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Journal of Statistical Theory and Practice

, Volume 4, Issue 2, pp 289–301 | Cite as

On Distribution-Free Runs Test for Symmetry using Extreme Ranked Set Sampling with an Application Involving Base Deficit Score

  • Hani M. Samawi
  • Robert Vogel
  • Christopher K. Senkowski
Article

Abstract

Most statistical inferences, which are essential for decision making and research in the area of biomedical sciences, are valid only under certain assumptions. One of the important assumptions in the literature is the symmetry of the underlying distribution of a study population. Several tests of symmetry are found in the literature. Most of these tests suffer from low statistical power which fails to detect a small but meaningful asymmetry in the population. Many investigators have attempted to improve the power of some of these tests. This paper examines several ranked set sample designs for the runs test of symmetry. Our investigation reveals that an optimal ranked set sample design for runs test of symmetry is the extreme ranked set sample (extreme ordered statistics sampling) (ERSS). This design of sampling increases the power and improves the performance of the runs test of symmetry and hence reduces the sample size needed in the study and the cost of the study. Intensive simulation is conducted to examine the power of the proposed optimal design for small sample sizes. Finally, base deficit values for patients subject to either blunt trauma or penetrating trauma are used to illustrate the procedures developed in this paper.

AMS Subject Classification

62G10 

Keywords

Base deficit Invariant test Ranked set sample Runs test Test of symmetry Power of the test 

