Direct Discrepancy Dynamic Fit Index Cutoffs for Arbitrary Covariance Structure Models

References

• Aguinis, H., Pierce, C. A., & Culpepper, S. A. (2009). Scale coarseness as a methodological artifact: Correcting correlation coefficients attenuated from using coarse scales. Organizational Research Methods, 12, 623–652. https://doi.org/10.1177/1094428108318065
   Web of Science ®Google Scholar
• Althouse, A. D. (2021). Post hoc power: Not empowering, just misleading. The Journal of Surgical Research, 259, A3–A6. https://doi.org/10.1016/j.jss.2019.10.049
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Beauducel, A., & Herzberg, P. Y. (2006). On the performance of maximum likelihood versus means and variance adjusted weighted least squares estimation in CFA. Structural Equation Modeling, 13, 186–203. https://doi.org/10.1207/s15328007sem1302_2
   Web of Science ®Google Scholar
• Bentler, P. M. (2006). EQS 6 Structural equations program manual. Multivariate Software, Inc.
   Google Scholar
• Bentler, P. M. (2007). On tests and indices for evaluating structural models. Personality and Individual Differences, 42, 825–829. https://doi.org/10.1016/j.paid.2006.09.024
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Bentler, P. M., & Yuan, K. H. (2000). On adding a mean structure to a covariance structure model. Educational and Psychological Measurement, 60, 326–339. https://doi.org/10.1177/00131640021970574
   Web of Science ®Google Scholar
• Beran, R., & Srivastava, M. S. (1985). Bootstrap tests and confidence regions for functions of a covariance matrix. The Annals of Statistics, 13, 95–115. https://doi.org/10.1214/aos/1176346579
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Briggs, N. E., & MacCallum, R. C. (2003). Recovery of weak common factors by maximum likelihood and ordinary least squares estimation. Multivariate Behavioral Research, 38, 25–56. https://doi.org/10.1207/S15327906MBR3801_2
   PubMed Web of Science ®Google Scholar
• 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.
   Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Bürkner, P. C. (2017). brms: An R package for Bayesian multilevel models using Stan. Journal of Statistical Software, 80, 28. https://doi.org/10.18637/jss.v080.i01
   Web of Science ®Google Scholar
• Cario, M. C., & Nelson, B. L. (1997). Modeling and generating random vectors with arbitrary marginal distributions and correlation matrix (pp. 1–19). Technical Report, Department of Industrial Engineering and Management Sciences, Northwestern University.
   Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for testing measurement invariance. Structural Equation Modeling: A Multidisciplinary Journal, 9, 233–255. https://doi.org/10.1207/S15328007SEM0902_5
   Web of Science ®Google Scholar
• Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Lawrence Erlbaum Associates.
   Google Scholar
• Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155–159. https://doi.org/10.1037//0033-2909.112.1.155
   PubMed Web of Science ®Google Scholar
• Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied multiple regression/correlation analysis for the behavioral sciences. Routledge.
   Google Scholar
• Correll, J., Mellinger, C., McClelland, G. H., & Judd, C. M. (2020). Avoid Cohen’s ‘small’, ‘medium’, and ‘large’ for power analysis. Trends in Cognitive Sciences, 24, 200–207. https://doi.org/10.1016/j.tics.2019.12.009
   PubMed Web of Science ®Google Scholar
• Cudeck, R., & Browne, M. W. (1992). Constructing a covariance matrix that yields a specified minimizer and a specified minimum discrepancy function value. Psychometrika, 57, 357–369. https://doi.org/10.1007/BF02295424
   Web of Science ®Google Scholar
• Davey, A. (2005). Issues in evaluating model fit with missing data. Structural Equation Modeling, 12, 578–597. https://doi.org/10.1207/s15328007sem1204_4
   Web of Science ®Google Scholar
• Demirtas, H. (2006). A method for multivariate ordinal data generation given marginal distributions and correlations. Journal of Statistical Computation and Simulation, 76, 1017–1025. https://doi.org/10.1080/10629360600569246
   Web of Science ®Google Scholar
• Edwards, M. C. (2013). Purple unicorns, true models, and other things i‘ve never seen. Measurement, 11, 107–111. https://doi.org/10.1080/15366367.2013.835178
   Google Scholar
• Enders, C. K. (2001). The impact of nonnormality on full information maximum-likelihood estimation for structural equation models with missing data. Psychological Methods, 6, 352–370. https://doi.org/10.1037/1082-989X.6.4.352
   PubMed Web of Science ®Google Scholar
• Enders, C. K. (2002). Applying the Bollen-Stine bootstrap for goodness-of-fit measures to structural equation models with missing data. Multivariate Behavioral Research, 37, 359–377. https://doi.org/10.1207/S15327906MBR3703_3
   PubMed Web of Science ®Google Scholar
• Enders, C. K. (2005). An SAS macro for implementing the modified Bollen-Stine bootstrap for missing data: Implementing the bootstrap using existing structural equation modeling software. Structural Equation Modeling, 12, 620–641. https://doi.org/10.1207/s15328007sem1204_6
   Web of Science ®Google Scholar
• Erdfelder, E., Faul, F., & Buchner, A. (1996). GPOWER: A general power analysis program. Behavior Research Methods, 28, 1–11.
