Psychometrika

, Volume 80, Issue 2, pp 365–378 | Cite as

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

Article

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.

Key words

standardized regression coefficients multiple regression ADF, confidence intervals 

Supplementary material

11336_2013_9380_MOESM1_ESM.pdf (4.7 mb)
(PDF 44 kB)
11336_2013_9380_MOESM2_ESM.pdf (55 kb)
(PDF 133 kB)

References

  1. Abadir, K.M., & Magnus, J.R. (2005). Matrix algebra. New York: Cambridge University Press. CrossRefGoogle Scholar
  2. 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
  3. Astin, H.S. (1967). Career development during the high school years. Journal of Counseling Psychology, 14, 94–98. CrossRefGoogle Scholar
  4. 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. CrossRefPubMedGoogle Scholar
  5. Becker, B., & Wu, M. (2007). The synthesis of regression slopes in meta-analysis. Statistical Science, 22, 414–429. CrossRefGoogle Scholar
  6. 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
  7. Bollen, K.A., & Stine, R. (1990). Direct and indirect effects: classical and bootstrap estimates of variability. Sociological Methodology, 20, 115–140. CrossRefGoogle Scholar
  8. Boomsma, A., & Hoogland, J.J. (2001). The robustness of LISREL modeling revisited. In R. Cudeck, S. Du Toit, & D. Sorbom (Eds.), Structural equation modeling: present and future (pp. 139–168). Chicago: Scientific Software International. Google Scholar
  9. Bring, J. (1994). How to standardize regression coefficients. American Statistician, 48, 209–213. Google Scholar
  10. Browne, M.W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24. Google Scholar
  11. Browne, M. (1982). Covariance structures. In D.M. Hawkins (Ed.), Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  12. Browne, M.W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical & Statistical Psychology, 37, 62–83. CrossRefGoogle Scholar
  13. Browne, M.W., Mels, G., & Cowan, M. (1994). Path analysis: Ramona: Systat for DOS advanced applications. SOFTWARE, Version 6:167–224. Google Scholar
  14. 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
  15. 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. CrossRefPubMedGoogle Scholar
  16. Casella, G., & Berger, R.L. (2001). Statistical inference. Belmont: Wadsworth. Google Scholar
  17. 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
  18. Cudeck, R. (1989). Analysis of correlation matrices using covariance structure models. Psychological Bulletin, 105, 317–327. CrossRefGoogle Scholar
  19. 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
  20. Efron, B. (1982). The jackknife, the bootstrap and other resampling plans. In CBMS-NSF regional conference series in applied mathematics (Vol. 38). Philadelphia: SIAM. Google Scholar
  21. Efron, B. (1987). Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397), 171–185. CrossRefGoogle Scholar
  22. Efron, B., & Tibshirani, R.J. (1994). Monographs on statistics & applied probability. An introduction to the bootstrap. Boca Raton: Chapman & Hall/CRC. Google Scholar
  23. Ferguson, T.S. (1996). A course in large sample theory. New York: Chapman & Hall. CrossRefGoogle Scholar
  24. Fleishman, A.I. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521–532. CrossRefGoogle Scholar
  25. Greene, W.H. (2003). Econometric analysis (5th ed.). Upper Saddle River: Prentice-Hall. Google Scholar
  26. 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. CrossRefPubMedGoogle Scholar
  27. Harris, R.J. (2001). A primer on multivariate statistics (3rd ed.). Mahwah: Lawrence Erlbaum Associates. Google Scholar
  28. Hays, W.L. (1994). Statistics (5th ed.). Worth Fort: Harcourt Brace College Publisher. Google Scholar
  29. Hsu, P.L. (1949). The limiting distribution of functions of sample means and application to testing hypotheses. In J. Neyman (Ed.), Proceedings of the first Berkeley symposium on mathematical statistics and probability (pp. 359–402). Berkeley: Univ. of California Press. Google Scholar
  30. Isserlis, L. (1916). On certain probable errors and correlation coefficients of multiple frequency distributions with skew regression. Biometrika, 11, 185–190. CrossRefGoogle Scholar
  31. Jamshidian, M., & Bentler, P.M. (2000). Improved standard errors of standardized parameters in covariance structure models: implications for construction explication. In R.D. Goffin & E. Helmes (Eds.), Problems and solutions in human assessment (pp. 73–94). Dordrecht: Kluwer Academic. CrossRefGoogle Scholar
