252
Views
0
CrossRef citations to date
0
Altmetric
Research Articles

Dynamic Fit Index Cutoffs for Hierarchical and Second-Order Factor Models

Pages 27-47 | Received 28 Mar 2023, Accepted 10 Jun 2023, Published online: 28 Jul 2023

References

  • American Educational Research Association, American Psychological Association, & National Council on Measurement in Education (AERA, APA, and NCME). (2014). Standards for educational and psychological testing. American Psychological Association.
  • Appelbaum, M., Cooper, H., Kline, R. B., Mayo-Wilson, E., Nezu, A. M., & Rao, S. M. (2018). Journal article reporting standards for quantitative research in psychology: The APA Publications and Communications Board task force report. The American Psychologist, 73, 3–25. https://doi.org/10.1037/amp0000191
  • Arend, M. G., & Schäfer, T. (2019). Statistical power in two-level models: A tutorial based on Monte Carlo simulation. Psychological Methods, 24, 1–19. https://doi.org/10.1037/met0000195
  • Barrett, P. (2007). Structural equation modelling: Adjudging model fit. Personality and Individual Differences, 42, 815–824. https://doi.org/10.1016/j.paid.2006.09.018
  • Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88, 588–606. https://doi.org/10.1037/0033-2909.88.3.588
  • Bollen, K. A. (1989). Structural equations with latent variables. Wiley.
  • Bollen, K. A., & Stine, R. A. (1992). Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods & Research, 21, 205–229. https://doi.org/10.1177/0049124192021002004
  • Brown, T. A. (2015). Confirmatory factor analysis for applied research. Guilford.
  • Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods & Research, 21, 230–258. https://doi.org/10.1177/0049124192021002005
  • 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). SAGE.
  • Browne, M. W., MacCallum, R. C., Kim, C. T., Andersen, B. L., & Glaser, R. (2002). When fit indices and residuals are incompatible. Psychological Methods, 7, 403–421. https://doi.org/10.1037//1082-989x.7.4.403
  • Brunner, M., Nagy, G., & Wilhelm, O. (2012). A tutorial on hierarchically structured constructs. Journal of Personality, 80, 796–846. https://doi.org/10.1111/j.1467-6494.2011.00749.x
  • Carroll, J. B. (1993). Human cognitive abilities: A survey of factor-analytical studies. Cambridge University Press.
  • Caspi, A., Houts, R. M., Belsky, D. W., Goldman-Mellor, S. J., Harrington, H., Israel, S., Meier, M. H., Ramrakha, S., Shalev, I., Poulton, R., & Moffitt, T. E. (2014). The p factor: One general psychopathology factor in the structure of psychiatric disorders? Clinical Psychological, 2, 119–137. https://doi.org/10.1177/2167702613497473
  • 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, 462–494. https://doi.org/10.1177/0049124108314720
  • D’Urso, D. E., Maassen, E., van Assen, M. A. L. M., Nuijten, M. B., De Roover, K., Wicherts, J. M. (2022). The dire disregard of measurement invariance testing in psychological science. PsyArXiv. Retrieved July 29, 2022, from https://psyar xiv. com/n3f5u/.
