Assessing the Size of Model Misfit in Structural Equation Models

Abstract

When a statistically significant mean difference is found, the magnitude of the difference is judged qualitatively using an effect size such as Cohen’s d. In contrast, in a structural equation model (SEM), the result of the statistical test of model fit is often disregarded if significant, and inferences are drawn using “close” models retained based on point estimates of sample statistics (goodness-of-fit indices). However, when a SEM cannot be retained using a test of exact fit, all substantive inferences drawn from it are suspect. It is therefore important to determine the size of the model misfit. Standardized residual covariances and residual correlations provide standardized effect sizes of the misfit of SEM models. They can be summarized using the Standardized Root Mean Squared Residual (SRMSR) and the Correlation Root Mean Squared Residual (CRMSR) which can be used as overall effect sizes of the misfit. Statistical theory is provided that allows the construction of confidence intervals and tests of close fit based on the SRMSR and CRMSR. It is hoped that the use of standardized effect sizes of misfit will help reconcile current practices in SEM and elsewhere in statistics.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    I am indebted to JBS Steenkamp for providing the data used in this example.

References

  1. Alden, D. L., Steenkamp, J. B. E. M., & Batra, R. (2006). Consumer attitudes toward marketplace globalization: Structure, antecedents and consequences. International Journal of Research in Marketing, 23(3), 227–239. doi:10.1016/j.ijresmar.2006.01.010.

    Article  Google Scholar 

  2. Barrett, P. (2007). Structural equation modelling: Adjudging model fit. Personality and Individual Differences, 42(5), 815–824. doi:10.1016/j.paid.2006.09.018.

    Article  Google Scholar 

  3. Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107(2), 238–246. doi:10.1037/0033-2909.107.2.238.

    Article  PubMed  Google Scholar 

  4. Bentler, P. M. (1995). EQS 5 [Computer Program]. Encino, CA: Multivariate Software Inc.

  5. Bollen, K. A. (1989). Structural equations with latent variables. New York: Wiley.

    Google Scholar 

  6. Bollen, K. A., & Arminger, G. (1991). Observational residuals in factor analysis and structural equation models. Sociological Methodology, 21, 235. doi:10.2307/270937.

    Article  Google Scholar 

  7. Bollen, K. A., & Long, J. S. (1993). Testing structural equation models. Newbury Park, CA: Sage.

    Google Scholar 

  8. Browne, M. W. (2000). Cross-validation methods. Journal of Mathematical Psychology, 44(1), 108–132. doi:10.1006/jmps.1999.1279.

    Article  PubMed  Google Scholar 

  9. Browne, M. W. (1982). Covariance structures. In D. M. Hawkins (Ed.), Topics in applied multivariate analysis (pp. 72–141). Cambridge: Cambridge University Press.

    Google Scholar 

  10. Browne, M. W. (1984). Asymptotically distribution-free methods for the analysis of covariance structures. British Journal of Mathematical and Statistical Psychology, 37(1), 62–83. doi:10.1111/j.2044-8317.1984.tb00789.x.

    Article  PubMed  Google Scholar 

  11. Browne, M. W., & Arminger, G. (1995). Specification and estimation of mean- and covariance-structure models. In G. Arminger, C. C. Clogg, & M. E. Sobel (Eds.), Handbook of statistical modeling for the social and behavioral sciences (pp. 185–249). New York: Plenum.

    Google Scholar 

  12. Browne, M. W., & Cudeck, R. (1989). Single sample cross-validation indices for covariance structures. Multivariate Behavioral Research, 24(4), 445–455. doi:10.1207/s15327906mbr2404_4.

    Article  PubMed  Google Scholar 

  13. 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. 136–162). Newbury Park, CA: Sage.

    Google Scholar 

  14. Chen, F., Curran, P. J., Bollen, K. A., Kirby, J., & Paxton, P. (2008). An empirical evaluation of the use of fixed cutoff points in rmsea test statistic in structural equation models. Sociological Methods & Research, 36(4), 462–494. doi:10.1177/0049124108314720.