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References

  1. Baklizi A., 2003. A conditional distribution free runs test for symmetry. Nonparametric Statistics, 15, 713–718.MathSciNetCrossRefGoogle Scholar
  2. Bohn, L.L., Wolfe, D.A., 1992. Nonparametric two-sample procedures for ranked-set samples data. Journal of American Statistical Association, 87, 522–561.CrossRefGoogle Scholar
  3. Bohn, L.L., Wolfe, D.A., 1994. The effect of imperfect judgment on ranked-set samples analog of the Mann-Whitney-Wilcoxon statistics. Journal of American Statistical Association, 89, 168–176.MathSciNetCrossRefGoogle Scholar
  4. Butler, C.C., 1969. A test for symmetry using sample distribution function. The Annals of Mathematical Statistics, 40, 2211–2214.CrossRefGoogle Scholar
  5. Chen, Z., 2001. Optimal ranked-set sampling scheme for inference on population quantiles. Statistica Sinica, 11, 23–37.MathSciNetzbMATHGoogle Scholar
  6. Davis, J.W., Mackersie, R.C., Holbrook, T.L., Hoyt, D.B., 1991. Base deficit as an indicator of significant abdominal injury. Annals Emerging Medicine, 20, 1406–1407.Google Scholar
  7. Davis J.W., Shackford, S.R., Mackersie, R.C., Hoyt, D.B., 1988. Base deficit as a guide to volume Resuscitation. Journal of Trauma, 28, 1464–1467.CrossRefGoogle Scholar
  8. Dell, T.R., Clutter, J.L., 1972. Ranked set sampling theory with order statistics background. Biometrics, 28, 545–555.CrossRefGoogle Scholar
  9. Halls, L.K., Dell, T.R., 1966. Trial of ranked set sampling for forage yields. Forest Science, 12, 22–26.Google Scholar
  10. Hettmansperger, T.P., 1984. Statistical Inference Based on Ranks. John Wiley & Sons, Inc.zbMATHGoogle Scholar
  11. Hettmansperger, T.P., 1995. The ranked-set sample sign test. Journal of Nonparametric Statistics, 4, 263–270MathSciNetCrossRefGoogle Scholar
  12. Hill, D.L., Rao, P.V., 1977. Test of Symmetry based on Cramer-Von Mises statistics. Biometrika, 64, 489–494.MathSciNetzbMATHGoogle Scholar
  13. Kaur, A., Patil, G. P., and Taillie, C., 2000. Optimal allocation for symmetric distributions in ranked set sampling. Annals of the Institute of Statistical Mathematics, 52, 239–254.MathSciNetCrossRefGoogle Scholar
  14. Kaur, A., Patil, G.P., Sinha A.K., Taillie, C., 1995. Ranked set sampling: An annotated bibliography. Environmental Ecological Statistics, 2, 25–54.CrossRefGoogle Scholar
  15. Koti, K.M., Babu, G.J., 1996. Sign test for ranked-set sampling. Communication in Statistics Theory and Methods, 25, 1617–1630.MathSciNetCrossRefGoogle Scholar
  16. Kvam, P.H., Samaniego, F.J., 1994. Nonparametric maximum likelihood estimation based on ranked set samples. Journal of American Statistical Association, 89, 526–537.MathSciNetCrossRefGoogle Scholar
  17. McIntyre, G.A., 1952. A method for unbiased selective sampling, using ranked sets. Australian Journal of Agriculture Research, 3, 385–90.CrossRefGoogle Scholar
  18. McWilliams, T.P., 1990. A distribution-free test of symmetry based on a runs statistic. Journal of American Statistical Association, 85, 1130–1133.MathSciNetCrossRefGoogle Scholar
  19. Modarres R., Gastwirth J.L., 1996. A modified runs test of symmetry. Statistics & Probability Letters, 31, 107–112.MathSciNetCrossRefGoogle Scholar
  20. Muttlak, H.A., 1997. Median ranked set sampling. Journal of applied Statistical Sciences, 6, 245–255.zbMATHGoogle Scholar
  21. Öztürk, O., 1999. One and two sample sign tests for ranked set samples with selective designs. Communication in Statistics, Theory and Methods, 28, 1231–1245.MathSciNetCrossRefGoogle Scholar
  22. Öztürk, O., Wolfe, D.A., 2000. Alternative ranked set sampling protocols for the sign test. Statistics & Probability Letters, 47, 15–23.MathSciNetCrossRefGoogle Scholar
  23. Öztürk, O., 2001. A nonparametric test of symmetry versus asymmetry for ranked-set samples. Communication in Statistics, Theory and Methods, 30, 2117–2133.MathSciNetCrossRefGoogle Scholar
  24. Patil, G.P., Sinha, A.K., Tillie C., 1999. Ranked set sampling: a bibliography. Environmental Ecological Statistics, 6, 91–98.CrossRefGoogle Scholar
  25. Ramberg, J.S., Schmeiser, B.W., 1974. An approximate method for generating asymmetric random variables. Communications of the ACM, 17, 78–82.MathSciNetCrossRefGoogle Scholar
  26. Rothman, E.D., Woodroofe, M.A, 1972. Cramer-Von Mises type statistic for testing symmetry. The Annals of Mathematical Statistics, 43, 2035–2038.MathSciNetCrossRefGoogle Scholar
  27. Samawi, H.M., 2001. On quantiles estimation using ranked samples with some applications. Journal of Korean Statistical Association, 30, 667–678.MathSciNetGoogle Scholar
  28. Samawi H.M. and Al-Saleh F.M. (2004). On bivariate ranked set sampling for distribution and quantile estimation and quantile interval estimation using ratio estimator. Communication in Statistics, Theory and Methods, 33, 1801–1819.MathSciNetCrossRefGoogle Scholar
  29. Samawi, H.M., Ahmed, M.S. and Abu-Dayyeh, W., 1996. Estimating the population mean using extreme ranked set sampling. Biometrical Journal, 38, 577–586.CrossRefGoogle Scholar
  30. Samawi, H.M., Abu-Dayyeh, W. (2003). More powerful sign test using median ranked set sample: Finite sample power comparison. Communication in Statistics, Computation and Simulation, 73, 697–708.MathSciNetCrossRefGoogle Scholar
  31. Tajuddin, I.H., 1994. Distribution-Free test for symmetry based on Wilcoxon two-sample test. Journal Applied Statistics, 21, 409–415.CrossRefGoogle Scholar
  32. Sinha, Arun K., 2005. On some recent developments in ranked set sampling. Bulletin of Informatics and Cybernetics, 37, 137–160.MathSciNetzbMATHGoogle Scholar
  33. Takahasi, K., Wakimoto, K., 1968. On unbiased estimates of the population mean based on the stratified sampling by means of ordering. Annals of the Institute of Statistical Mathematics, 20, 1–31.MathSciNetCrossRefGoogle Scholar
  34. Tremblay, L.N., Feliciano, D.V., Rozycki, G.S., 2002. Assessment of initial base deficit as a predictor of outcome: mechanism does make a difference. American Surgeon, 68, 689–694.Google Scholar

Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  • Hani M. Samawi
    • 1
  • Robert Vogel
    • 1
  • Christopher K. Senkowski
    • 2
  1. 1.Jiann-Ping Hsu College of Public HealthGeorgia Southern UniversityStatesboroUSA
  2. 2.ACI Surgical AssociatesMemorial University Medical CenterSavannahUSA

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