  • DeYoung, C. G. (2006). Higher-order factors of the Big Five in a multi-informant sample. Journal of Personality and Social Psychology, 91, 1138–1151. https://doi.org/10.1037/0022-3514.91.6.1138
  • Digman, J. M. (1997). Higher-order factors of the Big Five. Journal of Personality and Social Psychology, 73, 1246–1256. https://doi.org/10.1037//0022-3514.73.6.1246
  • Fan, X., & Sivo, S. A. (2005). Sensitivity of fit indexes to misspecified structural or measurement model components: Rationale of two-index strategy revisited. Structural Equation Modeling: A Multidisciplinary Journal, 12, 343–367. https://doi.org/10.1207/s15328007sem1203_1
  • Fan, X., & Sivo, S. A. (2007). Sensitivity of fit indices to model misspecification and model types. Multivariate Behavioral Research, 42, 509–529. https://doi.org/10.1080/00273170701382864
  • Flake, J. K., Pek, J., & Hehman, E. (2017). Construct validation in social and personality research: Current practice and recommendations. Social Psychological and Personality Science, 8, 370–378. https://doi.org/10.1177/1948550617693063
  • Green, P., & MacLeod, C. J. (2016). SIMR: An R package for power analysis of generalized linear mixed models by simulation. Methods in Ecology and Evolution, 7, 493–498. https://doi.org/10.1111/2041-210X.12504
  • Greiff, S., & Heene, M. (2017). Why psychological assessment needs to start worrying about model fit. European Journal of Psychological Assessment, 33, 313–317. https://doi.org/10.1027/1015-5759/a000450
  • Groskurth, K., Bhaktha, N., Lechner, C. (2022). Making model judgments ROC(K) solid: Tailored cutoffs for fit indices through simulation and ROC analysis in structural equation modeling. PsyArXiv. Retrieved May 16, 2023, from https://psyarxiv.com/62j89/.
  • Hancock, G. R., & Mueller, R. O. (2011). The reliability paradox in assessing structural relations within covariance structure models. Educational and Psychological Measurement, 71, 306–324. https://doi.org/10.1177/0013164410384856
  • Heene, M., Hilbert, S., Draxler, C., Ziegler, M., & Buhner, M. (2011). Masking misfit in confirmatory factor analysis by increasing unique variances: A cautionary note on the usefulness of cutoff values of fit indices. Psychological Methods, 16, 319–336. https://doi.org/10.1037/a0024917
  • Hu, L. T., & Bentler, P. M. (1998). Fit indices in covariance structure modeling: Sensitivity to underparameterized model misspecification. Psychological Methods, 3, 424–453. https://doi.org/10.1037/1082-989X.3.4.424
  • Hu, L., & Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1–55. https://doi.org/10.1080/10705519909540118
  • Hu, L. T., Bentler, P. M., & Kano, Y. (1992). Can test statistics in covariance structure analysis be trusted? Psychological Bulletin, 112, 351–362. https://doi.org/10.1037/0033-2909.112.2.351
  • Jackson, D. L., Gillaspy, J. A., & Purc-Stephenson, R. (2009). Reporting practices in confirmatory factor analysis: An overview and some recommendations. Psychological Methods, 14, 6–23. https://doi.org/10.1037/a0014694
  • Johnson, D. R., Kaufman, J. C., Baker, B. S., Patterson, J. D., Barbot, B., Green, A. E., van Hell, J., Kennedy, E., Sullivan, G. F., Taylor, C. L., Ward, T., & Beaty, R. E. (2022). Divergent semantic integration (DSI): Extracting creativity from narratives with distributional semantic modeling. Behavior Research Methods, advance online publication. https://doi.org/10.3758/s13428-022-01986-2
  • Kane, M. (2013). The argument-based approach to validation. School Psychology Review, 42, 448–457. https://doi.org/10.1080/02796015.2013.12087465
  • Kim, H., & Millsap, R. (2014). Using the Bollen-Stine bootstrapping method for evaluating approximate fit indices. Multivariate Behavioral Research, 49, 581–596. https://doi.org/10.1080/00273171.2014.947352
  • Kline, R. B. (2010). Principles and practice of structural equation modeling (3rd ed.). Guilford.
