Abstract
This paper details a Bayesian alternative to the use of least squares and equal weighting coefficients in regression. An equal weight prior distribution for the linear regression parameters is described with regard to the conditional normal regression model, and resulting posterior distributions for these parameters are detailed. Some interesting connections between this Bayesian procedure and several other methods for estimating optimal weighting coefficients are discussed. In addition, results are presented of a Monte Carlo investigation which compared the effectiveness of the Bayesian procedure relative to least squares, equal weight, ridge, and Bayesian exchangeability estimations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Reference notes
Fischer, G. W.Four methods for assessing multi-attribute utilities: An experimental validation. Engineering Psychology Laboratory University of Michigan, 1972.
Laughlin, J. E.Equal weighting coefficients as an alternative to least squares in regression. Unpublished manuscript. Columbia: Department of Psychology, University of South Carolina, 1978.
Ramsay, J. O. & Novick, M. R.PLU Robust Bayesian decision theory: Point estimation. Unpublished manuscript. McGill University, 1977.
References
Beckwith, N. E. & Lehmann, D. R. The importance of differential weights in multiple attribute models of consumer attitude.Journal of Marketing Research, 1973,10, 141–145.
Darlington, R. B. Reduced-variance regression.Psychological Bulletin, 1978,85, 1238–1255.
Dawes, R. M. & Corrigan, B. Linear models in decision making.Psychological Bulletin, 1974,81, 95–106.
Dempster, A. P., Schatzoff, M. & Wermuth, N. A simulation study of alternatives to ordinary least squares.Journal of the American Statistical Association, 1977,72, 77–91.
Einhorn, H. J. & Hogarth, R. M. Unit weighting schemes for decision making.Organizational Behavior and Human Performance, 1975,13, 171–192.
Hoerl, A. E. & Kennard, R. W. Ridge regression: Biased estimation for nonorthogonal problems.Technometrics, 1970,12, 55–67.
Hoerl, A. E., Kennard, R. W. & Baldwin, K. F. Ridge regression: some simulations.Communications in Statistics, 1975,4, 105–123.
Laughlin, J. E. Comments on “Estimating coefficients in linear models: It don't make no nevermind”.Psychological Bulletin, 1978,85, 247–253.
Lawshe, C. H. & Shucker, R. E. The relative efficiency of four test weighting methods in multiple regression.Educational and Psychological Measurement, 1959,19, 103–114.
Lehmann, D. R. Television show preference: Application of a choice model.Journal of Marketing Research, 1971,8, 47–55.
Lindley, D. V. & Smith, A. F. M. Bayes estimates for the linear model.Journal of the Royal Statistical Society, Series B, 1972,33, 1–41.
Novick, M. R. & Jackson, P. H.Statistical methods for educational and psychological research. New York: McGraw-Hill, 1974.
O'Hagan, A. On posterior joint and marginal modes.Biometrika, 1976,63, 329–333.
Price, B. Ridge regression: Application to nonexperimental data.Psychological Bulletin, 1977,84, 759–766.
Schmidt, F. L. The relative efficiency of regression and simple unit predictor weights in applied differential psychology.Educational and Psychological Measurement, 1971,31, 699–714.
Schmidt, F. L. The reliability of differences between linear regression weights in applied differential psychology.Educational and Psychological Measurement, 1972,32, 879–886.
Trattner, M. H. Comparison of three methods for assembling aptitude test batteries.Personnel Psychology, 1963,16, 221–232.
Tucker, L. R, Koopman, R. F. & Linn, R. L. Evaluation of factor analytic research procedures by means of simulated correlation matrices.Psychometrika, 1969,34, 421–459.
Wainer, H. Estimating coefficients in linear models: It don't make no nevermind.Psychological Bulletin, 1976,83, 213–217.
Wainer, H. On the Sensitivity of regression and regressors.Psychological Bulletin, 1978,85, 267–273.
Wainer, H. & Thissen, D. Three steps toward robust regression.Psychometrika, 1976,41, 9–34.
Wesman, A. G. & Bennett, G. K. Multiple regression vs. simple addition of scores in prediction of college grades.Educational and Psychological Measurement, 1959,19, 243–246.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Laughlin, J.E. A bayesian alternative to least squares and equal weighting coefficients in regression. Psychometrika 44, 271–288 (1979). https://doi.org/10.1007/BF02294693
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294693