
[32] Keskin, S. (2002). Varyansların homojenliğini test etmede kullanılan bazı yöntemlerin ı. tip hata ve testin gücü bakımından irdelenmesi, Yayımlanmamış Doktora Tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü, Ankara.
[33] Öztuna, D., Elhan, A. H., and Tuccar, E. (2006). Investigation of four different normality tests in terms of type 1 error rate and power under different distributions. Tübitak Academic Journals, 36, 171-176.
[34] Thadewald, T., and Büning, H. (2007). Jarque-bera test and its competitors for testing normality-a power comparison. Journal of Applied Statistics, 34, 87-105.
[35] Mendeş, M., and Pala A. (2003). Type I error rate and power of three normality tests. Pakistan Journal of Information and Technology, 2, 135-139.
[36] Shapiro, S. S., Wilk, M. B., and Chen, H. J. (1968). A comparative study of various tests for normality. Journal of the American Statistical Association, 63, 1343-1372.
[37] Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons. Journal of the America Statistical Association, 69, 730-737.
[38] Seier, E. (2002). Comparison of tests for univariate normality. site file:///C:/Users/ASUS/ Downloads/Comparison_of_tests_of_univariate_normality.pdf. Access Date: 20.12.2021.
[39] Razali, N. M., and Yap, B. W. (2011). Power comparisons of shapiro-wilk, kolmogorov-smirnov, lilliefors and anderson-darling tests. Journal of Statistical Modeling and Analytics, 2, 21-33.