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The Normal-Theory and Asymptotic Distribution-Free (ADF) Covariance Matrix of Standardized Regression Coefficients: Theoretical Extensions and Finite Sample Behavior

Published online by Cambridge University Press:  01 January 2025

Jeff A. Jones  and
Niels G. Waller
Jeff A. Jones
Affiliation:
University of Minnesota-Twin Cities
Niels G. Waller*
Affiliation:
University of Minnesota-Twin Cities
*
Requests for reprints should be sent to Niels G. Waller, Department of Psychology, University of Minnesota, 75 East River Road, Minneapolis, MN, 55455, USA. E-mail: nwaller@umn.edu
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Abstract

Yuan and Chan (Psychometrika, 76, 670–690, 2011) recently showed how to compute the covariance matrix of standardized regression coefficients from covariances. In this paper, we describe a method for computing this covariance matrix from correlations. Next, we describe an asymptotic distribution-free (ADF; Browne in British Journal of Mathematical and Statistical Psychology, 37, 62–83, 1984) method for computing the covariance matrix of standardized regression coefficients. We show that the ADF method works well with nonnormal data in moderate-to-large samples using both simulated and real-data examples. R code (R Development Core Team, 2012) is available from the authors or through the Psychometrika online repository for supplementary materials.

Keywords

standardized regression coefficientsmultiple regressionADFconfidence intervals
Type
Original Paper
Information
Psychometrika , Volume 80 , Issue 2 , June 2015 , pp. 365 - 378
Copyright
Copyright © 2013 The Psychometric Society

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Footnotes

Electronic Supplementary Material The online version of this article (doi:10.1007/s11336-013-9380-y) contains supplementary material, which is available to authorized users.