    Article  Google Scholar 

  15. Coffman, D., & Millsap, R. (2006). Evaluating latent growth curve models using individual fit statistics. Structural Equation Modeling, 13(1), 37–41. doi:10.1207/s15328007sem1301_1.

    Article  Google Scholar 

  16. Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.

    Google Scholar 

  17. Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2003). Applied multiple regression/correlation analysis for behavioral sciences (3rd ed.). Mahwah, NJ: Erlbaum.

    Google Scholar 

  18. Edwards, J. R., & Lambert, L. S. (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological Methods, 12(1), 1–22. doi:10.1037/1082-989X.12.1.1.

    Article  PubMed  Google Scholar 

  19. Hildreth, L. (2013). Residual analysis for structural equation modeling. Ph.D. dissertation, Statistics Department, Iowa State University.

  20. Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55. doi:10.1080/10705519909540118.

    Article  Google Scholar 

  21. Jöreskog, K. G. (1993). Testing structural equation models. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 294–316). Newbury Park, CA: Sage.

    Google Scholar 

  22. Jöreskog, K. G., & Sörbom, D. (1981). Analysis of linear structural relationships by maximum likelihood and least squares methods. Research report 81-8, Uppsala, Sweden.

  23. Jöreskog, K. G., & Sörbom, D. (1988). LISREL 7. A guide to the program and applications (2nd ed.). Chicago, IL: International Education Services.

    Google Scholar 

  24. Kenny, D. A., & McCoach, D. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling: A Multidisciplinary Journal, 10(3), 333–351. doi:10.1207/S15328007SEM1003_1.

    Article  Google Scholar 

  25. Lee, S. Y., & Jennrich, R. I. (1979). A study of algorithms for covariance structure analysis with specific comparisons using factor analysis. Psychometrika, 44(1), 99–113. doi:10.1007/BF02293789.

    Article  Google Scholar 

  26. MacCallum, R. C., Wegener, D. T., Uchino, B. N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114(1), 185–199. doi:10.1037/0033-2909.114.1.185.

    Article  PubMed  Google Scholar 

  27. Marsh, H., Hau, K., & Grayson, D. (2005). Goodness of fit in structural equation models. In A. Maydeu-Olivares & J. J. Mcardle (Eds.), Contemporary psychometrics: A Festschrift for Roderick P. McDonald (pp. 275–340). Mahwah, NJ: Erlbaum.

    Google Scholar 

  28. Marsh, H. W., Hau, K., & Wen, Z. (2004). In search of golden rules: Comment on hypothesis-testing approaches to setting cutoff values for fit indexes and dangers in overgeneralizing Hu and Bentler’s (1999) findings. Structural Equation Modeling: A Multidisciplinary Journal, 11(3), 320–341. doi:10.1207/s15328007sem1103_2.

    Article  Google Scholar 

  29. Maydeu-Olivares, A. (2006). Limited information estimation and testing of discretized multivariate normal structural models. Psychometrika, 71(1), 57–77. doi:10.1007/s11336-005-0773-4.

    Article  Google Scholar 

  30. Maydeu-Olivares, A. (2015). Evaluating fit in IRT models. In S. P. Reise & D. A. Revicki (Eds.), Handbook of item response theory modeling: Applications to typical performance assessment (pp. 111–127). New York: Routledge.

    Google Scholar 

  31. Maydeu-Olivares, A., & Shi, D. (2017). Effect sizes of model misfit in structural equation models: Standardized residual covariances and residual correlations. Methodology: European Journal of Research Methods for the Behavioral and Social Sciences.

  32. McDonald, R. P., & Ho, M.-H. R. (2002). Principles and practice in reporting structural equation analyses. Psychological Methods, 7(1), 64–82. doi:10.1037/1082-989X.7.1.64.

    Article  PubMed  Google Scholar 

  33. Muthén, B. (1984). A general structural equation model with dichotomous, ordered categorical, and continuous latent variable indicators. Psychometrika, 49(1), 115–132. doi:10.1007/BF02294210.

