, Volume 84, Issue 1, pp 84–104 | Cite as

Frequentist Model Averaging in Structural Equation Modelling

  • Shaobo JinEmail author
  • Sebastian Ankargren


Model selection from a set of candidate models plays an important role in many structural equation modelling applications. However, traditional model selection methods introduce extra randomness that is not accounted for by post-model selection inference. In the current study, we propose a model averaging technique within the frequentist statistical framework. Instead of selecting an optimal model, the contributions of all candidate models are acknowledged. Valid confidence intervals and a \(\chi ^2\) test statistic are proposed. A simulation study shows that the proposed method is able to produce a robust mean-squared error, a better coverage probability, and a better goodness-of-fit test compared to model selection. It is an interesting compromise between model selection and the full model.


model selection post-selection inference coverage probability local asymptotic goodness-of-fit 



We would like to thank the reviewers for providing valuable comments. Shaobo Jin was partly supported by Vetenskapsrådet (Swedish Research Council) under the contract 2017-01175.

Supplementary material

11336_2018_9624_MOESM1_ESM.r (38 kb)
Supplementary material 1 (R 37 KB)


  1. Akaike, H. (1973). Information theory as an extension of the maximum likelihood principle. In B. N. Petrov & F. Csaki (Eds.), Second international symposium on information theory. Budapest: Akademiai Kiado.Google Scholar
  2. Ankargren, S., & Jin, S. (2018). On the least squares model averaging interval estimator. Communications in Statistics: Theory and Methods, 47, 118–132.CrossRefGoogle Scholar
  3. Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107, 238–246.CrossRefGoogle Scholar
  4. Berk, R., Brown, L., Buja, A., Zhang, K., & Zhao, L. (2013). Valid post-selection inference. Annals of Statistics, 41, 802–837.CrossRefGoogle Scholar
  5. Browne, M. W. (1974). Generalized least squares estimators in the analysis of covariance structures. South African Statistical Journal, 8, 1–24. Reprinted in 1977 in D. J. Aigner & A. S. Goldberger (Eds.). Latent variables in socioeconomic models (pp. 205–226). Amsterdam: North Holland.Google Scholar
  6. Browne, M. W. (1984). Asymptotically distribution-free methods in the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37, 62–83.CrossRefGoogle Scholar
  7. Browne, M. W. (1987). Robustness of statistical inference in factor analysis and related models. Biometrika, 74, 375–384.CrossRefGoogle Scholar
  8. Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–161). Newbury Park: Sage.Google Scholar
  9. Buckland, S. T., Burnham, K. P. K. P., & Augustin, H. (1997). Model selection: An integral part of inference. Biometrics, 53, 603–618.CrossRefGoogle Scholar
  10. Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference understanding AIC and BIC in model selection. Sociological Methods & Research, 33, 261–304.CrossRefGoogle Scholar
  11. Charkhi, A., Claeskens, G., & Hansen, B. E. (2016). Minimum mean square error model averaging in likelihood models. Statistica Sinica, 26, 809–840.Google Scholar
  12. Fletcher, D., & Dillingham, P. W. (2011). Model-averaged confidence intervals for factorial experiments. Computational Statistics & Data Analysis, 55, 3041–3048.CrossRefGoogle Scholar
  13. Fletcher, D., & Turek, D. (2011). Model-averaged profile likelihood intervals. Journal of Agricultural, Biological and Environmental Statistics, 17, 38–51.CrossRefGoogle Scholar
  14. Hansen, B. E. (2007). Least squares model averaging. Econometrica, 75, 1175–1189.CrossRefGoogle Scholar
  15. Hansen, B. E. (2014). Model averaging, asymptotic risk, and regression groups. Quantitative Economics, 5, 495–530.CrossRefGoogle Scholar
  16. Hansen, B. E., & Racine, J. S. (2012). Jackknife model averaging. Journal of Econometrics, 167, 38–46.CrossRefGoogle Scholar
  17. Hjort, N. L., & Claeskens, G. (2003a). Frequentist model average estimators. Journal of the American Statistical Association, 98, 879–899.CrossRefGoogle Scholar
  18. Hjort, N. L., & Claeskens, G. (2003b). Rejoinder. Journal of the American Statistical Association, 98, 938–945.CrossRefGoogle Scholar
  19. Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial (with discussion). Statistical Science, 14, 382–417.CrossRefGoogle Scholar
  20. Ishwaran, H., & Rao, J. S. (2003). Discussion. Journal of the American Statistical Association, 98, 922–925.CrossRefGoogle Scholar
  21. Kabaila, P. (1995). The effect of model selection on confidence regions and prediction regions. Econometric Theory, 11, 537–549.CrossRefGoogle Scholar
