Statisticians are constantly looking for methods of statistical inference that would be both effective and would require meeting as few assumptions as possible. Permutation tests seem to fit here, as using them makes it possible to perform statistical inference in situations where classical parametric tests do not work. Permutation tests appear to be comparably powerful to parametric tests, but require meeting fewer assumptions, e.g. regarding the size of the sample or the from of distribution of the tested variable in a population. The presented tests make it possible to verify the overall hypothesis about the identity of both location and scale parameters in the studied populations. In literature, the Lepage test and the Cucconi test are most often referred to in this context. The paper considers various forms of test statistics, and presents a simulation study carried out to determine the size and power of the tests under normality. As the study demonstrated, the advantage of the proposed method is that it can be applied to small-size samples. A nonparametric, complex procedure was used to assess the overall ASL (achieved significance level) value by applying the permutation principle. For comparative purposes, the results for the permutation Lepage test and the permutation Cucconi test are also presented.

permutation tests, comparing populations, test power, the Lepage test, the Cucconi test

ANDERSON, M. J., WALSH, D. C. I., CLARKE, K.R., GORLEY, R. N., GUERRA–CASTRO, E., (2017). Permutational Multivariate Analysis of Variance (PERMANOVA), Wiley StatsRef: Statistics Reference Online, pp. 1–15.

ANSARI, A. R., BRADLEY, R. A., (1960). Rank–sum tests for dispersions. Annals of Mathematical Statistics 31, pp. 1174–1189.

BALAKRISHNAN, N., MA, C. W., (1990). A comparative study of various tests for the equality of two population variances, Journal of Statistical Computation and Simulation, 35, pp. 41–89.

BONNINI, S., CORAIN, L., MAROZZI, M., SALMASO, L., (2014). Nonparametric Hypothesis Testing Rank and Permutation Methods with Applications in R, John Wiley & Sons, Ltd.

CHANG, C.–H., PAL, N., (2008). A Revisit to the Behrens–Fisher Problem: Comparison of Five Test Methods. Communications in Statistics – Simulation and Computation, 37, (6), pp. 1064–1085.

CONOVER, W. J., JOHNSON, M. E., JOHNSON, M. M., (1981). A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data, Technometrics, 23, pp. 351–361.

CUCCONI, O., (1968). Un nuovo test non parametrico per it confronto tra due gruppi campionori. Giornale degli Economisti, XXVII, pp. 225–248.

DURAN, B. S., TSAI, W. S., LEWIS, T. O., (1976). A class of location-scale tests, Biometrika, 63, pp. 173–176.

FISHER, R. A., (1932). Statistical Methods for Research Workers, 4 ed., Edinburgh: Oliver & Boyd.

GENG, S., WANG, W. J., MILLER, C., (1979). Small sample size comparisons of tests for homogeneity of variances by Monte-Carlo. Communications in Statistics – Simulation and Computation, 8, pp. 379–389.

GOGOI, P., GOGOI, B., (2017). Some Tests Procedures for Scale Differences. International Advanced Research Journal in Science, Engineering and Technology, Vol. 4, Issue 11, pp. 155–166.

HALL, I. J., (1972). Some comparisons of tests for equality of variances, Journal of Statistical Computation and Simulation, 1, pp. 183–194.

JANSSEN, A., PAULS, T., (2005). A Monte Carlo comparison of studentized bootstrap and permutation tests for heteroscedastic two–sample problems. Computational Statistics, 20 (3), pp. 369–383.

KESELMAN, H. J., GAMES, P. A., CLINCH, J. J., (1979). Tests for homogeneity of variance. Communications in Statistics – Simulation and Computation, 8, pp. 113–119.

KOŃCZAK, G., (2016). Testy permutacyjne, Teoria i zastosowania, Katowice: Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach.

LEPAGE, Y., (1971). A combination of Wilcoxon’s and Ansari–Bradley’s statistics, Biometrika 58, pp. 213–217.

