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Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation

Received: 24 February 2017     Accepted: 1 March 2017     Published: 9 June 2017
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Abstract

This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.

Published in American Journal of Theoretical and Applied Statistics (Volume 6, Issue 5-1)

This article belongs to the Special Issue Statistical Distributions and Modeling in Applied Mathematics

DOI 10.11648/j.ajtas.s.2017060501.18
Page(s) 51-61
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Error Rate, Power-of-Test, Normality, Sensitivity, and Simulation

References
[1] Abbas M. (2013): Robust Goodness of Fit Test Based on the Forward Search. American Journal of App. Mathematics and Statistics, 2013, Vol. 1, No. 1, 6-10.
[2] Anderson, T. W. (1962): On the Distribution of the Two-Sample Cramer–von Mises Criterion. The Annals of Mathematical Statistics (Institute of Mathematical Statistics) 33 (3): 1148–1159. doi:10.1214/aoms/1177704477. ISSN 0003-4851.
[3] Baghban A. A., Younespour S., Jambarsang S., Yousefi M., Zayeri F., and Jalilian F. A. (2013): How to test normality distribution for a variable: a real example and a simulation study. Journal of Paramedical Sciences (JPS). Vol.4, No.1 ISSN 2008-4978.
[4] D’Agostino R. B. and Stephens M. A. (1986): Goodness-of-fit techniques. New York, Marcel Dekker.
[5] Derya O., Atilla H. E., and Ersoz T. (2006): Investigation of Four Different Normality Tests in Terms of Type I Error Rate and Power Under Different Distributions. Turk. J. Med. Sci. 2006; 36 (3): 171-176.
[6] Douglas G. B. and Edith S. (2002): A test of normality with high uniform power. Journal of Computational Statistics and Data Analysis 40 (2002) 435–445. www.elsevier.com/locate/csda.
[7] Everitt, B. S. (2006): The Cambridge Dictionary of Statistics. Third Edition. Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo. Pp. 240-241.
[8] Farrel P. J and Stewart K. R. (2006): Comprehensive Study of Tests for Normality and Symmetry; Extending the Spiegelhalter Test. Journal of Statistical Computation and Simulation. Vol. 76, No. 9, Pp 803-816.
[9] Frain J. C. (2006): Small Sample Power of Tests of Normality when the Alternative is an α-stable distribution.http://www.tcd.ie/Economics/staff/frainj/Stable_Distribution/normal.pdf
[10] Guner B. and Johnson J. T. (2007): Comparison of the Shapiro-Wilk and Kurtosis Tests for the Detection of Pulsed Sinusoidal Radio Frequency Interference. The Ohio State University, Department of Electrical and Computer Engineering and Electro-Science Laboratory, 1320 Kinnear Road, Columbus, OH 43210, USA.
[11] Gupta S. C. (2011): Fundamentals of Statistics. Sixth Revised and Enlarged Edition. Himilaya Publishing House PVT Ltd. Mumbai-400 004. Pp 16. 28-16.31
[12] Jarque, C. M. and Bera, A. K. (1980): Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters 6 (3): 255–259. doi:10.1016/0165-1765(80)90024-5.
[13] Jarque, C. M. and Bera, A. K. (1981): Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence. Economics Letters 7 (4): 313–318. doi:10.1016/0165-1765(81)900235-5.
[14] Jarque, C. M. and Bera, A. K. (1987): A test for normality of observations and regression residuals. International Statistical Review 55 (2): 163–172. JSTOR 1403192.
[15] Jason O. (2002): Notes on the use of data transformations. Practical Assessment, Research & Evaluation, 8(6). http://PAREonline.net/getvn.asp?v=8&n=6.
[16] Jason O. (2010): Improving your data transformations: Applying the Box-Cox transformation. Practical Assessment, Research & Evaluation, 15(12). Available online: http://pareonline.net/getvn.asp?v=15&n=12.
[17] Marmolejo-Ramos F. and Gonza´lez-Burgos J. (2012): A Power Comparison of Various Tests of Univariate Normality on Ex-Gaussian Distributions. European Journal of Research Methods for the Behavioural and Social Sciences. ISSN-Print 1614-1881• ISSN-Online 1614-2241. DOI: 10.1027/1614-2241/a000059. www.hogrefe.com/journals/methodology
[18] Mayette S. and Emily A. B. (2013): Empirical Power Comparison of Goodness of Fit Tests for Normality in the Presence of Outliers. Journal of Physics. Conference Series 435 (2013) 012041.doi:10.1088/1742-6596/435/1/012041.
[19] Molin P. and Abdi H. (2007): Lilliefors/Van Soest’s test of normality. Encyclopedia of measurement and Statistics. Pp. 1-10. www.utd.edu/~herve