   Google Scholar
• 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
   Web of Science ®Google Scholar
• Faul, F., Erdfelder, E., Lang, A. G., & Buchner, A. (2007). G* Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39, 175–191. https://doi.org/10.3758/bf03193146
   PubMed Web of Science ®Google Scholar
• Ferrari, A. J., Charlson, F. J., Norman, R. E., Flaxman, A. D., Patten, S. B., Vos, T., & Whiteford, H. A. (2013). The epidemiological modelling of major depressive disorder: Application for the global burden of disease study 2010. PLOS One, 8, e69637. https://doi.org/10.1371/journal.pone.0069637
   PubMed Web of Science ®Google Scholar
• Ferrari, P. A., & Barbiero, A. (2012). Simulating ordinal data. Multivariate Behavioral Research, 47, 566–589. https://doi.org/10.1080/00273171.2012.692630
   PubMed Web of Science ®Google Scholar
• Fialkowski, A. C. (2022). SimMultiCorrData: Simulation of correlated data with multiple variable types [Software]. https://cran.r-project.org/web/packages/SimMultiCorrData/
   Google Scholar
• Fialkowski, A., & Tiwari, H. (2019). SimCorrMix: Simulation of correlated data with multiple variable types including continuous and count mixture distributions. R Journal, 11, 250. https://doi.org/10.32614/RJ-2019-022
   Web of Science ®Google Scholar
• Field, A. P. (2005). Discovering statistics with SPSS (2nd ed.). Sage.
   Google Scholar
• 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
   Web of Science ®Google Scholar
• Fleishman, A. I. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521–532. https://doi.org/10.1007/BF02293811
   Web of Science ®Google Scholar
• Funder, D. C., & Ozer, D. J. (2019). Evaluating effect size in psychological research: Sense and nonsense. Advances in Methods and Practices in Psychological Science, 2, 156–168. https://doi.org/10.1177/2515245919847202
   Google Scholar
• Galeeva, R., Hoogland, J., Eydeland, A., & Stanley, M. (2007). Measuring correlation risk, https://www.bbk.ac.uk/cfc/pdfs/conference%20papers/Wed/Correlation_final_2.pdf
   Google Scholar
• García-Batista, Z. E., Guerra-Peña, K., Cano-Vindel, A., Herrera-Martínez, S. X., & Medrano, L. A. (2018). Validity and reliability of the Beck Depression Inventory (BDI-II) in general and hospital population of Dominican Republic. PLOS One, 13, e0199750. https://doi.org/10.1371/journal.pone.0199750
   PubMed Web of Science ®Google Scholar
• Gershgorin, S. (1931). Uber die Abgrenzung der Eigenwerte einer Matrix. Bulletin de l‘Académie des Sciences de l‘URSS. Classe Des Sciences Mathématiques, 6, 749–754.