  • Krueger, R. F., Markon, K. E., Patrick, C. J., Benning, S. D., & Kramer, M. D. (2007). Linking antisocial behavior, substance use, and personality: An integrative quantitative model of the adult externalizing spectrum. Journal of Abnormal Psychology, 116, 645–666. https://doi.org/10.1037/0021-843X.116.4.645
  • Markon, K. E., Krueger, R. F., & Watson, D. (2005). Delineating the structure of normal and abnormal personality: An integrative hierarchical approach. Journal of Personality and Social Psychology, 88, 139–157. https://doi.org/10.1037/0022-3514.88.1.139
  • Marsh, H. W., & Craven, R. G. (2006). Reciprocal effects of self-concept and performance from a multidimensional perspective: Beyond seductive pleasure and unidimensional perspectives. Perspectives on Psychological Science, 1, 133–163. https://doi.org/10.1111/j.1745-6916.2006.00010.x
  • Marsh, H. W., Hau, K. T., & Grayson, D. (2005). Goodness of fit evaluation in structural equation modeling. In A. Maydeu-Olivares & J. McArdle (Eds.), Psychometrics (pp. 275–340). Erlbaum Associates.
  • Marsh, H. W., Hau, K.-T., & 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, 11, 320–341. https://doi.org/10.1207/s15328007sem1103_2
  • Marsh, H. W., & Hocevar, D. (1985). Application of confirmatory factor analysis to the study of self-concept: First-and higher order factor models and their invariance across groups. Psychological Bulletin, 97, 562–582. https://doi.org/10.1037/0033-2909.97.3.562
  • Martin, A. J. (2007). Examining a multidimensional model of student motivation and engagement using a construct validation approach. The British Journal of Educational Psychology, 77, 413–440. https://doi.org/10.1348/000709906X118036
  • Maydeu-Olivares, A., Shi, D., & Rosseel, Y. (2018). Assessing fit in structural equation models: A Monte-Carlo evaluation of RMSEA versus SRMR confidence intervals and tests of close fit. Structural Equation Modeling, 25, 389–402. https://doi.org/10.1080/10705511.2017.1389611
  • McDonald, R. P., & Ho, M. H. R. (2002). Principles and practice in reporting structural equation analyses. Psychological Methods, 7, 64–82. https://doi.org/10.1037/1082-989x.7.1.64
  • McGrew, K. S. (2009). CHC theory and the human cognitive abilities project: Standing on the shoulders of the giants of psychometric intelligence research. Intelligence, 37, 1–10. https://doi.org/10.1016/j.intell.2008.08.004
  • McNeish, D., An, J., & Hancock, G. R. (2018). The thorny relation between measurement quality and fit index cutoffs in latent variable models. Journal of Personality Assessment, 100, 43–52. https://doi.org/10.1080/00223891.2017.1281286
  • McNeish, D. (2023). Generalizability of dynamic fit index, equivalence testing, and Hu & Bentler cutoffs for evaluating fit in factor analysis. Multivariate Behavioral Research, 58, 195–219. advance online publication. https://doi.org/10.1080/00273171.2022.2163477
  • McNeish, D., & Wolf, M. G. (2021). Dynamic fit index cutoffs for confirmatory factor analysis models. Psychological Methods, advance online publication.
  • McNeish, D., & Wolf, M. G. (2022). Dynamic fit cutoffs for one-factor models. Behavior Research Methods, advance online publication.
  • McNeish, D. (in press). Dynamic fit index cutoffs for factor analysis with Likert, ordinal, or binary responses. https://psyarxiv.com/tp35s/
  • Meehl, P. E. (1978). Theoretical risks and tabular asterisks: Sir Karl, Sir Ronald, and the slow progress of soft psychology. Journal of Consulting and Clinical Psychology, 46, 806–834. https://doi.org/10.1037/0022-006X.46.4.806
  • Miles, J., & Shevlin, M. (2007). A time and a place for incremental fit indices. Personality and Individual Differences, 42, 869–874. https://doi.org/10.1016/j.paid.2006.09.022
  • Millsap, R. E. (2007). Structural equation modeling made difficult. Personality and Individual Differences, 42, 875–881. https://doi.org/10.1016/j.paid.2006.09.021
  • Millsap, R. E. (2013). A simulation paradigm for evaluating model fit. In M. Edwards & R. MacCallum (Eds.), Current issues in the theory and application of latent variable models (pp. 165–182). Routledge.