  34. Ogasawara, H. (2001). Standard errors of fit indices using residuals in structural equation modeling. Psychometrika, 66(3), 421–436. doi:10.1007/BF02294443.

    Article  Google Scholar 

  35. Raykov, T. (2000). On sensitivity of structural equation modeling to latent relation misspecifications. Structural Equation Modeling, 7(4), 596–607. doi:10.1207/S15328007SEM0704_4.

    Article  Google Scholar 

  36. Reise, S. P., Scheines, R., Widaman, K. F., & Haviland, M. G. (2012). Multidimensionality and structural coefficient bias in structural equation modeling: A bifactor perspective. Educational and Psychological Measurement, 73(1), 5–26. doi:10.1177/0013164412449831.

    Article  Google Scholar 

  37. Rosseel, Y. (2012). lavaan: An R package for structural equation modeling. Journal of Statistical Software, 48(2), 1–36. doi:10.18637/jss.v048.i02.

    Article  Google Scholar 

  38. Saris, W. E., Satorra, A., & van der Veld, W. M. (2009). Testing structural equation models or detection of misspecifications? Structural Equation Modeling: A Multidisciplinary Journal, 16(4), 561–582. doi:10.1080/10705510903203433.

    Article  Google Scholar 

  39. Satorra, A. (1989). Alternative test criteria in covariance structure analysis: A unified approach. Psychometrika, 54(1), 131–151. doi:10.1007/BF02294453.

    Article  Google Scholar 

  40. Satorra, A. (2015). A comment on a paper by H. Wu and M. W. Browne (2014). Psychometrika, 80(3), 613–618. doi:10.1007/s11336-015-9455-z.

    Article  PubMed  Google Scholar 

  41. Satorra, A., & Bentler, P. M. (1994). Corrections to test statistics and standard errors in covariance structure analysis. In A. Von Eye & C. C. Clogg (Eds.), Latent variable analysis. Applications for developmental research (pp. 399–419). Thousand Oaks, CA: Sage.

    Google Scholar 

  42. Savalei, V. (2012). The relationship between root mean square error of approximation and model misspecification in confirmatory factor analysis models. Educational and Psychological Measurement, 72(6), 910–932. doi:10.1177/0013164412452564.

    Article  Google Scholar 

  43. Schmidt, F., & Hunter, J. (1997). Eight common but false objections to the discontinuation of significance testing in the analysis of research data. In L. L. Harlow, S. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests (pp. 37–64). Mahwah, NJ: Lawrence Erlbaum.

    Google Scholar 

  44. Schott, J. R. (1997). Matrix analysis for statistics. New York: Wiley.

    Google Scholar 

  45. Steiger, J. H. (1989). EzPATH: A supplementary module for SYSTAT and SYGRAPH [Computer program manual]. Evanston, IL: Systat Inc.

  46. Steiger, J. H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25(2), 173–180. doi:10.1207/s15327906mbr2502_4.

    Article  PubMed  Google Scholar 

  47. Steiger, J. H. (2000). Point estimation, hypothesis testing, and interval estimation using the RMSEA: Some comments and a reply to Hayduck and Glaser. Structural Equation Modeling, 7(2), 149–162. doi:10.1207/S15328007SEM0702_1.

    Article  Google Scholar 

  48. Steiger, J. H., & Fouladi, R. (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). Mahwah, NJ: Erlbaum.

    Google Scholar 

  49. Steiger, J. H., & Lind, J. C. (1980). Statistically-based tests for the number of common factors. Paper presented at the Annual Meeting of the Psychometric Society, Iowa.

  50. Tanaka, J. S., & Huba, G. J. (1989). A general coefficient of determination for covariance structure models under arbitrary GLS estimation. British Journal of Mathematical and Statistical Psychology, 42(2), 233–239. doi:10.1111/j.2044-8317.1989.tb00912.x.

    Article  Google Scholar 

  51. Wu, H., & Browne, M. W. (2015). Quantifying adventitious error in a covariance structure as a random effect. Psychometrika, 80(3), 571–600. doi:10.1007/s11336-015-9451-3.

