Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank

Yıl 2019, Cilt: 32 Sayı: 2, 718 - 735, 01.06.2019

Bülent Altunkaynak Hamza Gamgam

Öz

This paper
proposes a test statistic for ordered alternatives based on the Wilcoxon signed
rank statistic. One of the classical tests, Jonckheere-Terpstra’s J test, and the R test suggested by Chen et al. were used for type I error rate and
power comparisons. For data generated from the normal distribution, all of the
tests gave type I error rates close to nominal alpha. When the data were
generated from chi-square distribution, the proposed G test and J test for
type I error gave better results than the R
test, but the error rates of the J
test for Student’s t distribution are
better than those of the others. Power results of simulation study for normal
distributions showed that the proposed G
test was superior to all other considered tests. The G and J tests for the
data generated from Student’s t distributions performed well. When the data
were generated from chi-square distributions, the proposed G test is more powerful than the others. The simulation showed that
the R test was inferior to the other
tests for all cases.

Anahtar Kelimeler

Ordered alternatives, Wilcoxon signed rank statistic, Simulation, Power

Kaynakça

• Kruskal, W.H., Wallis, W.A., “Use of ranks in one-criterion variance analysis”, J Amer Statist Assoc., 47:583–621, (1952).
• Bhapkar, V.P., “A nonparametric test for the problem of several samples”, Annals of Mathematical Statistics, 32:1108–1117, (1961).
• Bishop, T.A., “Heteroscedastic ANOVA, MANOVA and multiple comparisons”, Phd.Thesis, The Ohio State University, Ohio, 15-35 (1976).
• Bishop, T.A., Dudewicz, E.J., “Exact analysis of variance with unequal variances: test procedures and tables”, Technometrics, 20:419–430, (1978).
• Bishop, T.A., Dudewicz, E.J. “Heteroscedastic ANOVA”, Sankhya, 43:40–57, (1981).
• Chen, S., Chen, J.H., “Single-stage analysis of variance under heteroscedasticity”, Comm Statist Simulation and Computation. 27(3): 641–666, (1998).
• Chen, S., “One-stage and two-stage statistical inference under heteroscedasticity”, Comm Statist Simulation and Computation, 30(4):991–1009, (2001).
• Gamage, J., Mathew, T., Weerahandi, S., “Generalized p-values and generalized confidence regions for the multivariate Behrens–Fisher problem and MANOVA”, J Multivariate Anal., 88:177–189, 2004.
• Lee, S., Ahn, C.H., “Modified ANOVA for unequal variances”, Comm Statist Simulation and Computation, 32:987–1004, (2003).
• Rice, W.R, Gaines, S.D., “One-way analysis of variance with unequal variances”, Proc Nat Acad Sci., 86:8183–8184, (1989).
• Weerahandi, S., “ANOVA under unequal error variances”, Biometrics, 51:589–599, (1995).
• Xu, L., Wang, S.A., “A new generalized p-value for ANOVA under heteroscedasticity”, Statistics & Probability Letters, 78(8):963–969, (2008).
• Terpstra, T.J., Chang, C.H., Magel, R.C., “On the use of Spearman’s correlation coefficient for testing ordered alternatives”, Journal of Statistical Computation and Simulation, 81(11):1381–1392, (2011).
• Terpstra, T.J., “The asymptotic normality and consistency of Kendall's test against trend, when ties are present in one ranking”, Indigationes Mathematicae, 14:327-333, (1952).
• Jonckheere, A.R., “A distribution-free k-sample test against ordered alternatives”, Biometrika, 41:133–145, (1954).
• Bartholomew, D.J., “Ordered tests in the analysis of variance”, Biometrika, 48:325–332, (1961).
• Chacko, V.J., “Testing homogeneity against ordered alternatives”, Ann Math Statist., 34: 945–956, (1963).
• Puri, M.L., “Some distribution-free k-sample rank tests of homogeneity against ordered alternatives”, Comm Pure Appl Math., 18:51–63, (1965).
• Odeh, R.E., “On Jonckheere's k-sample test against ordered alternatives”, Technometrics, 13:912–918, (1971).
• Archambault, W.A.T., Mack, G.A., Wolfe, D.A., “K-sample rank tests using pair-specific scoring functions”, Canadian Journal of Statistics, 5:195–207, (1977).
• Hettmansperger, T.P., Norton, R.M., “Tests for patterned alternatives in k-sample problems”, J Amer Statist Assoc., 82: 292–299, (1987).
• Beier, F., Buning, H., “An adaptive test against ordered alternatives”, Computational Statistics & Data Analysis, 25(4):441-452, (1997).
• Neuhauser, M., Liu, P.Y., Hothorn, L.A., “Nonparametric test for trend: Jonckheere’s test, a modification and a maximum test”, Biometrical Journal, 40:899–909, (1998).
• Chen, S., Chen, J.H., Chang, H.F., “A one-stage procedure for testing homogeneity of means against an ordered alternative under unequal variances”, Comm Statist Simulation and Computation, 33(1):49–67, (2004).
• Shan, G., Young, D., Kang, L., “A new powerful nonparametric rank test for ordered alternative problem”, Plos One, 9(11):1–10, (2014).
• Gaur, A., “A class of k-sample distribution-free test for location against ordered alternatives”, Comm Stat Theory and Methods, 46(5):2343–2353, (2017).
• Gibbons, J.D., Nonparametric Statistical Inference, McGraw-Hill, New York, (1971).
• Daniel, W.W., Applied Nonparametric Statistics, Houghton Mifflin, Boston, (1978).
• Bucchianico, A.D., “Computer algebra, combinatorics, and the Wilcoxon-Mann-Whitney statistic”, J Stat Plan Inf., 79: 349–364, (1999).
• Wiel, M.A., “Exact distributions of distribution-free test statistics” Phd.Thesis, Eindhoven University of Technology, The Netherlands, 27-43 (2000).
• Bradley, J.V., “Robustness?”, Br J Math Stat Psychol. 31:144–152, (1978).
• Fagerland, M.W., Sandvik, L., “Performance of five two-sample location tests for skewned distributions with unequal variances”, Contemporary Clinical Trials, 30:490-496, (2009).