[20] Narges A. (2013): Shapiro-Wilk Test in Evaluation of Asymptotic Distribution on Estimators of Measure of Kurtosis and Skew. International Mathematical Forum, Vol. 8, 2013, no. 12, 573–576. HIKARI Ltd, www.m-hikari.com
[21] Nor A. A, Teh S. Y., Abdul-Rahman O. and Che-Rohani Y. (2011): Sensitivity of Normality Tests to Non-Normal Data. Sains Malaysiana 40 (6) (2011): 637–641.
[22] Nor-Aishah H. and Shamsul R. A (2007): Robust Jacque-Bera Test of Normality. Proceedings of The 9th Islamic Countries Conference on Statistical Sciences 2007. ICCS-IX 12-14 Dec 2007
[23] Nornadiah M. R. and Yap B. W. (2011): Power Comparison of Shapiro-Wilk, Kolmogorov-Smirnov, Lillieforsand Anderson-Darling Tests. Journal of Statistical Modeling and Analytics. Vol. 2. No. 1, 21-33.2011. ISSN 978-967-363-157-5.
[24] Panagiotis M. (2010): Three Different Measures of Sample Skewness and Kurtosis and their Effects on the Jarque-Bera Test for Normality. Jonkoping International Business School Jönköping University JIBS, Sweden. Working Papers No. 2010-9.
[25] Piegorsch W. W and Bailer A. J. (2005): Analyzing Environmental Data. Page: 432. John Wiley & Sons, Ltd. The Atrian, South Gate, Chichester, West Sussex P019 85Q, England. ISBN: 0-470-84836-7 (HB).
[26] Pitchaya S., Dee A. B., and Kevin S. (2012): Exploring the Impact of Normality and Significance Tests in Architecture Experiments. Department of Computer Science, University of Virginia.
[27] Richard M. F. and Ciriaco V. F. (2010): Applied Probability and Stochastic Processes. Springer Heidelberg Dordrecht, London New York. Pp 95-96. ISBN: 978-3-642-05155-5. DOI: 10.1007/978-3-642-05158-6
[28] Rinnakorn C., and Kamon B. (2007): A Power Comparison of Goodness-of-fit Tests for Normality Based on the Likelihood Ratio and the Non-likelihood Ratio. Thailand Statistician, July 2007; 5: 57-68. http://statassoc.or.th.
[29] Royston J. P (1983): A Simple Method for Evaluating the Shapiro-Francia W' Test of Non-Normality. Journal of the Royal Statistical Society. Series D (The Statistician), Vol. 32, No. 3 (Sep.,1983), pp. 297-300. http://www.jstor.org/stable/2987935. Accessed: 26/07/2014.
[30] Ryan, T. A. and Joiner B. L. (1976): Normal Probability Plots and Tests for Normality, Technical Report, Statistics Department, The Pennsylvania State University.
[31] Ryan T. A. and Joiner B. L. (1990): Normal probability plots and tests for normality. Minitab Statistical Software: Technical Reports, November, 1-14.
[32] Schaffer M. (2010): Procedure for Monte Carlo Simulation. SGPE QM Lab 3. Monte Carlos Mark version of 4.10.2010.
[33] Seth R. (2008): Transform your data; Statistics column. Nutrition 24, Pp 492–494. Elsevier. doi:10.1016/j.nut.2008.01.00
[34] Shigekazu N., Hiroki H. and Naoto N. (2012): A Measure of Skewness for Testing Departures from Normality. Journal of Computational Statistics and Data Analysis. XIV: 1202.5093V1.
[35] Siddik K. (2006): Comparison of Several Univariate Normality Tests Regarding Type I Error Rate and Power of the Test in Simulation based Small Samples Journal of Applied Science Research 2 (5): 296-300, 2006. © 2006, INSInet Publication.
[36] Stephen M. A (1974): EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association. Theories and Methods Section, Vol. 69, No 347, Pp 730-737. http://www.jstor.org.
[37] Urzúa, C. M. (1996): On the correct use of omnibus tests for normality, Economics Letters 53: 247-251. http://www.sciencedirect.com/science/article/pii/S0165176596009238
[38] Weurtz, D. and Katzgraber H. G (2005). Precise finite-sample quantiles of the Jarque-Bera adjusted Lagrange multiplier test. Swiss Federal Institute of Technology, Institute for Theoretical Physics, ETH H¨onggerberg, C-8093 Zurich.
[39] Yap B. W. and Sim C. H. (2011): Comparisons of various types of normality tests. Journal of Statistical Computation and Simulation 81:12, 2141-2155, DOI:10.1080/00949655.2010.520163.
[40] Zvi D., Ofir T. and Dawit Z. (2008): A modified Kolmogorov-Smirnov test for Normality. MPRA Paper No. 14385. http://mpra.ub.uni-muenchen.de/14385/.
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    Ukponmwan H. Nosakhare, Ajibade F. Bright. (2017). Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. American Journal of Theoretical and Applied Statistics, 6(5-1), 51-61. https://doi.org/10.11648/j.ajtas.s.2017060501.18