    Article  PubMed  PubMed Central  Google Scholar 

  52. Yuan, K.-H., & Bentler, P. M. (1997). Mean and covariance structure analysis: Theoretical and practical improvements. Journal of the American Statistical Association, 92, 767–774. doi:10.1080/01621459.1997.10474029.

    Article  Google Scholar 

  53. Yuan, K.-H., & Bentler, P. M. (1998). Normal theory based test statistics in structural equation modelling. The British Journal of Mathematical and Statistical Psychology, 51(2), 289–309. doi:10.1111/j.2044-8317.1998.tb00682.x.

    Article  PubMed  Google Scholar 

  54. Yuan, K.-H., & Bentler, P. M. (1999). \(F\) tests for mean and covariance structure analysis. Journal of Educational and Behavioral Statistics, 24(3), 225–243. doi:10.3102/10769986024003225.

    Article  Google Scholar 

  55. Yuan, K.-H., & Hayashi, K. (2010). Fitting data to model: Structural equation modeling diagnosis using two scatter plots. Psychological Methods, 15(4), 335–351. doi:10.1037/a0020140.

    Article  PubMed  Google Scholar 

  56. Yuan, K.-H., Marshall, L. L., & Bentler, P. M. (2003). Assessing the effect of model misspecifications on parameter estimates in structural equation models. Sociological Methodology, 33(1), 241–265. doi:10.1111/j.0081-1750.2003.00132.x.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Alberto Maydeu-Olivares.

Additional information

Presidential Address to the Psychometric Society, delivered at the annual meeting in Madison (WI), July 2014. This research was supported by an ICREA-Academia Award and Grant SGR 2014 1500 from the Catalan Government and Grant PSI2012-33601 from the Spanish Ministry of Education. I am indebted to Peter Bentler, Ke-Hai Yuan, Albert Satorra, Jim Steiger, Haruhiko Ogasawara, and Yves Rosseel for their helpful comments. I am also most thankful to Yves Rosseel for implementing these methods in the Lavaan package in R.

Appendix

Appendix

Asymptotic Distribution of Standardized Residual Covariances and Residual Correlations

From standard theory (e.g., Satorra, 1989)

$$\begin{aligned} \sqrt{N}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}}}})\mathop =\limits ^a \mathbf{H} \sqrt{N}\left( {\mathbf{s}-{\varvec{{\upsigma }}}_0} \right) \end{aligned}$$
(34)

and the asymptotic distribution of the unstandardized (raw) residual covariances follows from (14) under parameter drift assumptions. The asymptotic distribution of the standardized residuals is obtained from a first-order Taylor expansion of \({{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})\) at \(\mathbf{s}={\varvec{{\upsigma }}} \) and \({\hat{{\varvec{{\upsigma }}}}}={\varvec{{\upsigma }}}\)

$$\begin{aligned}&{{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}}}})\approx \left. {{{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})} \right| _{\mathbf{s}={\varvec{{\upsigma }}} ,{\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} } +\left. {\frac{\partial {{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})}{\partial \mathbf{{s}'}}} \right| _{\mathbf{s}={\varvec{{\upsigma }}} } (\mathbf{s}-{\varvec{{\upsigma }}} )+\left. {\frac{\partial {{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})}{\partial {{\hat{{\varvec{{\upsigma }}}}}}'}} \right| _{{\hat{{\varvec{{\upsigma }}} }} ={\varvec{{\upsigma }}}} ({\hat{{\varvec{{\upsigma }}} }}-{\varvec{{\upsigma }}} )\nonumber \\&\quad =\mathbf{G}(\mathbf{s}-{\varvec{{\upsigma }}} )-\mathbf{G}({\hat{{\varvec{{\upsigma }}} }}-{\varvec{{\upsigma }}} )=\mathbf{G}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }}), \end{aligned}$$
(35)

since \(\left. {{{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})} \right| _{\mathbf{s}={\varvec{{\upsigma }}} ,{\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} } =\mathbf{0}\), \(\left. {\frac{\partial {{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})}{\partial \mathbf{{s}'}}} \right| _{\mathbf{s}={\varvec{{\upsigma }}} } =\mathbf{G}\) and \(\left. {\frac{\partial {{\hat{\mathbf{G}}}}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }})}{\partial {{\hat{{\varvec{{\upsigma }}}}}}'}} \right| _{{\hat{{\varvec{{\upsigma }}}}}={\varvec{{\upsigma }}}} =-\mathbf{G}\). Equation (16) follows from (14), (34), and (35).

