The week's pick
Reseach Article
Empirical Investigation of Type 1 Error Rate of Univariate Tests of Normality
| International Journal of Computer Applications |
| Foundation of Computer Science (FCS), NY, USA |
| Volume 148 - Number 8 |
| Year of Publication: 2016 |
| Authors: Kayode Ayinde, John Olatunde Kuranga, Gbenga Sunday Solomon |
10.5120/ijca2016911253
|
Kayode Ayinde, John Olatunde Kuranga, Gbenga Sunday Solomon . Empirical Investigation of Type 1 Error Rate of Univariate Tests of Normality. International Journal of Computer Applications. 148, 8 ( Aug 2016), 24-31. DOI=10.5120/ijca2016911253
Abstract
Normality assumption is important in univariate parametric statistical tests. Either the varibles or the error terms in the model have to be normally distributed before valid statistical conclusions could be made. Various tests of univariate normality including that of Pearson, Kolmogorov–Smirnov, Anderson-Darling, Shapiro–Wilk, Lilliefor, D’Agostino and Pearson, Jarque-Bera, Shapiro-Franca, Energy and Cramer-von Mises tests have been developed. However, when applied in practice, they hardly give the same result. Thus, this research work aims at investigating the Type 1 error rate of these tests so as to identify the best one and suggest the same for statistics users. The tests were compared by conducting Monte Carlo experiments five thousand (5000) times with six sample sizes at three pre-selected levels of significance. A test was adjudged good at a particular level of significance if its empirical Type 1 error rate approximated the true error rates most often. It is best if its number of counts at which it was good over the sample sizes and levels of significance was the highest. Results reveal that Type 1 error rate of all the univariate tests are good except that of Kolmogorov–Smirnov, Pearson Unadjusted and Jarque-Bera. Moreover, those of Anderson-Darling, Shapiro-Wilk, Energy and Cramer-von Mises tests are relatively best. They are therefore recommended for testing the assumption of normality in any univariate data set.
References
- Pearson, K. 1900. 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, Philosophical Magazine Series, 550 (302), 157–175.
- Pearson, K. 1905. On general theory of skew correlation and Non-linear regression, London: Dulau and Co.
- Kolmogorov, A. N. 1933. Sulla determinazione empirica di una lagge di distribuzione, Giornale dell Instituto Italiano degli Attuari, 4, 83-91.
- Anderson, T. W. and Darling, D. A. 1954. A Test of Goodness of Fit, Journal of the Anerican statistical Association, 49 (268), 765-769.
- Shapiro, S. S. and Wilk, M. B. 1965. An Analysis of variance Test for Normalty, Biometrika, 52 (3), 591 – 611.
- Lilliefors, H. W. 1967. On the Kolmogorov – Smirnov Test for Normality with mean andvariance unknown, Journal of American statistical Assocition, 62 (318), 399- 402.
- D’Agostino, 1970. R. B. Transformation to normality of the null distribution of g1, Biometrika, 57, 679–681.
- Agostino, R. D and Pearson, E. S. 1973. Test for Departure from Normality. Empirical Results for the Distributions of b_2 and √(b_1 ), Biometrika, 60 (3), 613-622.
- Jarque, C. M. and Bera, A. K. 1987. A test for Normality of observations and regression residual, Internat. Statst., 55 (2), 163 – 172.
- Shapiro, S. S. and Francia, R. S. 1972. An Approximate analysis of varinace test for normality, Journal of American Statistical Association, 67, 215-216.
- Szekely, G. J. and Rizzo, M. L. 2005. A New Test for Multivariate Normality, Journal of Multivariate Analysis, 93 (1), 58-80.
- Thadewald, T. and Buning, H. 2007. Jarque – Bera and its Competitors for Testing Normality, Journal of Applied Statistics, 34 (1), 87 – 91.
- Anderson, T. W. 1962. On the Distribution of the Two-Sample Cramer–von Mises Criterion, The Annals of Mathematical Statistics (Institute of Mathematical Statistics) doi:10.1214/aoms/1177704477. ISSN 0003-4851, 33 (3), 1148–1159.
- Althous, I. A., Ware W. B. and Ferron, J. M. 1998. Detecting departures from Normality; A Monte carlo on simulation of A New omibus Test base Moments, in Annual Meeting of the American Education Research Association, San Diego, CA.
- D’Agostino, R. B. 1971. An omnibus test of normality for moderate and large size samples, Biometrika,52 ( 2), 341–348.
- Romao, X., Delgado, R. and Costa, A. 2010.An Empirical Power Comparison of Univariate Goodness-of-Fit Test for Normality, Journal of Statistical Computation and Simulation, 80 (5), 545-591.
- Kuranga, J. O. 2015. Type 1 Error Rate and Power Comparison of Some Normality Test Statistics, Ogbomoso: Unpublished M.Phil. Thesis, Department of Statistics, Ladoke Akintola University of Technology.
- Ayinde, K., Solomon, G. S. and Adejumo, A. O. 2016. A Comparative Study of the Type 1 Error Rates of Some Multivariate Tests of Normality, Nigerian Association of Mathematical Physics, 34 (1), 185-198.
Index Terms
Keywords