LEPAGE, Y., (1973). A table for a combined Wilcoxon Ansari–Bradley statistic, Biometrika 60, pp. 113–116.

LIM, T.S., LOH, W. Y., (1996). A comparison of tests of equality of variances, Computational Statistics and Data Analysis, 22, pp. 287–301.

LIPTAK, I., (1958). On the combination of independent tests. Magyar Tudomanyos Akademia Matematikai Kutato Intezenek Kozlomenyei 3, pp. 127–141.

LUNDE, A., TIMMERMANN, A., (2004). Duration dependence in stock prices: an analysis of bull and bear markets, Journal of Business and Economic Statistics, 22, pp. 253–273.

MANN, H., WHITNEY, D., (1947). On a test of whether one of two random variables is stochastically larger than the other, Annals of Mathematical Statistics, 18, (1), pp. 50–60.

MAROZZI, M., (2004). A bi-aspect nonparametric test for the two-sample location problem, Computational Statistics and Data Analysis, 44, pp. 639–648.

MAROZZI, M., (2007). Multivariate tri–aspect non-parametric testing, Journal of Nonparametric Statistics, 19, pp. 269–282.

MAROZZI, M., (2008). The Lepage location–scale test revisited, Far East Journal of Theoretical Statistics 24, pp. 137–155.

MAROZZI, M., (2009). Some notes on the location–scale Cucconi test, Journal of Nonparametric Statistics, 21, 5, pp. 629–647.

MAROZZI, M., (2011). Levene type tests for the ratio of two scales, Journal of Statistical Computation and Simulation, 81, pp. 815–826.

MAROZZI, M., (2012a). A distribution free test for the equality of scales. Communication in Statistics – Simulation and Computation, 41, pp. 878–889.

MAROZZI, M., (2012b). A combined test for differences in scale based on the interquantile range. Statistical Papers, 53, pp. 61–72. MOOD, A. M., (1954). On the asymptotic efficiency of certain nonparametric two–sample tests. Ann Math Stat 25, pp. 514–522.

MUCCIOLI, C., BELFORD, R., PODGOR, M., SAMPAIO, P., DE SMET, M., NUSSENBLATT, R., (1996). The diagnosis of intraocular inflammation and cytomegalovirus retinitis in HIV–infected patients by laser flare photometry, Ocular Immunology and Inflammation, 4, pp. 75–81.

MURAKAMI, H., (2007). Lepage type statistic based on the modified Baumgartner statistic, Computational Statistics & Data Analysis, 51, pp. 5061–5067.

NEUHAUSER, M., LEUCHS, A.-K., BALL, D., (2011). A new location-scale test based on a combination of the ideas of Levene and Lepage, Biometrical Journal, 53, pp. 525–534.

O’BRIEN, R. G., (1979). A general ANOVA method for robust test of additive models for variance, Journal of the American Statistical Association, 74, pp. 877–880.

PARK, H-I., (2015a). Simultaneous Tests with Combining Functions under Normality, Communications for Statistical Applications and Methods, Vol. 22, No. 6, pp. 639–646.

PARK, H-I., (2015b). Nonparametric Simultaneous Test Procedures, Revista Colombiana de Estadística, 38(1), pp. 107–121.

PESARIN, F., (2001). Multivariate Permutation Test with Applications in Biostatistics, Chichester: Wiley.

SALMASO, L., SOLARI, A., (2005). Multiple aspect testing for case-control designs, Metrika, 62, pp. 331–340.

TIPPETT, L. H. C., (1931). The Methods of Statistics, London: Williams and Norgate.

WILCOXON, F., (1949). Some rapid approximate statistical procedures, Stamford, CT: Stamford Research Laboratories, American Cyanamid Corporation.

YONETANI, T., GORDON, H. B., (2001). Abrupt changes as indicators of decadal climate variability, Climate Dynamics, 17, pp. 249–258.