   Google Scholar
• Giner-Sorolla, R., Montoya, A., Aberson, C., Carpenter, T., Lewis, N., Bostyn, D. H., & Soderberg, C. K. (2019). Power to detect what? Considerations for planning and evaluating sample size, PsyArXiv. https://osf.io/preprints/psyarxiv/rv3kw/
   Google Scholar
• 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
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Groskurth, K., Bluemke, M., & Lechner, C. M. (2023). Why we need to abandon fixed cutoffs for goodness-of-fit indices: An extensive simulation and possible solutions. Behavior Research Methods. Advance online publication. https://doi.org/10.3758/s13428-023-02193-3
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Hardin, J., Garcia, S. R., & Golan, D. (2013). A method for generating realistic correlation matrices. The Annals of Applied Statistics, 7, 1733–1762. https://doi.org/10.1214/13-AOAS638
   Web of Science ®Google Scholar
• Headrick, T. C., & Sawilowsky, S. S. (1999). Simulating correlated multivariate nonnormal distributions: Extending the Fleishman power method. Psychometrika, 64, 25–35. https://doi.org/10.1007/BF02294537
   Web of Science ®Google Scholar
• Heene, M., Hilbert, S., Draxler, C., Ziegler, M., & Bühner, 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
   PubMed Web of Science ®Google Scholar
• Higham, N. J. (2002). Computing the nearest correlation matrix—a problem from finance. IMA Journal of Numerical Analysis, 22, 329–343. https://doi.org/10.1093/imanum/22.3.329
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Jackson, D. L. (2007). The effect of the number of observations per parameter in misspecified confirmatory factor analytic models. Structural Equation Modeling, 14, 48–76. https://doi.org/10.1080/10705510709336736
   Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Joregenson, T. D., Pornprasertmanit, S., Schoemann, A. M., & Rosseel, Y. (2022). semTools: Useful tools for structural equation modeling [Software]. https://cran.r-project.org/web/packages/semTools
   Google Scholar
• Jöreskog, K. G. (1969). A general approach to confirmatory maximum likelihood factor analysis. Psychometrika, 34, 183–202. https://doi.org/10.1007/BF02289343
   Web of Science ®Google Scholar
• Jöreskog, K. G., & Sörbom, D. (1982). Recent developments in structural equation modeling. Journal of Marketing Research, 19, 404–416. https://doi.org/10.1177/002224378201900402
   Web of Science ®Google Scholar
• Kang, Y., McNeish, D., & Hancock, G. R. (2016). The role of measurement quality on practical guidelines for assessing measurement and structural invariance. Educational and Psychological Measurement, 76, 533–561. https://doi.org/10.1177/0013164415603764
   PubMed Web of Science ®Google Scholar
• Kelley, K., & Preacher, K. J. (2012). On effect size. Psychological Methods, 17, 137–152. https://doi.org/10.1037/a0028086
   PubMed Web of Science ®Google Scholar
• Kenny, D. A., & McCoach, D. B. (2003). Effect of the number of variables on measures of fit in structural equation modeling. Structural Equation Modeling, 10, 333–351. https://doi.org/10.1207/S15328007SEM1003_1
   Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Kirmayer, L. J. (2001). Cultural variations in the clinical presentation of depression and anxiety: Implications for diagnosis and treatment. The Journal of Clinical Psychiatry, 62, 22–30.
   PubMed Web of Science ®Google Scholar
• Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford.
