Advertisement

Reexamining the goodness-of-fit problem for interval-scale scores

  • Richard A. ChechileEmail author
Statistics: Research And Teaching
  • 165 Downloads

Abstract

A classic data-analytic problem is the statistical evaluation of the distributional form of interval-scale scores. The investigator may need to know whether the scores originate from a single Gaussian distribution or from a mixture of Gaussian distributions or from a different probability distribution. The relative merits of extant goodness-of-fit metrics are discussed. Monte Carlo power analyses are provided for several of the more powerful goodness-of-fit metrics.

Keywords

Idealize Marker Alternative Distribution Nonstandard Distribution Theoretical Cumulative Distribution Function Journal Ofthe American Statistical Association 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Bagby, R. J. (1995). Calculating normal probabilities.American Mathematical Monthly,102, 46–49.CrossRefGoogle Scholar
  2. Batchelder, W. H. (1991). Getting wise about minimum distance measures.Journal of Mathematical Psychology,35, 267–273.CrossRefGoogle Scholar
  3. Benard, A., &Bos-Levenbach, E. C. (1953). The plotting of observations on probability paper.Statisica Neerlandica,7, 163–173.CrossRefGoogle Scholar
  4. Bickel, P. J., &Breiman, L. (1983). Sums of functions of nearestneighbor distances, moment bounds, limit theorem, and a goodness-of-fit test.Annals of Probability,11, 185–214.CrossRefGoogle Scholar
  5. Blom, G. (1958).Statistical estimates and transformed beta variables. New York: Wiley.Google Scholar
  6. Bowker, A. H., &Lieberman, G. J. (1972).Engineering statistics (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  7. Chechile, R. A. (1997).A vector-based goodness-of-fit metric for interval data. Manuscript submitted for publication.Google Scholar
  8. Cressie, N., &Read, T. R. C. (1984). Multinominal goodness-of-fit tests.Journal of the Royal Statistical Society: Series B,46, 440–464.Google Scholar
  9. Dallal, G. E., &Wilkinson, L. (1986). An approximation to the distribution of Lilliefors’s test statistic for normality.American Statistician,40, 294–296.CrossRefGoogle Scholar
  10. Filliben, J. J. (1975). The probability plot correlation coefficient test for normality.Technometrics,17, 111–118.CrossRefGoogle Scholar
  11. Fishman, G. S. (1996).Monte Carlo: Concepts, algorithms, and applications. New York: Springer-Verlag.Google Scholar
  12. Gringorten, I. I. (1963). A plotting rule for extreme probability paper.Journal of Geophysical Research,68, 813–814.CrossRefGoogle Scholar
  13. Gumbel, E. J. (1964).Statistical theory of extreme values. Washington, DC: National Bureau of Standards.Google Scholar
  14. Hahn, G. J., &Shapiro, S. S. (1967).Statistical models in engineering. New York: Wiley.Google Scholar
  15. Harter, H. L., &Balakrishnan, N. (1996).CRC handbook of tables for the use of order statistics in estimation. Boca Raton, FL: CRC Press.Google Scholar
  16. Kelley, T. L. (1948).The Kelley statistical tables. Cambridge, MA: Harvard University Press.Google Scholar
  17. Kolmogorov, A. N. (1933). Sulla determinazione empirica di una legge di distribuizione.Giornale dell’Istituto Italiano degli Attuari,4, 83–91.Google Scholar
  18. Larsen, R. I., Curran, T. C., &Hunt, W. F. (1980). An air quality data analysis system for interrelating effects, standards, and needed source reduction: Part 6. Calculating concentration reductions needed to achieve the new national ozone standard.Journal of the Air Pollution Control Association,30, 662–669.Google Scholar
  19. Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov test for normality with mean and variance unknown.Journal of the American Statistical Association,62, 399–402.CrossRefGoogle Scholar
  20. Lilliefors, H. W. (1969). On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown.Journal of the American Statistical Association,64, 387–389.CrossRefGoogle Scholar
  21. Looney, S. W., &Gulledge, T. R. (1985). Use of the correlation coefficient with normal probability plots.American Statistician,39, 75–79.CrossRefGoogle Scholar
  22. Luce, R. D. (1986).Response times: Their role in inferring elementary mental organization. New York: Oxford University Press.Google Scholar
  23. Massey, F. J. (1951). The Kolmogorov-Smirnov test for goodness-offit.Journal of the American Statistical Association,46, 68–78.CrossRefGoogle Scholar
  24. Neyman, J., &Pearson, E. S. (1928). On the use and interpretation of certain test criteria for the purposes of statistical inference.Biometrika,20A (Pt. I), 175–240; (Pt. II), 263–294.Google Scholar
  25. Pearson, E. S., D’Agostino, R. B., &Bowman, K. O. (1977). Tests for departure from normality: Comparison of powers.Biometrika,64, 231–246.CrossRefGoogle Scholar
  26. Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling.Philosophy Magazine,50, 157–172.Google Scholar
  27. Pyke, R. (1965). Spacings.Journal of the Royal Statistical Society: Series A,109, 85–110.Google Scholar
  28. Schilling, M. F. (1983). Goodness-of-fit testing in Rm based on the weighted empirical distribution of certain nearest-neighbor statistics.Annals of Statistics,11, 1–12.CrossRefGoogle Scholar
  29. Shapiro, S. S., &Francia, R. S. (1972). An approximate analysis of variance test for normality.Journal of the American Statistical Association,67, 215–216.CrossRefGoogle Scholar
  30. Shapiro, S. S., &Wilk, M. B. (1965). An analysis of variance test for normality (complete samples).Biometrika,52, 591–611.Google Scholar
  31. Shapiro, S. S., &Wilk, M. B. (1972). An analysis of variance test for the exponential distribution (complete samples).Technometrics,14, 355–370.CrossRefGoogle Scholar
  32. Shapiro, S. S., Wilk, M. B., &Chen, H. J. (1968). A comparative study of various tests for normality.Journal of the American Statistical Association,63, 1343–1372.CrossRefGoogle Scholar
  33. Siegel, S. (1956).Nonparametric statistics for the behavioral sciences. New York: McGraw-Hill.Google Scholar
  34. Siegel, S., &Castellan, N. J., Jr. (1988).Nonparametric statistics for the behavioral sciences (2nd ed.). New York: McGraw-Hill.Google Scholar
  35. Smirnov, N. (1939). Ob uklonenijah empiričeskoi krivoi raspredelenija [On the deviations of the empirical distributional curve].Matematiceskii Sbornik,48, 3–26.Google Scholar
  36. Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons.Journal of the American Statistical Association,69, 730–737.CrossRefGoogle Scholar

Copyright information

© Psychonomic Society, Inc. 1998

Authors and Affiliations

  1. 1.Psychology DepartmentTufts UniversityMedford

Personalised recommendations