A New Maximum Test via the Dependent Samples t-Test and the Wilcoxon Signed-Ranks Test

Saverpierre Maggio, Shlomo S. Sawilowsky

Department of Evaluation and Research, Wayne State University, Detroit, USA.
Department of Psychology, University of Windsor, Windsor, Canada;Department of Evaluation and Research, Wayne State University, Detroit, USA.

DOI: 10.4236/am.2014.51013                

Abstract

A maximum test in lieu of forcing a choice between the two dependent samples t-test and Wilcoxon signed-ranks test is proposed. The maximum test, which requires a new table of critical values, maintains nominal α while guaranteeing the maximum power of the two constituent tests. Critical values, obtained via Monte Carlo methods, are uniformly smaller than the Bonferroni-Dunn adjustment, giving it power superiority when testing for treatment alternatives of shift in location parameter when data are sampled from non-normal distributions.

Keywords

Maximum Test; Dependent Samples t-Test; Wilcoxon Signed-Ranks Test; Bonferroni-Dunn Adjustment; Experiment-Wise Type I Error; Inferential Statistics; Monte Carlo Method

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. R. Blair, “Combining Two Nonparametric Tests of Location,” Journal of Modern Applied Statistical Methods, Vol. 1, No. 1, 2002, pp. 13-18.
[2]	D. R. Cox, “The Role of Significance Tests,” Scandinavian Journal of Statistics, Vol. 4, No. 2, 1977, pp. 49-70.
[3]	R. C. Blair and J. J. Higgins, “Comparison of the Power of the Paired Samples t test to that of Wilcoxon’s Sign-Ranks Test Under Various Population Shapes,” Psychological Bulletin, Vol. 97, No. 1, 1985, pp. 119-128. http://dx.doi.org/10.1037/0033-2909.97.1.119
[4]	H. J. Arnold, “Small Sample Power of the One Sample Wilcoxon Test for Non-Normal Shift Alternatives,” The Annals of Mathematical Statistics, Vol. 36, No. 6, 1965, pp. 1767-1778. http://dx.doi.org/10.1214/aoms/1177699805
[5]	R. Randles and D. Wolfe, “Introduction to the Theory of Nonparametric Statistics,” John Wiley & Sons, New York, 1979.
[6]	R. C. Blair and J. J. Higgins, “The Power of t and Wilcoxon Statistics: A Comparison,” Evaluation Review, Vol. 4, No. 5, 1980, pp. 645-656. http://dx.doi.org/10.1177/0193841X8000400506
[7]	T. A. Gerke and H. A. Randles, “A Method for Resolving Ties in Asymptotic Relative Efficiency,” Statistics and Probability Letter, Vol. 80, No. 13-14, 2010, pp. 1065-1069. http://dx.doi.org/10.1016/j.spl.2010.02.021
[8]	W. T. Wiederman and R. W. Alexandrowicz, “A Modified Normal Scores Test for Paired Data,” European Journal of Re- search Methods for the Behavioral and Social Sciences, Vol. 7, No. 1, 2011, pp. 25-38. http://dx.doi.org/10.1027/1614-2241/a000020
[9]	S. S. Sawilowsky and G. F. Fahoome, “Statistics through Monte Carlo Simulation with FORTRAN,” Journal of Modern Applied Statistical Methods Inc., Michigan, 2003.
[10]	J. Algina, R. C. Blair and W. T. Coombs, “A Maximum Test for Scale: Type I Error Rates and Power,” Journal of Educational and Behavioral Statistics, Vol. 20, No. 1, 1995, pp. 27-39.
[11]	L. H. C. Tippett, “The Methods of Statistics,” Williams and Norgate, England, 1934.
[12]	“International Mathematical and Statistical Libraries,” IMSL Library, Houston, 1980.
[13]	S. S. Sawilowsky, R. C. Blair and J. J. Higgins, “An Investigation of the Type I Error and Power Properties of the Rank Trans- form in Factorial ANOVA,” Communications in Statistics, Vol. 14, No. 3, 1989, pp. 255-267.
[14]	R. C. Blair and J. J. Higgins (unpublished, 1992) as referred to in S. S Sawilowsky and G. F. Fahoome, “Statistics through Monte Carlo Simulation with FORTRAN,” Journal of Modern Applied Statistical Methods Inc., Michigan, 2003.
[15]	R. J. Simes, “An Improved Bonferroni Procedure for Multiple Tests of Significance,” Biometrika, Vol. 73, No. 3, 1986, pp. 751-754. http://dx.doi.org/10.1093/biomet/73.3.751
[16]	Y. Hochberg, “A Sharper Bonferroni Procedure for Multiple Tests of Significance,” Biometrika, Vol. 75, No. 4, 1988, pp. 800-802. http://dx.doi.org/10.1093/biomet/75.4.800
[17]	J. D. Gibbons and S. Chakraborti, “Comparisons of the Mann-Whitney, Students t, and Alternate t-Tests for Means of Normal Distributions,” Journal of Experimental Education, Vol. 59, 1991, pp. 258-267.