   Google Scholar
• Kraft, M. A. (2020). Interpreting effect sizes of education interventions. Educational Researcher, 49, 241–253. https://doi.org/10.3102/0013189X20912798
   Web of Science ®Google Scholar
• Lai, K., & Green, S. B. (2016). The problem with having two watches: Assessment of fit when RMSEA and CFI disagree. Multivariate Behavioral Research, 51, 220–239. https://doi.org/10.1080/00273171.2015.1134306
   PubMed Web of Science ®Google Scholar
• Levine, M., & Ensom, M. H. (2001). Post hoc power analysis: An idea whose time has passed? Pharmacotherapy, 21, 405–409. https://doi.org/10.1592/phco.21.5.405.34503
   PubMed Web of Science ®Google Scholar
• Lewandowski, D., Kurowicka, D., & Joe, H. (2009). Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis, 100, 1989–2001. https://doi.org/10.1016/j.jmva.2009.04.008
   Web of Science ®Google Scholar
• Li, C. H. (2016a). Confirmatory factor analysis with ordinal data: Comparing robust maximum likelihood and diagonally weighted least squares. Behavior Research Methods, 48, 936–949. https://doi.org/10.3758/s13428-015-0619-7
   PubMed Web of Science ®Google Scholar
• Li, C. H. (2016b). The performance of ML, DWLS, and ULS estimation with robust corrections in structural equation models with ordinal variables. Psychological Methods, 21, 369–387. https://doi.org/10.1037/met0000093
   PubMed Web of Science ®Google Scholar
• MacCallum, R. C. (2003). Working with imperfect models. Multivariate Behavioral Research, 38, 113–139. https://doi.org/10.1207/S15327906MBR3801_5
   PubMed Web of Science ®Google Scholar
• MacCallum, R. C., & Tucker, L. R. (1991). Representing sources of error in the common-factor model: Implications for theory and practice. Psychological Bulletin, 109, 502–511. https://doi.org/10.1037/0033-2909.109.3.502
   Web of Science ®Google Scholar
• MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1, 130–149. https://doi.org/10.1037/1082-989X.1.2.130
   Web of Science ®Google Scholar
• 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, 185–199. https://doi.org/10.1037/0033-2909.114.1.185
   PubMed Web of Science ®Google Scholar
• Makalic, E., & Schmidt, D. F. (2022). An efficient algorithm for sampling from sin k (x) for generating random correlation matrices. Communications in Statistics, 51, 2731–2735. https://doi.org/10.1080/03610918.2019.1700277
   Google Scholar
• Marcoulides, K. M., & Yuan, K. H. (2017). New ways to evaluate goodness of fit: A note on using equivalence testing to assess structural equation models. Structural Equation Modeling, 24, 148–153. https://doi.org/10.1080/10705511.2016.1225260
   Web of Science ®Google Scholar
• Markland, D. (2007). The golden rule is that there are no golden rules: A commentary on Paul Barrett’s recommendations for reporting model fit in structural equation modelling. Personality and Individual Differences, 42, 851–858. https://doi.org/10.1016/j.paid.2006.09.023
   Web of Science ®Google Scholar
• 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: A festschrift to Roderick P. McDonald (pp. 275–340). Erlbaum Associates.
   Google Scholar
• Marsh, H. W., Hau, K. T., Balla, J. R., & Grayson, D. (1998). Is more ever too much? The number of indicators per factor in confirmatory factor analysis. Multivariate Behavioral Research, 33, 181–220. https://doi.org/10.1207/s15327906mbr3302_1
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Martin, S. R. (2020, August 27). Is the LKJ(1) prior uniform? “Yes”. http://srmart.in/is-the-lkj1-prior-uniform-yes
   Google Scholar
• Maydeu-Olivares, A. (2017). Assessing the size of model misfit in structural equation models. Psychometrika, 82, 533–558. https://doi.org/10.1007/s11336-016-9552-7
   Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• McIntosh, C. N. (2007). Rethinking fit assessment in structural equation modelling: A commentary and elaboration on Barrett (2007). Personality and Individual Differences, 42, 859–867. https://doi.org/10.1016/j.paid.2006.09.020
   Web of Science ®Google Scholar
• McNeish, D. (2023a). Dynamic fit index cutoffs for categorical factor analysis with Likert-type, ordinal, or binary responses. The American Psychologist, 78, 1061–1075. https://doi.org/10.1037/amp0001213
   PubMed Web of Science ®Google Scholar
• McNeish, D. (2023b). Generalizability of dynamic fit index, equivalence testing, and Hu & Bentler cutoffs for evaluating fit in factor analysis. Multivariate Behavioral Research, 58, 195–219. https://doi.org/10.1080/00273171.2022.2163477
   PubMed Web of Science ®Google Scholar
• McNeish, D. (2024). Dynamic fit index cutoffs for treating Likert items as continuous. PsyArXiv, https://osf.io/preprints/psyarxiv/fgukr.
   Google Scholar
• McNeish, D., & Manapat, P. D. (2023). Dynamic fit index cutoffs for hierarchical and second-order factor models. Structural Equation Modeling. Advance online publication. https://doi.org/10.1080/10705511.2023.2225132
   Web of Science ®Google Scholar
• McNeish, D., & Wolf, M. G. (2023a). Dynamic fit index cutoffs for confirmatory factor analysis models. Psychological Methods, 28, 61–88. https://doi.org/10.1037/met0000425
   PubMed Web of Science ®Google Scholar
• McNeish, D., & Wolf, M. G. (2023b). Dynamic fit cutoffs for one-factor models. Behavior Research Methods, 55, 1157–1174. https://doi.org/10.3758/s13428-022-01847-y
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• 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.
