The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Sample Size Determination of Univariate and Bivariate Ordinal Outcomes by Nonparametric Wilcoxon Tests

단변량 및 이변량 순위변수의 비모수적 윌콕슨 검정법에 의한 표본수 결정방법

  • Park, Hae-Gang (Department of Biostatistics, Medical College, The Catholic University of Korea) ;
  • Song, Hae-Hiang (Department of Biostatistics, Medical College, The Catholic University of Korea)
  • 박해강 (가톨릭대학교 의과대학 의학통계학과) ;
  • 송혜향 (가톨릭대학교 의과대학 의학통계학과)
  • Received : 20090800
  • Accepted : 20091000
  • Published : 2009.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

The power function in sample size determination has to be characterized by an appropriate statistical test for the hypothesis of interest. Nonparametric tests are suitable in the analysis of ordinal data or frequency data with ordered categories which appear frequently in the biomedical research literature. In this paper, we study sample size calculation methods for the Wilcoxon-Mann-Whitney test for one- and two-dimensional ordinal outcomes. While the sample size formula for the univariate outcome which is based on the variances of the test statistic under both null and alternative hypothesis perform well, this formula requires additional information on probability estimates that appear in the variance of the test statistic under alternative hypothesis, and the values of these probabilities are generally unknown. We study the advantages and disadvantages of different sample size formulas with simulations. Sample sizes are calculated for the two-dimensional ordinal outcomes of efficacy and safety, for which bivariate Wilcoxon-Mann-Whitney test is appropriate than the multivariate parametric test.

표본수 결정에서 요구되는 검정력 함수는 연구가설에 상응하는 가장 적절한 검정방법에 의한 것이어야 한다. 의학연구의 논문에 자주 나타나는 순위자료 또는 범주형 빈도자료의 분석에는 비모수적 방법이 적절하며, 본 논문에서는 단변량 및 이변량 순위변수에 대한 윌콕슨-만-휘트니(Wilcoxon-Mann-Whitney; WMW) 검정법에 의한 표본수 결정방법을 제시한다. 단변량 순위변수의 윌콕슨 검정에서는 귀무가설과 대립가설 하의 분산을 이용한 표본수 공식이 귀무가설 하의 분산만 이용한 표본수 공식보다 정확하지만, 대립가설 하의 분산식에 나타나는 확률값이 일반적으로 알려져 있지 않으므로 이 확률값의 추정이 문제가 된다. 모의실험으로 두 방법에 대한 장, 단점을 알아본다. 효능과 안전성의 이변량 순위변수에서는 이변량 WMW 검정법에 의한 표본수 결정방법이 모수적 검정법에 의한 표본수 결정방법보다 더욱 바람직하다.

Keywords

References

  1. 이현학, 송혜향 (2009). 이변량 효능과 안전성 이항변수의 표본수 결정방법, <응용통계 연구>, 22, 341–353
  2. Agresti, A. (1984). Analysis of Ordinal Categorical Data, John Wiley& Sons, New York
  3. Delong, E. R., Delong, D. M. and Clarke-Pearson, D. L. (1988). Comparing the areas under two or more correlated receiver operating characteristic curves: A nonparametric approach, Biometrics, 44, 837–845
  4. Graubard, B. I. and Korn, E. L. (1987). Choice of column scores for testing independence in ordered 2 $\times$ K contingency tables, Biometrics, 43, 471–476
  5. Haldane, J. B. S. and Smith, C. A. B. (1948). A simple exact test for birth-order effect, Annals Eugen, 14, 117–124
  6. Mann, H. B. and Whitney, D. R. (1947). On a test whether one of two random variables is stochastically larger than the other, Annals of Mathematical Statistics, 18, 50–60 https://doi.org/10.1214/aoms/1177730491
  7. Nabulsi, N. N., Tamim, H., Mahfoud, Z., Itani, M., Sabra, R., Chamseddine, F. and Mikati, M. (2006). Alternating ibuprofen and acetaminophen in the treatment of febrile children: A pilot study, BMC Medicine, 4, 4 https://doi.org/10.1186/1741-7015-4-4
  8. Noether, G. E. (1987). Sample size determination for some common nonparametric tests, Journal of American Statistical Association, 82, 645–647 https://doi.org/10.2307/2289477
  9. Sundrum, R. M. (1953). The power of Wilcoxon's 2-sample test, Journal of the Royal Statistical Society, 15, 246–252
  10. Thall, P. F. and Cheng, S. (1999). Treatment comparisons based on two dimensional safety and efficacy alternatives in oncology trials, Biometrics, 55, 746–753 https://doi.org/10.1111/j.0006-341X.1999.00746
  11. Van Dantzig, D. (1951). On the consistency and the power of Wilcoxon's two sample test, Koninklijke Nederlandse Akademie Van Wetenschappen, Proceedings Serise A, 54, 1–9
  12. Vollandt, R. and Horn, M. (1997). Evaluation of Noether's method of sample size determination for the Wilcoxon-Mann-Whitney test, Biometrical Journal, 39, 823–829 https://doi.org/10.1002/bimj.4710390709
  13. Wang, H., Chen, B. and Chow, S. C. (2003). Sample size determination based on rank tests in clinical trials, Journal of Biopharmaceutical Statistics, 13, 735–751 https://doi.org/10.1081/BIP-120024206
  14. Wetherill, G. B. (1960). The Wilcoxon test and non-null hypotheses, Journal of the Royal Statistical Society. Series B (Methodological), 22, 402–418
  15. Zhao, Y. D., Rahardja, D. and Qu, Y. (2008). Sample size calculation for the Wilcoxon-Mann-Whitney test adjusting for ties, Statistics in Medicine, 27, 462–468 https://doi.org/10.1002/sim.2912