The asymptotic distribution of the residual correlations is obtained in a similar fashion. Using a first-order Taylor expansion of \(\mathbf{r}-{\hat{{\varvec{\uprho }} }}\) at \(\mathbf{s}={\varvec{{\upsigma }}} \) and \({\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} \)

$$\begin{aligned}&(\mathbf{r}-{\hat{{\varvec{\uprho }} }})\approx \left. {(\mathbf{r}-{\hat{{\varvec{\uprho }} }})} \right| _{\mathbf{s}={\varvec{{\upsigma }}} ,{\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} } +\left. {\frac{\partial (\mathbf{r}-{\hat{{\varvec{\uprho }} }})}{\partial \mathbf{{s}'}}} \right| _{\mathbf{s}={\varvec{{\upsigma }}}} (\mathbf{s}-{\varvec{{\upsigma }}} )+\left. {\frac{\partial (\mathbf{r}-{\hat{{\varvec{\uprho }} }})}{\partial {{\hat{{\varvec{{\upsigma }}} }}}'}} \right| _{{\hat{{\varvec{{\upsigma }}}}}={\varvec{{\upsigma }}}} ({\hat{{\varvec{{\upsigma }}} }}-{\varvec{{\upsigma }}})\nonumber \\&\quad =\mathbf{F}(\mathbf{s}-{\varvec{{\upsigma }}} )-\mathbf{F}({\hat{{\varvec{{\upsigma }}} }}-{\varvec{{\upsigma }}} )=\mathbf{F}(\mathbf{s}-{\hat{{\varvec{{\upsigma }}} }}), \end{aligned}$$
(36)

where \(\left. {(\mathbf{r}-{\hat{{\varvec{\uprho }}}})} \right| _{\mathbf{s}={\varvec{{\upsigma }}} ,{\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} } =\mathbf{0}\), \(\left. {\frac{\partial (\mathbf{r}-{\hat{{\varvec{\uprho }}}})}{\partial {\mathbf{s}'}}} \right| _{\mathbf{s}={\varvec{{\upsigma }}} } =\mathbf{F}\), \(\left. {\frac{\partial (\mathbf{r}-{\hat{{\varvec{\uprho }} }})}{\partial {{\hat{{\varvec{{\upsigma }}} }}}'}} \right| _{{\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}}} =-\mathbf{F}\). The nonzero elements of F, \({\partial (r_{ij} -{\hat{\rho }}_{ij})}\big /{\partial {\hat{{\sigma } }}_{lk} }\) evaluated at \({\hat{{\varvec{{\upsigma }}} }}={\varvec{{\upsigma }}} \), are \(-{{\sigma }_{ij}}\big /{\left( {2{\sigma }_{ii}^{3/2} {\sigma }_{jj}^{1/2} } \right) }=\frac{\rho _{ij} }{2{\sigma }_{jj} }\) if \(i \ne { k, j = l}, -{{\sigma }_{ij} }\big /{\left( {2{\sigma }_{ii}^{1/2} {\sigma }_{jj}^{3/2} } \right) }=\frac{\rho _{ij} }{2{\sigma }_{ii} }\) if \({ i = k, j \ne l}\), and \(1\big /{\left( {{\sigma }_{ii}^{1/2} {\sigma }_{jj}^{1/2}} \right) }\) if \((i, j) = (k, l)\) . Notice that F can be written as \(\mathbf{F }=\mathbf{J G}\), where the nonzero elements of J are \(-\frac{\rho _{ij} }{2}\) if \({i \ne k, j = l}\) or \({i = k, j \ne l}\), and 1 if \((i, j) = (k, l)\).

