Three Steps Towards Robust Regression | Psychometrika | Cambridge Core (original) (raw)

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.

References

Anderson, N. H. Scales and statistics: Parametric and non-parametric. Psychological Bulletin, 1961, 58, 305–316.10.1037/h0042576CrossRef Google Scholar

Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H., and Tukey, J. W. Robust estimates of location, 1972, Princeton, N. J.: Princeton University Press.Google Scholar

Bock, R. D. Multivariate statistical methods in behavioral research, 1975, New York: McGraw-Hill.Google Scholar

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 PubMed

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 PubMed

Box, G. E. P., and Tiao, G. C. A Bayesian approach to some outlier problems. Biometrika, 1968, 55, 119–129.10.1093/biomet/55.1.119CrossRef Google Scholar PubMed

Czuber, E. Theorie der beobachtungsfehler. Leipzig, 1891.Google Scholar

David, H. A. Gini's mean difference rediscovered. Biometrika, 1968, 55, 573–574.Google Scholar

Devlin, S. J., Gnanadesikan, R., and Kettenring, J. R. Robust estimation and outlier detection with correlation coefficients. Biometrika, 1975, in press. (a)10.1093/biomet/62.3.531CrossRef Google Scholar

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

Downton, F. Linear estimates with polynomial coefficients. Biometrika, 1966, 53, 129–141.10.1093/biomet/53.1-2.129CrossRef Google Scholar PubMed

Gini, C. Variabilita e mutabilita, contributo allo studio delle distribuzione e relazione statistiche. Sudi-Economico-Giuridici della R. Universita di Cagliari, 1912.Google Scholar

Gnanadesikan, R., and Kettenring, J. R. Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics, 1972, 28, 81–124.10.2307/2528963CrossRef Google Scholar

Green, B. F. Parameter sensitivity in multivariate methods, 1974, Baltimore: Department of Psychology, Johns Hopkins University.Google Scholar

Helmert, F. R. Die Berechnung des wahrscheinlichen Beobachtungs fehlers aus den ersten Potenzen der Differenzen gleichgenauer directer Beobachtungen. Astronomische Nachrichten, 1876, 88, 257–272.10.1002/asna.18760880802CrossRef Google Scholar

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, 909–927.10.1080/01621459.1974.10480225CrossRef Google Scholar

Hogg, R. V., and Randles, R. Adaptive distribution-free regression methods. Technometrics, 1975, in press.10.1080/00401706.1975.10489366CrossRef Google Scholar

Hotelling, H., and Pabst, M. R. Rank correlation and tests of significance involving no assumption of normality. Annals of Mathematical Statistics, 1936, 7, 29–43.10.1214/aoms/1177732543CrossRef Google Scholar

Huber, P. J. Robust statistics: A review. Annals of Mathematical Statistics, 1972, 43, 1041–1067.10.1214/aoms/1177692459CrossRef Google Scholar

Knuth, D. E. The art of computer programming (Vol. 2). Reading, Mass.: Addison-Wesley. 1969, 1–112.Google Scholar

Mood, A. M. Introduction to the theory of statistics, 1950, New York: McGraw-Hill.Google Scholar

Roche, A. F., Wainer, H., and Thissen, D. Predicting adult stature for individuals, 1975, Basel, Switz.: Karger.Google Scholar PubMed

Samejima, F. Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 1969, No. 17.Google Scholar

Singleton, R. C. An efficient algorithm for sorting with minimal storage. Communications of the Association for Computing Machinery, 1969, 12, 185–187.10.1145/362875.362901CrossRef Google Scholar

Tukey, J. W. Exploratory data analysis (limited preliminary edition, Vol. 3), 1970, Reading, Mass.: Addison-Wesley.Google Scholar

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, A 25, 331–352.Google Scholar

von Andrae, Uber die Bestimmung des wahrscheinlichen Fehlers durch die gegebenen Differenzen vom gleich genauen Beobachtungen einer Unbekannten. Astronomische Nachrichten, 1872, 79, 257–272.10.1002/asna.18720791702CrossRef Google Scholar

Wainer, H. Predicting the outcome of the Senate trial of Richard M. Nixon. Behavioral Science, 1974, 19, 404–406.10.1002/bs.3830190607CrossRef Google Scholar

Wainer, H. Estimating coefficients in linear models: It don't make no nevermind. Psychological Bulletin, 1975, in press.Google Scholar

Wainer, H., Gruvaeus, G., and Zill, N. Senatorial decision making: I. The determination of structure. Behavioral Science, 1973, 18, 7–19.10.1002/bs.3830180103CrossRef Google Scholar

Wainer, H., and Thissen, D. Multivariate semi-metric smoothing in multiple prediction. Journal of the American Statistical Association, 1975, 70. (a)Google Scholar

Wainer, H., and Thissen, D. When jackknifing fails (or does it?). Psychometrika, 1975, 40, 113–114.10.1007/BF02291483CrossRef Google Scholar

Wainer, H., Zill, N., and Gruvaeus, G. Senatorial decision making: II. Prediction. Behavioral Science, 1973, 18, 20–26.CrossRef Google Scholar

Wright, S. Evolution and the genetics of populations. Vol. 1, Genetic and biometric foundations, 1968, Chicago: University of Chicago Press.Google Scholar