  • Mulaik, S. A. (2009). Foundations of factor analysis. McGraw-Hill.
  • Murray, A. L., & Johnson, W. (2013). The limitations of model fit in comparing the bi-factor versus higher-order models of human cognitive ability structure. Intelligence, 41, 407–422. https://doi.org/10.1016/j.intell.2013.06.004
  • Nieman, T., Mai, R. (2023). FCO: Flexible cutoffs for model fit evaluation in covariance-based structural models. [Software]. Retrieved from CRAN. https://cran.r-project.org/web/packages/FCO/index.html
  • Niemand, T., & Mai, R. (2018). Flexible cutoff values for fit indices in the evaluation of structural equation models. Journal of the Academy of Marketing Science, 46, 1148–1172. https://doi.org/10.1007/s11747-018-0602-9
  • Nye, C. D., & Drasgow, F. (2011). Assessing goodness of fit: Simple rules of thumb simply do not work. Organizational Research Methods, 14, 548–570. https://doi.org/10.1177/1094428110368562
  • Paxton, P., Curran, P. J., Bollen, K. A., Kirby, J., & Chen, F. (2001). Monte Carlo experiments: Design and implementation. Structural Equation Modeling, 8, 287–312. https://doi.org/10.1207/S15328007SEM0802_7
  • Pornprasertmanit, S., Wu, W., & Little, T. D. (2013). Using a Monte Carlo approach for nested model comparisons in structural equation modeling. In R. E. Millsap, L. A. van der Ark, D. M. Bolt, & C. M. Woods (Eds.), New developments in quantitative psychology (pp. 187–197). Springer.
  • Pornprasertmanit, S., Miller, P., Schoemann, A., Jorgensen, T., Quick, C. (2022). SIMulated structural equation modeling (R package version 0.5–16). http://CRAN.R-project.org/package=simsem
  • Reynolds, M. R., & Keith, T. Z. (2017). Multi-group and hierarchical confirmatory factor analysis of the Wechsler Intelligence Scale for Children—Fifth Edition: What does it measure? Intelligence, 62, 31–47. https://doi.org/10.1016/j.intell.2017.02.005
  • Rhemtulla, M., Brosseau-Liard, P. É., & Savalei, V. (2012). When can categorical variables be treated as continuous? A comparison of robust continuous and categorical SEM estimation methods under suboptimal conditions. Psychological Methods, 17, 354–373. https://doi.org/10.1037/a0029315
  • Rijmen, F. (2010). Formal relations and an empirical comparison among the bi‐factor, the testlet, and a second‐order multidimensional IRT model. Journal of Educational Measurement, 47, 361–372. https://doi.org/10.1111/j.1745-3984.2010.00118.x
  • Robitzsch, A. (2020, October) Why ordinal variables can (almost) always be treated as continuous variables: Clarifying assumptions of robust continuous and ordinal factor analysis estimation methods. Frontiers in Education, 5, 589965. https://doi.org/10.3389/feduc.2020.589965
  • Ropovik, I. (2015). A cautionary note on testing latent variable models. Frontiers in Psychology, 6, 1715. https://doi.org/10.3389/fpsyg.2015.01715
  • Saris, W. E., Satorra, A., & Sörbom, D. (1987). The detection and correction of specification errors in structural equation models. Sociological Methodology, 17, 105–129. https://doi.org/10.2307/271030
  • Saris, W. E., Satorra, A., & Van der Veld, W. M. (2009). Testing structural equation models or detection of misspecifications? Structural Equation Modeling, 16, 561–582. https://doi.org/10.1080/10705510903203433
  • 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, 910–932. https://doi.org/10.1177/0013164412452564
  • Schönemann, P. H. (1981). Power as a function of communality in factor analysis. Bulletin of the Psychonomic Society, 17, 57–60. https://doi.org/10.3758/BF03333667