References

Abadir, K.M., & Magnus, J.R. (2005). Matrix algebra. New York: Cambridge University Press.CrossRefGoogle Scholar
Algina, J., & Moulder, B.C. (2001). Sample sizes for confidence intervals on the increase in the squared multiple correlation coefficient. Educational and Psychological Measurement, 61, 633–649.CrossRefGoogle Scholar
Astin, H.S. (1967). Career development during the high school years. Journal of Counseling Psychology, 14, 94–98.CrossRefGoogle Scholar
Austin, J.T., & Hanisch, K.A. (1990). Occupational attainment as a function of abilities and interests: a longitudinal analysis using project talent data. Journal of Applied Psychology, 75(1), 77–86.CrossRefGoogle ScholarPubMed
Becker, B., & Wu, M. (2007). The synthesis of regression slopes in meta-analysis. Statistical Science, 22, 414–429.CrossRefGoogle Scholar
Bentler, P., & Lee, S.Y. (1983). Covariance structures under polynomial constraints: applications to correlation and alpha-type structural models. Journal of Educational and Behavioral Statistics, 8(3), 207.CrossRefGoogle Scholar
Bollen, K.A., & Stine, R. (1990). Direct and indirect effects: classical and bootstrap estimates of variability. Sociological Methodology, 20, 115–140.CrossRefGoogle Scholar
Boomsma, A., & Hoogland, J.J. (2001). The robustness of LISREL modeling revisited. In Cudeck, R., Du Toit, S., & Sorbom, D. (Eds.), Structural equation modeling: present and future (pp. 139–168). Chicago: Scientific Software International.Google Scholar
Bring, J. (1994). How to standardize regression coefficients. American Statistician, 48, 209–213.CrossRefGoogle Scholar
Browne, M.W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24.Google Scholar
Browne, M. (1982). Covariance structures. In Hawkins, D.M. (Ed.), Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical & Statistical Psychology, 37, 62–83.CrossRefGoogle ScholarPubMed
Browne, M.W., Mels, G., & Cowan, M. (1994). Path analysis: Ramona: Systat for DOS advanced applications. SOFTWARE, Version 6:167–224.Google Scholar
Browne, M.W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and Its Applications, 82, 169–176.CrossRefGoogle Scholar
Card, J.J. (1987). Epidemiology of post-traumatic stress disorder in a national cohort of Vietnam veterans. Journal of Clinical Psychology, 43(1), 6–17.3.0.CO;2-X>CrossRefGoogle Scholar
Casella, G., & Berger, R.L. (2001). Statistical inference. Belmont: Wadsworth.Google Scholar
Cohen, J., Cohen, P., West, S.G., & Aiken, L.S. (2003). Applied multiple regression/correlation analysis for the behavioral sciences (3rd ed.). Mahwah: Lawrence Erlbaum Associates, Inc..Google Scholar
Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317–327.CrossRefGoogle Scholar
Curran, P.J. (1994). The robustness of confirmatory factor analysis to model misspecification and violations of normality. Unpublished Doctoral Dissertation, Arizona State University.Google Scholar
Efron, B. (1982). The jackknife, the bootstrap and other resampling plans. CBMS-NSF regional conference series in applied mathematics. Philadelphia: SIAM.Google Scholar
Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397), 171–185.CrossRefGoogle Scholar
Efron, B., & Tibshirani, R.J. (1994). An introduction to the bootstrap. Boca Raton: Chapman & Hall/CRC.CrossRefGoogle Scholar
Ferguson, T.S. (1996). A course in large sample theory. New York: Chapman & Hall.CrossRefGoogle Scholar
Fleishman, A.I. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521–532.CrossRefGoogle Scholar
Greene, W.H. (2003). Econometric analysis (5th ed.). Upper Saddle River: Prentice-Hall.Google Scholar
Greenland, S., Maclure, M., Schlesselman, J.J., Poole, C., & Morgenstern, H. (1991). Standardized regression coefficients: a further critique and review of some alternatives. Epidemiology, 2, 387–392.CrossRefGoogle ScholarPubMed
Harris, R.J. (2001). A primer on multivariate statistics (3rd ed.). Mahwah: Lawrence Erlbaum Associates.CrossRefGoogle Scholar
Hays, W.L. (1994). Statistics (5th ed.). Worth Fort: Harcourt Brace College Publisher.Google Scholar
Hsu, P.L. (1949). The limiting distribution of functions of sample means and application to testing hypotheses. In Neyman, J. (Ed.), Proceedings of the first Berkeley symposium on mathematical statistics and probability (pp. 359–402). Berkeley: Univ. of California Press.Google Scholar
Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression. Biometrika, 11, 185–190.CrossRefGoogle Scholar
Jamshidian, M., & Bentler, P.M. (2000). Improved standard errors of standardized parameters in covariance structure models: implications for construction explication. In Goffin, R.D., & Helmes, E. (Eds.), Problems and solutions in human assessment (pp. 73–94). Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Jones, J.A., & Waller, N.G. (in press). Computing confidence intervals for standardized regression coefficients. Psychological Methods. doi:https://doi.org/10.1037/a0033269.CrossRefGoogle Scholar
Jones, J.A., & Waller, N.G. (2013). The normal-theory and asymptotic distribution-free (ADF) covariance matrix of standardized regression coefficients: theoretical extensions and finite sample behavior. Minneapolis: University of Minnesota. Retrieved from http://www.psych.umn.edu/faculty/waller/downloads/techreports/TR052913.pdf.Google ScholarPubMed
Kano, Y., Berkane, M., & Bentler, P.M. (1993). Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations. Journal of the American Statistical Association, 88(421), 135–143.CrossRefGoogle Scholar
