Research Article
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Year 2019, Volume: 32 Issue: 2, 718 - 735, 01.06.2019

Abstract

References

  • 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).

Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank

Year 2019, Volume: 32 Issue: 2, 718 - 735, 01.06.2019

Abstract

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.

References

  • 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).
There are 32 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Statistics
Authors

Bülent Altunkaynak 0000-0002-7571-2155

Hamza Gamgam 0000-0002-9595-9315

Publication Date June 1, 2019
Published in Issue Year 2019 Volume: 32 Issue: 2

Cite

APA Altunkaynak, B., & Gamgam, H. (2019). Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank. Gazi University Journal of Science, 32(2), 718-735.
AMA Altunkaynak B, Gamgam H. Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank. Gazi University Journal of Science. June 2019;32(2):718-735.
Chicago Altunkaynak, Bülent, and Hamza Gamgam. “Test Statistic for Ordered Alternatives Based on Wilcoxon Signed Rank”. Gazi University Journal of Science 32, no. 2 (June 2019): 718-35.
EndNote Altunkaynak B, Gamgam H (June 1, 2019) Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank. Gazi University Journal of Science 32 2 718–735.
IEEE B. Altunkaynak and H. Gamgam, “Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank”, Gazi University Journal of Science, vol. 32, no. 2, pp. 718–735, 2019.
ISNAD Altunkaynak, Bülent - Gamgam, Hamza. “Test Statistic for Ordered Alternatives Based on Wilcoxon Signed Rank”. Gazi University Journal of Science 32/2 (June 2019), 718-735.
JAMA Altunkaynak B, Gamgam H. Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank. Gazi University Journal of Science. 2019;32:718–735.
MLA Altunkaynak, Bülent and Hamza Gamgam. “Test Statistic for Ordered Alternatives Based on Wilcoxon Signed Rank”. Gazi University Journal of Science, vol. 32, no. 2, 2019, pp. 718-35.
Vancouver Altunkaynak B, Gamgam H. Test Statistic for Ordered Alternatives based on Wilcoxon Signed Rank. Gazi University Journal of Science. 2019;32(2):718-35.