  22. Kabaila, P., & Leeb, H. (2006). On the large-sample minimal coverage probability of confidence intervals after model selection. Journal of the American Statistical Association, 101, 619–629.CrossRefGoogle Scholar
  23. Kabaila, P., Welsh, A. H., & Abeysekera, W. (2016). Model-averaged confidence intervals. Scandinavian Journal of Statistics, 43, 35–48.CrossRefGoogle Scholar
  24. Karatzoglou, A., Smola, A., Hornik, K., & Zeileis, A. (2004). kernlab—An S4 package for kernel methods in R. Journal of Statistical Software, 11, 1–20.CrossRefGoogle Scholar
  25. Kember, D., & Leung, D. Y. P. (2009). Development of a questionnaire for assessing students’ perceptions of the teaching and learning environment and its use in quality assurance. Learning Environments Research, 12, 15–29.CrossRefGoogle Scholar
  26. Kember, D., & Leung, D. Y. P. (2011). Disciplinary differences in student rating of teaching quality. Research in Higher Education, 52, 278–299.CrossRefGoogle Scholar
  27. Kim, J., & Pollard, D. (1990). Cube root asymptotics. The Annals of Statistics, 18, 191–219.CrossRefGoogle Scholar
  28. Knight, K., & Fu, W. (2000). Aasymptotic for lasso-type estimators. The Annals of Statistics, 28, 1356–1378.CrossRefGoogle Scholar
  29. Lee, W. W. S., Leung, D. Y. P., & Lo, K. C. H. (2013). Development of generic capabilities in teaching and learning eenvironment at associate degree level. In M. S. Khine (Ed.), Application of structural equation modeling in educational research and practice (pp. 169–184). Rotterdam: Sense Publishers.CrossRefGoogle Scholar
  30. Leeb, H., & Pötscher, B. M. (2005). Model selection and inference: Facts and fiction. Econometric Theory, 21, 21–59.CrossRefGoogle Scholar
  31. Liu, C.-A. (2015). Distribution theory of the least squares averaging estimator. Journal of Econometrics, 186, 142–159.CrossRefGoogle Scholar
  32. Liu, C.-A., & Kuo, B.-S. (2016). Model averaging in predictive regressions. The Econometrics Journal, 19, 203–231.CrossRefGoogle Scholar
  33. Liu, Q., & Okui, R. (2013). Heteroscedasticity-robust \(C_p\) model averaging. The Econometrics Journal, 16, 463–472.CrossRefGoogle Scholar
  34. MacCallum, R. C. (1986). Specification searches in covariance structure modeling. Quantittative Methods in Psychology, 100, 107–120.Google Scholar
  35. MacCallum, R. C., Roznowski, M., & Necowitz, L. B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Quantittative Methods in Psychology, 111, 490–504.Google Scholar
  36. Madigan, D., & Raftery, A. E. (1994). Model selection and accounting for model uncertainty in graphical models using Occam’s window. Journal of the American Statistical Association, 89, 1535–1546.CrossRefGoogle Scholar
  37. Magnus, J. R., & Neudecker, H. (1986). Symmetry, 0–1 matrices and Jacobians: A review. Econometric Theory, 2, 157–190.CrossRefGoogle Scholar
  38. Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
  39. Muthén, B., du Toit, S. H. C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes. Retrieved from Accessed 12 Sept 2013.
  40. Pötscher, B. M. (1991). Effects of model selection on inference. Econometric Theory, 7, 163–185.CrossRefGoogle Scholar
  41. Raftery, A. E., & Zheng, Y. (2003). Discussion: Performance of Bayesian model averaging. Journal of the American Statistical Association, 98, 931–938.CrossRefGoogle Scholar
  42. Sarris, W. E., Satorra, A., & Sorbom, D. (1987). The detection and correction of specification errors in structural equation models. Sociological Methodology, 17, 105–129.CrossRefGoogle Scholar
  43. Schomaker, M., & Heumann, C. (2011). Model averaging in factor analysis: An analysis of Olympic decathlon data. Journal of Quantitative Analysis in Sports, 7, Article 4.Google Scholar
  44. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.CrossRefGoogle Scholar
  45. Sörbom, D. (1989). Model modification. Psychometrika, 54, 371–384.CrossRefGoogle Scholar
  46. Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38, 1–10.CrossRefGoogle Scholar
  47. Turek, D., & Fletcher, D. (2012). Model-averaged Wald confidence intervals. Computational Statistics & Data Analysis, 56, 2809–2815.CrossRefGoogle Scholar
  48. Turlach, B. A., & Weingessel, A. (2013). quadprog: Functions to solve quadratic programming problems. R package version 1.5-5.Google Scholar
  49. Vanderbei, R. J. (1999). LOQO: An interior point code for quadratic programming. Optimization Methods and Software, 11, 451–484.CrossRefGoogle Scholar
  50. Wang, H., & Zhou, S. Z. F. (2013). Interval estimation by frequentist model averaging. Communications in Statistics: Theory and Methods, 42, 4342–4356.CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2018

Authors and Affiliations

  1. 1.Department of StatisticsUppsala UniversityUppsalaSweden

Personalised recommendations