  32. Jones, J.A., & Waller, N.G. (in press). Computing confidence intervals for standardized regression coefficients. Psychological Methods. doi:10.1037/a0033269.
  33. 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 Scholar
  34. 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. Google Scholar
  35. Kelley, K. (2007). Confidence intervals for standardized effect sizes: theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. CrossRefGoogle Scholar
  36. 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. CrossRefPubMedGoogle Scholar
  37. 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
  38. Kwan, J.L., & Chan, W. (2011). Comparing standardized coefficients in structural equation modeling: a model reparameterization approach. Behavior Research Methods, 43(3), 730–745. CrossRefPubMedGoogle Scholar
  39. 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
  40. Mardia, K.V. (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 57, 519–530. CrossRefGoogle Scholar
  41. Micceri, T. (1989). The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105, 156–166. CrossRefGoogle Scholar
  42. Muthén, B., & Muthén, L. (2012). Mplus users guide (7th ed.). Los Angeles: Muthen & Muthen. Google Scholar
  43. 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
  44. Oehlert, G.W. (1992). A note on the delta method. American Statistician, 46, 27–29. Google Scholar
  45. 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
  46. Ogasawara, H. (2008). Asymptotic expansion in reduced rank regression under normality and nonnormality. Communications in Statistics. Theory and Methods, 37, 1051–1070. CrossRefGoogle Scholar
  47. Olkin, I., & Finn, J.D. (1995). Correlations redux. Psychological Bulletin, 118, 155–164. CrossRefGoogle Scholar
  48. Olkin, I., & Siotani, M. (1976). Asymptotic distribution of functions of a correlation matrix. In S. Ideka (Ed.), Essays in probability and statistics (pp. 235–251). Tokyo: Shinko, Tsusho Co., Ltd. Google Scholar
  49. 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. CrossRefGoogle Scholar
  50. Peterson, R., & Brown, S. (2005). On the use of beta coefficients in meta-analysis. Journal of Applied Psychology, 90, 175–181. CrossRefPubMedGoogle Scholar
  51. R Development Core Team (2012). R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing. ISBN 3-900051-07-0. http://www.R-project.org/. Google Scholar
  52. Rencher, A.C. (2008). Linear models in statistics (2nd ed.). Hoboken: Wiley. Google Scholar
  53. Steiger, J.H. (1995). Structural equation modeling (SEPATH). Tulsa: Statsoft Inc. Google Scholar
  54. Steiger, J.H., & Fouladi, R.T. (1997). Noncentrality interval estimation and the evaluation of statistical models. In L.L. Harlow, S.A. Mulaik, & J.H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). London: Taylor and Francis. Google Scholar
  55. 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
  56. 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
  57. Tukey, J.W. (1954). Causation, regression, and path analysis. In O. Kempthorne, T.A. Bancroft, J.W. Gowen, & J.L. Lush (Eds.), Statistics and mathematics in biology (pp. 35–66). Ames: Iowa State College Press. Google Scholar
  58. Vale, C.D., & Maurelli, V.A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48, 465–471. CrossRefGoogle Scholar
  59. Waller, N.G. (2011). The geometry of enhancement in multiple regression. Psychometrika, 76, 634–649. CrossRefGoogle Scholar
  60. Waller, N.G., & Jones, J.A. (2010). Correlation weights in multiple regression. Psychometrika, 75(1), 58–69. CrossRefGoogle Scholar
  61. 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
  62. 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 Scholar
  63. West, S.G., Aiken, L.S., Wu, W., & Taylor, A.B. (2007). Multiple regression: applications of the basics and beyond in personality research. In R. Robins, R.C. Fraley, & R.F. Krueger (Eds.), Handbook of research methods in personality psychology (pp. 573–601). New York: Guilford. Google Scholar
  64. Wise, L.L., McLaughlin, D.H., & Steel, L. (1979). The project TALENT data bank handbook. Palo Alto: American Institutes for Research. Google Scholar
  65. Yuan, K.H., & Chan, W. (2011). Biases and standard errors of standardized regression coefficients. Psychometrika, 76, 670–690. CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2013

Authors and Affiliations

  1. 1.Department of PsychologyUniversity of MinnesotaMinneapolisUSA

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