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    Ukponmwan H. Nosakhare; Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am. J. Theor. Appl. Stat. 2017, 6(5-1), 51-61. doi: 10.11648/j.ajtas.s.2017060501.18

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    Ukponmwan H. Nosakhare, Ajibade F. Bright. Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation. Am J Theor Appl Stat. 2017;6(5-1):51-61. doi: 10.11648/j.ajtas.s.2017060501.18

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  • @article{10.11648/j.ajtas.s.2017060501.18,
      author = {Ukponmwan H. Nosakhare and Ajibade F. Bright},
      title = {Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {6},
      number = {5-1},
      pages = {51-61},
      doi = {10.11648/j.ajtas.s.2017060501.18},
      url = {https://doi.org/10.11648/j.ajtas.s.2017060501.18},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.s.2017060501.18},
      abstract = {This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.},
     year = {2017}
    }
    

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    T1  - Evaluation of Techniques for Univariate Normality Test Using Monte Carlo Simulation
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    AU  - Ajibade F. Bright
    Y1  - 2017/06/09
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    JF  - American Journal of Theoretical and Applied Statistics
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    AB  - This paper examines the sensitivity of nine normality test statistics; W/S, Jaque-Bera, Adjusted Jaque-Bera, D’Agostino, Shapiro-Wilk, Shapiro-Francia, Ryan-Joiner, Lilliefors’and Anderson Darlings test statistics, with a view to determining the effectiveness of the techniques to accurately determine whether a set of data is from normal distribution or not. Simulated data of sizes 5, 10, …, 100 is used for the study and each test is repeated 100 times for increased reliability. Data from normal distributions (N (2, 1) and N (0, 1)) and non-normal distributions (asymmetric and symmetric distributions: Weibull, Chi-Square, Cauchy and t-distributions) are simulated and tested for normality using the nine normality test statistics. To ensure uniformity of results, one statistical software is used in all the data computations to eliminate variations due to statistical software. The error rate of each of the test statistic is computed; the error rate for the normal distribution is the type I error and that for non-normal distribution is type II error. Power of test is computed for the non-normal distributions and use to determine the strength of the methods. The ranking of the nine normality test statistics in order of superiority for small sample sizes is; Adjusted Jarque-Bera, Lilliefor’s, D’Agostino, Ryan-Joiner, Shapiro-Francia, Shapiro-Wilk, W/S, Jarque-Bera and Anderson-Darling test statistics while for large sample sizes, we have; D’Agostino, Ryan-Joiner, Shapiro-Francia, Jarque-Bera, Anderson-Darling, Lilliefor’s, Adjusted Jarque-Bera, Shapiro-Wilk and W/S test statistics. Hence, only D’Agostino test statistic is classified as Uniformly Most Powerful since it is effective for both small and large sample sizes. Other methods are Locally Most Powerful. Shapiro-Francia, an improvement of Shapiro-Wilk is more sensitive for both small and large samples, hence should replace Shapiro-Wilk while the Adjsted Jarque-Bera and the Jarque-Bera should both be retained for small and large samples respectively.
    VL  - 6
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    ER  - 

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Author Information
  • Department of General Studies, Mathematics and Computer Science Unit, Petroleum Training Institute, Warri, Nigeria

  • Department of General Studies, Mathematics and Computer Science Unit, Petroleum Training Institute, Warri, Nigeria

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