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Journal of Statistical Theory and Practice

, Volume 6, Issue 4, pp 681–697 | Cite as

Nonparametric Predictive Inference for Accuracy of Ordinal Diagnostic Tests

  • Faiza F. Elkhafifi
  • Frank P. A. Coolen
Article

Abstract

We introduce nonparametric predictive inference (NPI) for accuracy of diagnostic tests with ordinal outcomes, with the inferences based on data for a disease group and a non-disease group. We introduce empirical and NPI lower and upper receiver operating characteristic (ROC) curves and the corresponding areas under the curves, and we prove that these are nested, with the latter equal to the NPI lower and upper probabilities for correctly ordered future observations from the non-disease and disease groups. We discuss the use of the Youden index related to the NPI lower and upper ROC curves in order to determine the optimal cutoff point for the test.

Keywords

Accuracy of diagnostic tests Lower and upper probabilities Nonparametric predictive inference Ordinal data ROC curves 

AMS Subject Classification

60A99 62G99 62P10 

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References

  1. Agresti A., 2010. Analysis of ordinal categorical data. Hoboken, NJ, Wiley.CrossRefGoogle Scholar
  2. Arts, G. R. J., and F. P. A. Coolen 2008. Two nonparametric predictive control charts. J. Stat. Theory Prac., 2, 499–512.MathSciNetCrossRefGoogle Scholar
  3. Augustin, T., and F. P. A. Coolen. 2004. Nonparametric predictive inference and interval probability. J. Stat. Plan. Inference, 124, 251–272.MathSciNetCrossRefGoogle Scholar
  4. Baker, R. M., and F. P. A. Coolen. 2010. Nonparametric predictive category selection for multinomial data. J. Stat. Theory Pract., 4, 509–526.MathSciNetCrossRefGoogle Scholar
  5. Bamber, D. 1975. The area above the ordinal dominance graph and the area below the receiver operating graph. J. Math. Psychol., 12, 387–415.MathSciNetCrossRefGoogle Scholar
  6. Coolen, F. P. A. 1998. Low structure imprecise predictive inference for Bayes’ problem. Stat. Probability Lett., 36, 349–357.MathSciNetCrossRefGoogle Scholar
  7. Coolen, F. P. A., 2006. On nonparametric predictive inference and objective Bayesianism. J. Logic Language Information, 15, 21–47.MathSciNetCrossRefGoogle Scholar
  8. Coolen, F. P. A. 2011. Nonparametric predictive inference. In International encyclopedia of statistical science, ed. M. Lovring 968–970 Berlin, Springer.CrossRefGoogle Scholar
  9. Coolen F. P. A., and T. Augustin. 2009. A nonparametric predictive alternative to the imprecise Dirichlet model: the case of a known number of categories. Int. J. Approx. Reasoning, 50, 217–230.MathSciNetCrossRefGoogle Scholar
  10. Coolen, F. P. A., P. Coolen-Schrijner, T. Coolen-Maturi, and F. F. G. A. Elkhafifi. 2012. Nonparametric predictive inference for ordinal data. Commun. Stat. Theory Methods, in press.Google Scholar
  11. Coolen-Maturi, T. A., P. Coolen-Schrijner and F. P. A. Coolen. 2012a. Nonparametric predictive inference for diagnostic accuracy. J. Stat. Plan. Inference, 142, 1141–1150.MathSciNetCrossRefGoogle Scholar
  12. Coolen-Maturi, T. A., P. Coolen-Schrijner, and F. P. A. Coolen. 2012b. Nonparametric predictive inference for binary diagnostic tests. J. Stat. Theory Pract., 6, 665–680.MathSciNetCrossRefGoogle Scholar
