Experimental Comparison of the Sample Sizes of the Two-Sample Tests of Wilcoxon and Student under the Arcsine Distribution
Abstract
When statistical experiments are performed, a sample size should be chosen in some optimum way so that we should use a sample size no larger than necessary. This paper compares the minimum sample sizes of two-sample t-test and the Wilcoxon rank-sum test under the arcsine distribution, based on their power [1, 2]. To accomplish this task, some essential probabilities that are useful in the power and sample size determination of the Wilcoxon rank-sum test were derived by the author for computing the approximated formula given by Lehmann [2]. The composite numerical integration algorithm is used to compute these probabilities, which are related to the arcsine distribution. In this study, a computer program was built by the author to find the exact (simulated) minimum sample sizes n for any significant level and power by iterating on n with starting points for n provided by the approximated formulas of [2, 3]. The scientific novelty of this research paper is determining the minimum sample sizes by considering a new set of the arcsine distribution shift alternatives of the forms giving the left-hand endpoint of the displaced distribution as the quantile of order p, 0 < p < 1, of the second distribution rather than using alternatives that specify as the quantile of order p, p < 0.5. As considered by [1], a choice that prevents losing some important alternative hypotheses is an extension to the set of alternative hypotheses considered by Guenther [1]. The exact (simulated) minimum sample sizes were computed and compared with each other and with the corresponding approximated formulas given by Lehmann [2] and Guenther [3]. Numerical results showed that the approximated formulas are very accurate and the Wilcoxon rank-sum test is more efficient when the sample size is more than 45. Otherwise, the Student two-sample t-test is better.
Keywords: arcsine distribution, minimum sample size, simulation, relative efficiency, Wilcoxon rank-sum test, two-sample t-test.
Full Text:
PDFReferences
GUENTHER W. C. Power and Sample Size Determination When the Alternative Hypotheses Are Given in Quantiles. The American Statistician, 1977, 31(3): 117-118. https://doi.org/10.1080/00031305.1977.10479213
LEHMANN E. L. Nonparametrics: Statistical methods based on ranks. Holden-Day, San Francisco, California, 1975.
GUENTHER W. C. Sample Size Formulas for Normal Theory T Test. The American Statistician, 1981, 35(4): 243-244. https://doi.org/10.1080/00031305.1981.10479363
GUENTHER W. C. Sampling Inspection in Statistical Quality Control. Macmillan, New York, 1977.
OWEN D. B. The power of Student's t-test. Journal of the American Statistical Association, 1965, 60(309): 320-333. https://doi.org/10.1080/01621459.1965.10480794
SUNDRUM R. M. The Power of Wilcoxon’s 2-Sample Test. Journal of the Royal Statistical Society: Series B (Methodological), 1953, 15(2): 246-252. https://doi.org/10.1111/j.2517-6161.1953.tb00139.x
KEILSON J. and WELLNER J. A. Oscillating Brownian Motion. Journal of Applied Probability, 1978, 15(2): 300-310. https://doi.org/10.2307/3213403
FELLER W. An Introduction to Probability Theory and Its Applications, Vol. 1. 3rd ed. John Wiley and Sons, New York, 1968.
LEE Y. W. Statistical Theory of Communications. John Wiley and Sons, New York, 1960.
ARNOLD B. C. and GROENVELD R. A. Some Properties of the Arcsine Distribution. Journal of the American Statistical Association, 1980, 75(369): 173-175. https://doi.org/10.1080/01621459.1980.10477449
NORTON R. M. On Properties of the Arc Sine Law. Sankhyā: The Indian Journal of Statistics, Series A, 1975, 37(2): 306-308. https://www.jstor.org/stable/25049987
NORTON R M. Moments Properties and the Arcsine Law. Sankhyā: The Indian Journal of Statistics, Series A, 1978, 40(2): 192-198. http://www.jstor.org/stable/25050147
SHANTARAM R. A Characterization of the Arcsine Law. Sankhyā: The Indian Journal of Statistics, Series A, 1978, 40(2): 199-207. https://www.jstor.org/stable/25050148
KEMPERMAN J. H. B. and SKIBINSKY M. On the characterization of an interesting property of the arcsine distribution. Pacific Journal of Mathematics, 1982, 103(2): 457-465.
SCHMIDT K. M. and ZHGLJAVSKY A. A characterization of the arcsine distribution. Statistics and Probability Letters, 2009, 79(24): 2451-2455. https://doi.org/10.1016/j.spl.2009.08.018
ENCYCLOPEDIA OF MATHEMATICS. Arcsine distribution, 2014. http://encyclopediaofmath.org/index.php?title=Arcsine_distribution&oldid=33530
BURDEN L., FAIRES D., and REYNOLDS C. Numerical Analysis. 2nd ed. Prindle, Weber and Schmidt, Boston, Massachusetts, 1981.
JENSEN A. and ROLAND H. Methods of computation. Scott, Foresman and Company, 1975.
INTERNATIONAL MATHEMATICAL AND STATISTICAL LIBRARY. Vol. 6.5.2. Houston, Texas, 2015.
HOGG R. V. and CRAIG A. T. Introduction to Mathematical Statistics. 4th ed. MacMillan Publishing Company, New York, 1978.
HODGES JR. J. L. and LEHMANN E. L. The efficiency of some nonparametric competitors of the t-test. Annals of Mathematical Statistics, 1956, 27(2): 324-335. https://doi.org/10.1214/aoms/1177728261
VAN DER LAAN P. and OOSTERHOFF J. Monte Carlo estimation of powers of the distribution-free two-sample tests of Wilcoxon, Van der Waerden and Terry and the comparison of these powers. Statistica Neerlandica, 1965, 19(4): 265-276. https://doi.org/10.1111/j.1467-9574.1965.tb00958.x
VAN DER LAAN P. and OOSTERHOFF J. Experimental determination of the power functions of the two-sample rank tests of Wilcoxon, Van der Waerden and Terry by Monte Carlo techniques. I. Normal parent distributions. Statistica Neerlandica, 1967, 21(1): 55-68. https://doi.org/10.1111/j.1467-9574.1967.tb00545.x
Refbacks
- There are currently no refbacks.
