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Testing for normality

  • G. Barrie Wetherill
  • P. Duncombe
  • M. Kenward
  • J. Köllerström
  • S. R. Paul
  • B. J. Vowden
Chapter
Part of the Monographs on Statistics and Applied Probability book series (MSAP)

Abstract

Before setting out on a discussion of tests for normality, we must consider briefly the consequences of non-normality. Clearly, if we know that the distribution of the disturbances is, say gamma, then a different model, and different methods, should be used than those discussed in this volume. However, in many situations we may have at least ‘near’ normality, and we wish to know how serious the effect of deviation from normality might be. Box and Watson (1962) showed that this depends on the distribution of the explanatory variables. If the explanatory variables can be regarded as approximately normal, then the F-test for multiple regression is insensitive to non-normality. If the explanatory variables cannot be so regarded, then there can be a substantial effect on the ‘F-test’ distribution.

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References

  1. Anderson, T. W. and Darling, D. A. (1952) Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes. Ann. Math. Statist., 23, 193–212.CrossRefGoogle Scholar
  2. Barnett, F. C. Mullen, K. and Saw, J. G. (1967) Linear estimation of a population scale parameter. Biometrika, 54, 551–554.Google Scholar
  3. Beasley, J. D. and Springer, S. G. (1977) Algorithm 111. The percentage points of the normal distribution. Appl. Statist., 26, 118–121.CrossRefGoogle Scholar
  4. Bowman, K. O. and Shenton, L. R. (1975) Omnibus test contours for departures from normality based on (b1) and b2. Biometrika, 62, 243–250.Google Scholar
  5. Box, G. E. P. and Watson, G. S. (1962) Robustness to non-normality of regression tests. Biometrika, 49, 93–106.Google Scholar
  6. B.S.I. (1982) Draft BS2846: British Standard guide to statistical interpretation of data. Part 7: Tests for departure from normality (ISO/DIS 5479). Obtainable from British Standards Institution, Sales Department, Linfood Wood, Milton Keynes, MK14 6LR.Google Scholar
  7. D’Agostino, R. B. (1971) An omnibus test of normality for moderate and large size samples. Biometrika, 58, 341–348.CrossRefGoogle Scholar
  8. Green, J. R. and Hegazy, Y. A. S. (1976) Powerful modified-EDF goodness- of-fit tests. J. Amer. Statist. Assoc., 71, 204–209.CrossRefGoogle Scholar
  9. Gurland, J. and Dahiya, R. C. (1972) Tests of fit for continuous distributions based on generalized minimum chi-squared, in Statistical Papers in Honour of George W. Snedcor (ed. T. A. Bancroft ). State University Press, Iowa, pp. 115–128.Google Scholar
  10. Hahn, G. J. and Shapiro, S. S. (1967) Statistical Models in Engineering. Wiley, New York.Google Scholar
  11. Harter, H. L. (1961) Expected values of normal order statistics. Biometrika, 48, 151–165.Google Scholar
  12. Hastings, C. (1955) Approximation for Digital Computers. Princeton University Press, Prinction, NJ.Google Scholar
  13. Huang, C. J. and Bolch, B. W. (1974) On the testing of regression disturbances for normality. J. Amer. Statist. Assoc., 69, 330–335.CrossRefGoogle Scholar
  14. Kendall, M. G. and Stuart, A. (1969) The Advanced Theory of Statistics, Vol. I, 3rd edn. Griffin, London.Google Scholar
  15. Pearson, E. S., D’Agostino, R. B. and Bowman, K. O. (1977) Test for departure from normality: comparison of powers. Biometrika, 64, 231–246.CrossRefGoogle Scholar
  16. Pettit, A. N. (1977) Testing the normality of several independent samples using the Anderson-Darling statistic. Appl. Statist. 26, 156–161.CrossRefGoogle Scholar
  17. Pierce, D. A. and Gray, R. J. (1982) Testing normality of errors in regression models. Biometrika, 69, 233–236.CrossRefGoogle Scholar
  18. Royston, J. P. (1982a) An extension of Shapiro and Wilk’s test for normality to large samples. Appl. Statist., 31, 115–124.CrossRefGoogle Scholar
  19. Royston, J. P. (1982b) Algorithm AS181. The W-test for normality. Appl. Statist., 31, 176–180.CrossRefGoogle Scholar
  20. Royston, J. P. (1983). A simple method for evaluating the Shapiro-Francia W’-test for non-normality. J. Inst. Statist., 32, 297–300.Google Scholar
  21. Seber, G. A. F. (1977) Linear Regression Analysis. Wiley, New York.Google Scholar
  22. Shapiro, S. S. and Francia, R. S. (1972) An approximate analysis of variance test for normality. J. Amer. Statist. Assoc., 61, 215–216.CrossRefGoogle Scholar
  23. Shapiro, S. S. and Wilk, M. B. (1965) An analysis of variance test for normality (complete samples). Biometrika, 52, 591–611.Google Scholar
  24. Shapiro, S. S., Wilk, M. B. and Chen, H. J. (1968) A comparative study of various tests for normality. J. Amer. Statist. Assoc., 63, 1343–1372.CrossRefGoogle Scholar
  25. Stephens, M. A. (1974) EDF statistics for goodness of fit and some comparisons. J. Amer. Statist. Assoc., 69, 730–737.CrossRefGoogle Scholar
  26. Stephens, M. A. (1976) Asymptotic results for goodness-of-fit statistics with unknown parameters. Ann. Statist., 4, 357–369.CrossRefGoogle Scholar
  27. White, H. and Macdonald, G. M. (1980) Some large-sample tests for non- normality in the linear regression model. J. Amer. Statist. Assoc., 75, 16–28.CrossRefGoogle Scholar

Further Reading

  1. D’Agostino, R. B. (1972) Small sample probability points for the D test of normality. Biometrika, 59, 219–221.CrossRefGoogle Scholar
  2. Kennard, K. V. C. (1978) A review of the literature on omnibus tests of normality with particular reference to tests based on sample moments. M.Sc. dissertation, University of Kent.Google Scholar

Copyright information

© G. Barrie Wetherill 1986

Authors and Affiliations

  • G. Barrie Wetherill
    • 1
  • P. Duncombe
    • 2
  • M. Kenward
    • 3
  • J. Köllerström
    • 3
  • S. R. Paul
    • 4
  • B. J. Vowden
    • 3
  1. 1.Department of StatisticsThe University of Newcastle upon TyneUK
  2. 2.Applied Statistics Research UnitUniversity of Kent at CanterburyUK
  3. 3.Mathematical InstituteUniversity of Kent at CanterburyUK
  4. 4.Department of Mathematics and StatisticsUniversity of WindsorCanada

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