  13. Coolen-Schrijner P., T. A. Maturi, and F. P. A. Coolen. 2009. Nonparametric predictive precedence testing for two groups. J. Stat. Theory Pract., 3, 273–287.MathSciNetCrossRefGoogle Scholar
  14. De Finetti, B. 1974. Theory of probability. London, UK, Wiley.zbMATHGoogle Scholar
  15. Fluss, R., D. Faraggi, and B. Reiser. 2005. Estimation of the Youden index and its associated cutoff point. Biometrical J., 47, 458–472.MathSciNetCrossRefGoogle Scholar
  16. Greiner, M., D. Pfeiffer, and R. D. Smith. 2000. Principles and practical application of the receiver-operating characteristic analysis for diagnostic tests. Preven. Vet. Med., 45, 23–41.CrossRefGoogle Scholar
  17. Hill, B. M., 1968. Posterior distribution of percentiles: Bayes’ theorem for sampling from a population. J. Am. Stat. Assoc., 63, 677–691.MathSciNetzbMATHGoogle Scholar
  18. Hill, B. M. 1988. De Finetti’s theorem, induction, and A n, or Bayesian nonparametric predictive inference (with discussion). In Bayesian statistics 3, ed. D. V. Lindley, J. M. Bernardo, M. H. DeGroot, and A. F. M. Smith, 211–241. Oxford, UK: Oxford University Press.Google Scholar
  19. Lawless J. F., and M. Fredette. 2005. Frequentist prediction intervals and predictive distributions. Biometrika, 92, 529–542.MathSciNetCrossRefGoogle Scholar
  20. McNeil, B. J., R. Sanders, P. O. Alderson, S. J. Hessel, H. Finberg, S. S. Siegelman, D. F. Adams, and H. L. Abrams. 1981. A prospective study of computed tomography, ultrasound and gallium imaging in patients with fever. Radiology, 139, 647–653.CrossRefGoogle Scholar
  21. McNeil, B. J., and J. A. Hanley. 1984. Statistical approaches to the analysis of receiver operating characteristic (ROC) curves. Me. Decision Making, 4, 137–150.CrossRefGoogle Scholar
  22. Nakas, C. T., and C. T. Yiannoutsos. 2006. Ordered multiple class receiver operating characteristic (ROC) analysis. In Encyclopedia of biopharmaceutical statistics. Doi: 10.1081/E-EBS-120041740. Oxford, UK, Taylor and Francis.CrossRefGoogle Scholar
  23. Pepe, M. S., 2003. The statistical evaluation of medical tests for classification and prediction. Oxford, UK, Oxford University Press.zbMATHGoogle Scholar
  24. Schäfer, H. 1989. Constructing a cut-off point for a quantitative diagnostic test. Stat. Med., 8, 1381–1391.CrossRefGoogle Scholar
  25. Sukhatme, S., and C. A. Beam. 1994. Stratification in nonparametric ROC studies. Biometrics, 50, 149–163.CrossRefGoogle Scholar
  26. Walley, P. 1991. Statistical reasoning with imprecise probabilities. London, UK; Chapman and Hall.CrossRefGoogle Scholar
  27. Weichselberger, K. 2000. The theory of interval-probability as a unifying concept for uncertainty. Int. J. Approx. Reasoning, 24, 149–170.MathSciNetCrossRefGoogle Scholar
  28. Weichselberger, K. 2001. Elementare Grundbegriffe einer Allgemeineren Wahrscheinlichkeitsrechnung: I. Intervallwahrscheinlichkeit als Umfassendes Konzept. Heidelberg, Physika.CrossRefGoogle Scholar
  29. Weinstein, S., N. A. Obuchowski, and M. L. Lieber. 2005. Clinical evaluation of diagnostic tests. Ame. J. Roentgenol., 184, 14–19.CrossRefGoogle Scholar
  30. Youden, W. J. 1950. Index for rating diagnostic tests. Cancer, 3, 32–35.CrossRefGoogle Scholar
  31. Zhou, X. H., N. A. Obuchowski, and D. K. McClish. 2002. Statistical methods in diagnostic medicine. New York, Wiley.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUK
  2. 2.Benghazi UniversityBenghaziLibya

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