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 R 2 (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, R xx , and (2) the orientation of the correlation weights to the latent vectors of R xx .
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References
Bring, J. (1994). How to standardize regression coefficients. American Statistician, 48, 209–213.
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.
Claudy, J.G. (1972). A comparison of five variable weighting procedures. Educational and Psychological Measurement, 32, 311–322.
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.
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.
Dunnette, M.D., & Borman, W.C. (1979). Personnel selection and classification systems. Annual Review of Psychology, 30, 477–525.
Einhorn, H.J., & Hogarth, R.M. (1975). Unit weighting schemes for decision making. Organizational Behavior and Human Performance, 13, 171–192.
Ferguson, C.C. (1979). Intersections of ellipsoids and planes of arbitrary orientation and position. Mathematical Geology, 11, 329–336.
Fishman, G.S. (1996). Monte Carlo: Theory, algorithms and applications. New York: Springer.
Goldberg, L.R. (1972). Parameters of personality inventory construction and utilization: A comparison of prediction strategies and tactics. Multivariate Behavioral Research Monographs, 7(2).
Goldberger, A.S. (1968). Topics in regression analysis. New York: Macmillan.
Green, B.F. (1977). Parameter sensitivity in multivariate methods. Multivariate Behavioral Research, 12, 263–287.
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.
Hoerl, A.E., & Kennard, R.W. (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.
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.
Koopman, R.F. (1988). On the sensitivity of a composite to its weights. Psychometrika, 53, 547–552.
Laughlin, J.E. (1978). Comment on “Estimating coefficients in linear models: It don’t make no nevermind.” Psychological Bulletin, 85, 247–253.
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).
Marsaglia, G., & Olkin, I. (1984). Generating correlation matrices. SIAM: Journal of Scientific and Statistical Computing, 5, 470.
The Psychological Corporation (1997). WAIS-III WMS-III technical manual. San Antonio, TX: Author.
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.
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
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.
Rozeboom, W.W. (1979). Sensitivity of a linear composite of predictor items to differential item weighting. Psychometrika, 44, 289–296.
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.
Tatsuoka, M.M. (1988). Multivariate analysis (2nd ed.). New York: Wiley.
Wainer, H. (1976). Estimating coefficients in linear models: It don’t make no nevermind. Psychological Bulletin, 83, 213–217.
Wainer, H. (1978). On the sensitivity of regression and regressors. Psychological Bulletin, 85, 267–273.
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.
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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.
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Waller, N.G., Jones, J.A. Correlation Weights in Multiple Regression. Psychometrika 75, 58–69 (2010). https://doi.org/10.1007/s11336-009-9127-y
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DOI: https://doi.org/10.1007/s11336-009-9127-y