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Correlation Weights in Multiple Regression

Published online by Cambridge University Press:  01 January 2025

Niels G. Waller  and
Jeff A. Jones
Niels G. Waller*
Affiliation:
University of Minnesota
Jeff A. Jones
Affiliation:
University of Minnesota
*
Requests for reprints should be sent to Niels G. Waller, Department of Psychology, University of Minnesota, N218 Elliott Hall, 75 East River Road, Minneapolis, MN, 55455 USA. E-mail: [email protected]
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Abstract

A general theory on the use of correlation weights in linear prediction has yet to be proposed. In this paper we take initial steps in developing such a theory by describing the conditions under which correlation weights perform well in population regression models. Using OLS weights as a comparison, we define cases in which the two weighting systems yield maximally correlated composites and when they yield minimally similar weights. We then derive the least squares weights (for any set of predictors) that yield the largest drop in R2 (the coefficient of determination) when switching to correlation weights. Our findings suggest that two characteristics of a model/data combination are especially important in determining the effectiveness of correlation weights: (1) the condition number of the predictor correlation matrix, Rxx, and (2) the orientation of the correlation weights to the latent vectors of Rxx.

Keywords

multiple regressioncorrelation weightsalternate weightsparameter sensitivity
Type
Theory and Methods
Information
Psychometrika , Volume 75 , Issue 1 , March 2010 , pp. 58 - 69
Copyright
Copyright © 2010 The Psychometric Society

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Footnotes

The authors would like to express their appreciation to Drs. Will Grove, Bob Pruzek, Scott Vrieze, and Steve Nydick for helpful comments on a draft of this article.

References

Bring, J. (1994). How to standardize regression coefficients. American Statistician, 48, 209–213.CrossRefGoogle Scholar
Campbell, J.P. (1974). A Monte Carlo approach to some problems inherent in multivariate prediction with special reference to multiple regression (Technical Report 2002). Personnel Training Research Program. Washington DC: Office of Navel Research.Google Scholar
Claudy, J.G. (1972). A comparison of five variable weighting procedures. Educational and Psychological Measurement, 32, 311–322.CrossRefGoogle Scholar
Dana, J., & Dawes, R.M. (2004). The superiority of simple alternatives to regression for social science predictions. Journal of Educational and Behavior Statistics, 29, 317–331.CrossRefGoogle Scholar
Davis, K.R., & Sauser, W.I. (1991). Effects of alternate weighting methods in a policy-capturing approach to job evaluation: A review and empirical investigation. Personnel Psychology, 44, 85–127.CrossRefGoogle Scholar
Dunnette, M.D., & Borman, W.C. (1979). Personnel selection and classification systems. Annual Review of Psychology, 30, 477–525.CrossRefGoogle Scholar
Einhorn, H.J., & Hogarth, R.M. (1975). Unit weighting schemes for decision making. Organizational Behavior and Human Performance, 13, 171–192.CrossRefGoogle Scholar
Ferguson, C.C. (1979). Intersections of ellipsoids and planes of arbitrary orientation and position. Mathematical Geology, 11, 329–336.CrossRefGoogle Scholar
Fishman, G.S. (1996). Monte Carlo: Theory, algorithms and applications, New York: Springer.CrossRefGoogle Scholar
Goldberg, L.R. (1972). Parameters of personality inventory construction and utilization: A comparison of prediction strategies and tactics. Multivariate Behavioral Research Monographs, 7(2).Google Scholar
Goldberger, A.S. (1968). Topics in regression analysis, New York: Macmillan.Google Scholar
Green, B.F. (1977). Parameter sensitivity in multivariate methods. Multivariate Behavioral Research, 12, 263–287.CrossRefGoogle ScholarPubMed
Greenland, S., Schlesselman, J.J., & Criqui, M.H. (1986). The fallacy of employing standardized regression coefficients and correlations as measures of effect. American Journal of Epidemiology, 123, 203–208.CrossRefGoogle ScholarPubMed
Hoerl, A.E., & Kennard, R.W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.CrossRefGoogle Scholar
Keren, G., & Newman, J.R. (1978). Additional considerations with regard to multiple regression and equal weighting. Organizational Behavior and Human Decision Processes, 22, 143–164.CrossRefGoogle Scholar
Koopman, R.F. (1988). On the sensitivity of a composite to its weights. Psychometrika, 53, 547–552.CrossRefGoogle Scholar
Laughlin, J.E. (1978). Comment on “Estimating coefficients in linear models: It don’t make no nevermind.”. Psychological Bulletin, 85, 247–253.CrossRefGoogle Scholar
Marks, M.R. (1966). Two kinds of regression weights which are better than betas in crossed samples. Paper presented at the meeting of the American Psychological Association, New York (September).Google Scholar
Marsaglia, G., & Olkin, I. (1984). Generating correlation matrices. SIAM: Journal of Scientific and Statistical Computing, 5, 470.Google Scholar
The Psychological Corporation (1997). WAIS-III WMS-III technical manual. San Antonio, TX: Author.Google Scholar
Pruzek, R.M., & Fredrick, B.C. (1978). Weighting procedures in linear models: Alternatives to least squares and limitations of equal weights. Psychological Bulletin, 85, 254–266.CrossRefGoogle Scholar
R Development Core Team (2007). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN:3-900051-07-0, URL:http://www.R-project.org.Google Scholar
Raju, N.S., Bilgic, R., Edwards, J.E., & Fleer, P.F. (1997). Methodology review: Estimation of population validity and cross-validity, and the use of equal weights in prediction. Applied Psychological Measurement, 21, 291–305.CrossRefGoogle Scholar
Rozeboom, W.W. (1979). Sensitivity of a linear composite of predictor items to differential item weighting. Psychometrika, 44, 289–296.CrossRefGoogle Scholar
Schmidt, F.L. (1971). The relative efficiency of regression and simple unit predictor weights in applied differential psychology. Educational and Psychological Measurement, 31, 699–714.CrossRefGoogle Scholar
Tatsuoka, M.M. (1988). Multivariate analysis, (2nd ed.). New York: Wiley.Google Scholar
Wainer, H. (1976). Estimating coefficients in linear models: It don’t make no nevermind. Psychological Bulletin, 83, 213–217.CrossRefGoogle Scholar
Wainer, H. (1978). On the sensitivity of regression and regressors. Psychological Bulletin, 85, 267–273.CrossRefGoogle Scholar
Wesman, A.G., & Bennett, G.K. (1959). Multiple regression vs. simple addition of scores in prediction of college grades. Education and Psychological Measurement, 19, 243–246.CrossRefGoogle Scholar