Advertisement

Psychometrika

, Volume 41, Issue 1, pp 9–34 | Cite as

Three steps towards robust regression

  • Howard Wainer
  • David Thissen
Article

Abstract

The three most commonly used statistics, the arithmetic mean, variance, and the product-moment correlation, are most unfortunate choices when data are not strictly Gaussian. A new measure of correlation and a measure of scale are proposed which are substantially more robust than their least squares counterparts. An illustration shows how increased robustness can be obtained through the use of equal regression weights without severe loss in accuracy. The paper also shows how incorporating knowledge about the theoretical structure of the regression coefficients into their estimation can aid substantially in increasing their robustness.

Keywords

Regression Coefficient Public Policy Statistical Theory Regression Weight Severe Loss 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anderson, N. H. Scales and statistics: Parametric and non-parametric.Psychological Bulletin, 1961,58, 305–316.Google Scholar
  2. Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H., and Tukey, J. W.Robust estimates of location. Princeton, N. J.: Princeton University Press, 1972.Google Scholar
  3. Bock, R. D.Multivariate statistical methods in behavioral research. New York: McGraw-Hill, 1975.Google Scholar
  4. Bock, R. D., and Kolakowski, D. Further evidence of sex-linked major gene influence on human spatial visualizing ability.Americal Journal of Human Genetics, 1973,25, 1–14.Google Scholar
  5. Bock, R. D., Wainer, H., Thissen, D., Peterson, A., Murray, J., and Roche, A. F. A parameterization of individual human growth curves.Human Biology, 1973,45, 63–80.Google Scholar
  6. Box, G. E. P., and Tiao, G. C. A Bayesian approach to some outlier problems.Biometrika, 1968,55, 119–129.Google Scholar
  7. Czuber, E.Theorie der beobachtungsfehler. Leipzig, 1891.Google Scholar
  8. David, H. A. Gini's mean difference rediscovered.Biometrika, 1968,55, 573–574.Google Scholar
  9. Devlin, S. J., Gnanadesikan, R., and Kettenring, J. R. Robust estimation and outlier detection with correlation coefficients.Biometrika, 1975, in press. (a)Google Scholar
  10. Devlin, S. J., Gnanadesikan, R., and Kettenring, J. R.Robust estimation of correlation and covariance matrices. Paper presented at the spring meeting of the Psychometric Society, Iowa City, April 26, 1975. (b)Google Scholar
  11. Downton, F. Linear estimates with polynomial coefficients.Biometrika, 1966,53, 129–141.Google Scholar
  12. Gauss, C. F.Gottingsche gelehrte anzeigen, 1821.Google Scholar
  13. Gini, C. Variabilita e mutabilita, contributo allo studio delle distribuzione e relazione statistiche. Sudi-Economico-Giuridici della R. Universita di Cagliari, 1912.Google Scholar
  14. Gnanadesikan, R., and Kettenring, J. R. Robust estimates, residuals, and outlier detection with multiresponse data.Biometrics, 1972,28, 81–124.Google Scholar
  15. Green, B. F.Parameter sensitivity in multivariate methods. Mimeographed manuscript. Baltimore: Department of Psychology, Johns Hopkins University, 1974.Google Scholar
  16. Helmert, F. R. Die Berechnung des wahrscheinlichen Beobachtungs fehlers aus den ersten Potenzen der Differenzen gleichgenauer directer Beobachtungen.Astronomische Nachrichten, 1876,88, 257–272.Google Scholar
  17. Hogg, R. V. Adaptive robust procedures: A partial review and some suggestions for further applications and theory.Journal of the American Statistical Association, 1974,69, 909–927.Google Scholar
  18. Hogg, R. V., and Randles, R. Adaptive distribution-free regression methods.Technometrics, 1975, in press.Google Scholar
  19. Hotelling, H., and Pabst, M. R. Rank correlation and tests of significance involving no assumption of normality.Annals of Mathematical Statistics, 1936,7, 29–43.Google Scholar
  20. Huber, P. J. Robust statistics: A review.Annals of Mathematical Statistics, 1972,43, 1041–1067.Google Scholar
  21. Knuth, D. E.The art of computer programming (Vol. 2). Reading, Mass.: Addison-Wesley, 1969, 1–112.Google Scholar
  22. Mood, A. M.Introduction to the theory of statistics. New York: McGraw-Hill, 1950.Google Scholar
  23. Roche, A. F., Wainer, H., and Thissen, D.Predicting adult stature for individuals. Basel, Switz.: Karger, 1975.Google Scholar
  24. Samejima, F. Estimation of latent ability using a response pattern of graded scores.Psychometrika Monograph Supplement, 1969, No. 17.Google Scholar
  25. Singleton, R. C. An efficient algorithm for sorting with minimal storage.Communications of the Association for Computing Machinery, 1969,12, 185–187.Google Scholar
  26. Tukey, J. W.Exploratory data analysis (limited preliminary edition, Vol. 3). Reading, Mass.: Addison-Wesley, 1970.Google Scholar
  27. Tukey, J. W., and McLaughlin, D. H. Less vulnerable confidence and significance procedures for location based upon a single sample: Trimming/Winsorization 1.Sankhyā, 1963, A25, 331–352.Google Scholar
  28. von Andrae. Uber die Bestimmung des wahrscheinlichen Fehlers durch die gegebenen Differenzen vom gleich genauen Beobachtungen einer Unbekannten.Astronomische Nachrichten, 1872,79, 257–272.Google Scholar
  29. Wainer, H. Predicting the outcome of the Senate trial of Richard M. Nixon.Behavioral Science, 1974,19, 404–406.Google Scholar
  30. Wainer, H. Estimating coefficients in linear models: It don't make no nevermind.Psychological Bulletin, 1975, in press.Google Scholar
  31. Wainer, H., Gruvaeus, G., and Zill, N. Senatorial decision making: I. The determination of structure.Behavioral Science, 1973,18, 7–19. (a)Google Scholar
  32. Wainer, H., and Thissen, D. Multivariate semi-metric smoothing in multiple prediction.Journal of the American Statistical Association, 1975,70. (a)Google Scholar
  33. Wainer, H., and Thissen, D. When jackknifing fails (or does it?)Psychometrika, 1975,40, 113–114. (b)Google Scholar
  34. Wainer, H., Zill, N., and Gruvaeus, G. Senatorial decision making: II. Prediction.Behavioral Science, 1973,18, 20–26. (b)Google Scholar
  35. Wright, S.Evolution and the genetics of populations. Vol. 1, Genetic and biometric foundations, Chicago: University of Chicago Press, 1968.Google Scholar

Copyright information

© Psychometric Society 1976

Authors and Affiliations

  • Howard Wainer
    • 1
  • David Thissen
  1. 1.The University of ChicagoUSA

Personalised recommendations