  • Schreiber, J. B. (2008). Core reporting practices in structural equation modeling. Research in Social & Administrative Pharmacy, 4, 83–97. https://doi.org/10.1016/j.sapharm.2007.04.003
  • Shi, D., Lee, T., & Maydeu-Olivares, A. (2019). Understanding the model size effect on SEM fit indices. Educational and Psychological Measurement, 79, 310–334. https://doi.org/10.1177/0013164418783530
  • Shi, D., Lee, T., & Terry, R. A. (2018). Revisiting the model size effect in structural equation modeling. Structural Equation Modeling, 25, 21–40. https://doi.org/10.1080/10705511.2017.1369088
  • Silvia, P. J. (2008). Another look at creativity and intelligence: Exploring higher-order models and probable confounds. Personality and Individual Differences, 44, 1012–1021. https://doi.org/10.1016/j.paid.2007.10.027
  • Sivo, S. A., Fan, X., Witta, E. L., & Willse, J. T. (2006). The search for “optimal” cutoff properties: Fit index criteria in structural equation modeling. The Journal of Experimental Education, 74, 267–288. https://doi.org/10.3200/JEXE.74.3.267-288
  • Tanaka, J. S. (1987). “How big is big enough?”: Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58, 134–146. https://doi.org/10.2307/1130296
  • Thoemmes, F., Rosseel, Y., & Textor, J. (2018). Local fit evaluation of structural equation models using graphical criteria. Psychological Methods, 23, 27–41. https://doi.org/10.1037/met0000147
  • Tomarken, A. J., & Waller, N. G. (2003). Potential problems with “well fitting” models. Journal of Abnormal Psychology, 112, 578–598. https://doi.org/10.1037/0021-843X.112.4.578
  • West, S. G., Wu, W., McNeish, D., & Savord, A. (2023). Model fit in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed., pp. 184–205) Guilford Press.
  • Wolf, M. G., & McNeish, D. (2023). dynamic: An R package for deriving dynamic fit index cutoffs for factor analysis. Multivariate Behavioral Research, 58, 189–194. https://doi.org/10.1080/00273171.2022.2163476
  • Wolf, M. G., McNeish, D. (2021). Dynamic Model Fit (version 1.1.0). [Software]. Available from www.dynamicfit.app.
  • Wolf, M. G., McNeish, D. (2022). dynamic: DFI cutoffs for latent variables models (version 1.1.0). [Software]. Retrieved from CRAN. https://cran.r-project.org/web/packages/dynamic.
  • Wright, A. G., Lukowitsky, M. R., Pincus, A. L., & Conroy, D. E. (2010). The higher order factor structure and gender invariance of the Pathological Narcissism Inventory. Assessment, 17, 467–483. https://doi.org/10.1177/1073191110373227
  • Yung, Y. F., Thissen, D., & McLeod, L. D. (1999). On the relationship between the higher-order factor model and the hierarchical factor model. Psychometrika, 64, 113–128. https://doi.org/10.1007/BF02294531
  • Zhang, M. F., Dawson, J. F., & Kline, R. B. (2021). Evaluating the use of covariance‐based structural equation modelling with reflective measurement in organizational and management research: A review and recommendations for best practice. British Journal of Management, 32, 257–272. https://doi.org/10.1111/1467-8551.12415
  • Zinbarg, R. E., Barlow, D. H., & Brown, T. A. (1997). Hierarchical structure and general factor saturation of the Anxiety Sensitivity Index: Evidence and implications. Psychological Assessment, 9, 277–284. https://doi.org/10.1037/1040-3590.9.3.277
  • Zyphur, M. J., Bonner, C. V., & Tay, L. (2023). Structural equation modeling in organizational research: The state of our science and some proposals for its future. Annual Review of Organizational Psychology and Organizational Behavior, 10, 495–517. https://doi.org/10.1146/annurev-orgpsych-041621-031401

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.