Kelley, K. (2007). Confidence intervals for standardized effect sizes: theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24.CrossRefGoogle Scholar
Kelley, K., & Maxwell, S.E. (2003). Sample size for multiple regression: obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8, 305–321.CrossRefGoogle Scholar
Kim, R.S. (2011). Standardized regression coefficients as indices of effect sizes in meta analysis. Unpublished Doctoral Dissertation, Florida State University, Tallahassee, Florida.Google Scholar
Kwan, J.L., & Chan, W. (2011). Comparing standardized coefficients in structural equation modeling: a model reparameterization approach. Behavior Research Methods, 43(3), 730–745.CrossRefGoogle Scholar
King, G. (1985). How not to lie with statistics: avoiding common mistakes in quantitative political science. American Journal of Political Science, 30, 666–687.CrossRefGoogle Scholar
Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530.CrossRefGoogle Scholar
Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166.CrossRefGoogle Scholar
Muthén, B., & Muthén, L. (2012). Mplus users guide (7th ed.). Los Angeles: Muthen & Muthen.Google Scholar
Nel, D.G. (1985). A matrix derivation of the asymptotic covariance matrix of sample correlation coefficients. Linear Algebra and Its Applications, 67, 137–145.CrossRefGoogle Scholar
Oehlert, G.W. (1992). A note on the delta method. American Statistician, 46, 27–29.CrossRefGoogle Scholar
Ogasawara, H. (2007). Asymptotic expansion and asymptotic robustness of the normal-theory estimators in the random regression model. Journal of Statistical Computation and Simulation, 77(10), 821–838.CrossRefGoogle Scholar
Ogasawara, H. (2008). Asymptotic expansion in reduced rank regression under normality and nonnormality. Communications in Statistics. Theory and Methods, 37, 1051–1070.CrossRefGoogle Scholar
Olkin, I., & Finn, J.D. (1995). Correlations redux. Psychological Bulletin, 118, 155–164.CrossRefGoogle Scholar
Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. In Ideka, S. (Ed.), Essays in probability and statistics (pp. 235–251). Tokyo: Shinko, Tsusho Co., Ltd..Google Scholar
Pearson, K., & Filon, L. (1898). On the probable errors of frequency constants and on the influence of random selection on variation and correlation. Philosophical Transactions of the Royal Society of London, 191, 229–311.Google Scholar
Peterson, R., & Brown, S. (2005). On the use of beta coefficients in meta-analysis. Journal of Applied Psychology, 90, 175–181.CrossRefGoogle ScholarPubMed
R Development Core Team (2012). R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. http://www.R-project.org/.Google Scholar
Rencher, A.C. (2008). Linear models in statistics (2nd ed.). Hoboken: Wiley.Google Scholar
Steiger, J.H. (1995). Structural equation modeling (SEPATH). Tulsa: Statsoft Inc..Google Scholar
Steiger, J.H., & Fouladi, R.T. (1997). Noncentrality interval estimation and the evaluation of statistical models. In Harlow, L.L., Mulaik, S.A., & Steiger, J.H. (Eds.), What if there were no significance tests? (pp. 221–257). London: Taylor and Francis.Google Scholar
Steiger, J.H., & Hakstian, A. (1980). The asymptotic distribution of elements of a correlation matrix (Technical Report No. 80 3 (May)). Institute of Applied Mathematics and Statistics, University of British Columbia.Google Scholar
Steiger, J., & Hakstian, A. (1982). The asymptotic distribution of elements of a correlation matrix: theory and application. British Journal of Mathematical & Statistical Psychology, 35, 208–215.CrossRefGoogle Scholar
Tukey, J.W. (1954). Causation, regression, and path analysis. In Kempthorne, O., Bancroft, T.A., Gowen, J.W., & Lush, J.L. (Eds.), Statistics and mathematics in biology (pp. 35–66). Ames: Iowa State College Press.Google Scholar
Vale, C.D., & Maurelli, V.A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48, 465–471.CrossRefGoogle Scholar
Waller, N.G. (2011). The geometry of enhancement in multiple regression. Psychometrika, 76, 634–649.CrossRefGoogle ScholarPubMed
Waller, N.G., & Jones, J.A. (2010). Correlation weights in multiple regression. Psychometrika, 75(1), 58–69.CrossRefGoogle Scholar
Waller, N.G., & Jones, J.A. (2011). Investigating the performance of alternate regression weights by studying all possible criteria in regression models with a fixed set of predictors. Psychometrika, 76, 410–439.CrossRefGoogle Scholar
Waller, N.G., Underhill, M., & Kaiser, H.A. (1999). A method for generating simulated plasmodes and artificial test clusters with user-defined shape, size, and orientation. Multivariate Behavioral Research, 34, 123–142.CrossRefGoogle ScholarPubMed
West, S.G., Aiken, L.S., Wu, W., & Taylor, A.B. (2007). Multiple regression: applications of the basics and beyond in personality research. In Robins, R., Fraley, R.C., & Krueger, R.F. (Eds.), Handbook of research methods in personality psychology (pp. 573–601). New York: Guilford.Google Scholar
Wise, L.L., McLaughlin, D.H., & Steel, L. (1979). The project TALENT data bank handbook. Palo Alto: American Institutes for Research.Google Scholar
Yuan, K.H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76, 670–690.CrossRefGoogle ScholarPubMed
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The Normal-Theory and Asymptotic Distribution-Free (ADF) Covariance Matrix of Standardized Regression Coefficients: Theoretical Extensions and Finite Sample Behavior
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