Asymptotic Mean and Variance of the Overall Effect Size Estimators

The asymptotic mean and variance of \(T=T_{s}\) or \(T_{r}\) under the parameter drift assumptions given in Eqs. (20) and (21) follow from standard results in quadratic forms of normal variates (e.g., Schott, 1997: Theorem 9.22).

The asymptotic mean and variance of \(\tilde{{{\iota }} }\) follow from standard results on Taylor expansions of moments of functions of random variables: Let T be a random variable with \({\mu }_T \) and \({\sigma }_T^2 \), then

$$\begin{aligned} E[f(T)]\approx & {} f({\mu }_T )+\frac{{f}''({\mu }_T )}{2}{{\sigma }}_T^2 , \end{aligned}$$
(37)
$$\begin{aligned} \hbox {var}[f(T)]\approx & {} \left[ {{f}'({\mu }_T )} \right] ^{2}{\sigma }_T^2 . \end{aligned}$$
(38)

For the \(\tilde{{{\iota }}}_{s}\) based on the SRMR, \(f(T)=\sqrt{\frac{T-c}{t}}\) and \({f}'(T)=\frac{1}{2t}\left( {\frac{T-c}{t}} \right) ^{-\frac{1}{2}}\) and \({f}''(T)=-\frac{1}{4t^{2}}\left( {\frac{T-c}{t}} \right) ^{-\frac{3}{2}}\). For the \(\tilde{{{\iota }} }_\mathrm{r} \) based on the CRMR, \(t-p\) is used instead of t. Equations (22) and (23) are obtained from (37) and (38) evaluating these derivatives at \({\mu }_T \) given in Eq. (20).

The asymptotic mean and variance of \({\hat{{{\iota }}}}=k^{-1}\tilde{{{\iota }} }\) are obtained from the asymptotic mean and variance of \(\tilde{{{\iota }} }\) again using (37) and (38): \(\mathrm{E}[{\hat{{{\iota }} }}]={{\iota }}, \hbox {var}[{\hat{{{\iota }} }}]=k^{-2}\hbox {var}[\tilde{{{\iota }}}]\).

Asymptotic Mean and Variance of the Sample SRMR and CRMR Under the Null Hypothesis of Exact Fit

Consider the statistic \(T^{*}=\widehat{\hbox {SRMR}}\hbox { or }\widehat{\hbox {CRMR}}\) given in Eqs. (7) and (8). We write \(T^{*}\) as \(f(T)=\sqrt{T/t}\), where \(T=T_{s}\) or \(T_{r}\), and for \(T_{r}\,t-p\) is used instead of t. Using (37) and (38) with \({f}'(T)=\frac{1}{2t}\left( {\frac{T}{t}} \right) ^{-\frac{1}{2}}\) and \({f}''(T)=-\frac{1}{4T^{2}}\left( {\frac{T}{t}} \right) ^{\frac{1}{2}}\),

$$\begin{aligned} \hbox {E}(T^{*})=\sqrt{\frac{{\mu }_T }{t}}\frac{8{\mu }_T^2 -{\sigma }_T^2 }{8{\mu }_T^2 }, \quad \hbox {var}(T^{*})=\frac{{\sigma }_T^2 }{4t\times {\mu }_T^2}. \end{aligned}$$
(39)

Now, under the null hypothesis of exact fit

$$\begin{aligned} \hbox {E}(T)={\mu }_T =\hbox {tr}({\varvec{\Xi }} ), \quad \hbox {var}(T)={\sigma }_{T}^{2}=2\hbox {tr}({\varvec{\Xi }} ^{2}). \end{aligned}$$
(40)

Equations (28) and (29) follow from (39) and (40).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Maydeu-Olivares, A. Assessing the Size of Model Misfit in Structural Equation Models. Psychometrika 82, 533–558 (2017). https://doi.org/10.1007/s11336-016-9552-7

Download citation

Keywords

  • goodness-of-fit
  • RMSEA
  • effect size