   Google Scholar
• Monroe, S., & Cai, L. (2015). Evaluating structural equation models for categorical outcomes: A new test statistic and a practical challenge of interpretation. Multivariate Behavioral Research, 50, 569–583. https://doi.org/10.1080/00273171.2015.1032398
   PubMed Web of Science ®Google Scholar
• Moshagen, M. (2012). The model size effect in SEM: Inflated goodness-of-fit statistics are due to the size of the covariance matrix. Structural Equation Modeling, 19, 86–98. https://doi.org/10.1080/10705511.2012.634724
   Web of Science ®Google Scholar
• Moshagen, M., & Auerswald, M. (2018). On congruence and incongruence of measures of fit in structural equation modeling. Psychological Methods, 23, 318–336. https://doi.org/10.1037/met0000122
   PubMed Web of Science ®Google Scholar
• Mulaik, S. A. (2009). Foundations of factor analysis. McGraw-Hill.
   Google Scholar
• Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of nonnormal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171–189. https://doi.org/10.1111/j.2044-8317.1985.tb00832.x
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Niemand, T., & Mai, R. (2023). FCO: Flexible cutoffs for model fit evaluation in covariance-based structural models. [Software]. https://cran.r-project.org/web/packages/FCO/index.html
   Google Scholar
• Nordahl-Hansen, A., Cogo-Moreira, H., Panjeh, S., & Quintana, D. S. (2024). Redefining effect size interpretations for psychotherapy RCTs in depression. Journal of Psychiatric Research, 169, 38–41. https://doi.org/10.1016/j.jpsychires.2023.11.009
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Olvera Astivia, O. L., & Zumbo, B. D. (2015). A cautionary note on the use of the Vale and Maurelli method to generate multivariate, nonnormal data for simulation purposes. Educational and Psychological Measurement, 75, 541–567. https://doi.org/10.1177/0013164414548894
   PubMed Web of Science ®Google Scholar
• Olvera Astivia, O. L., Kroc, E., & Zumbo, B. D. (2023). Simultaneous estimation of the intermediate correlation matrix for arbitrary marginal densities. Behavior Research Methods. Advance online publication. https://doi.org/10.3758/s13428-023-02123-3
   PubMed Web of Science ®Google Scholar
• Opdyke, J. D. (2020). Full probabilistic control for direct and robust, generalized and targeted stressing of the correlation matrix (Even When Eigenvalues are Empirically Challenging). Talk Presented at QuantMinds International Conference, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3673362
   Google Scholar
• Patel, V. (2017). Talking sensibly about depression. PLOS Medicine, 14, e1002257. https://doi.org/10.1371/journal.pmed.1002257
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Peugh, J. L., & Enders, C. K. (2004). Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research, 74, 525–556. https://doi.org/10.3102/00346543074004525
   Web of Science ®Google Scholar
• Peugh, J. L., Litson, K., & Feldon, D. F. (2023). Equivalence testing to judge model fit: A Monte Carlo simulation. Psychological Methods, Advance online publication. https://psycnet.apa.org/doi/101037/met0000591
   PubMed Web of Science ®Google Scholar
• Pinheiro, J. C., & Bates, D. M. (1996). Unconstrained parametrizations for variance-covariance matrices. Statistics and Computing, 6, 289–296. https://doi.org/10.1007/BF00140873
   Web of Science ®Google Scholar
• 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
   Google Scholar
• 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.
   Google Scholar
• Pourahmadi, M., & Wang, X. (2015). Distribution of random correlation matrices: Hyperspherical parameterization of the Cholesky factor. Statistics & Probability Letters, 106, 5–12. https://doi.org/10.1016/j.spl.2015.06.015
   Web of Science ®Google Scholar
• Rapisarda, F., Brigo, D., & Mercurio, F. (2007). Parameterizing correlations: A geometric interpretation. IMA Journal of Management Mathematics, 18, 55–73. https://doi.org/10.1093/imaman/dpl010
   Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Ropovik, I. (2015). A cautionary note on testing latent variable models. Frontiers in Psychology, 6, 1715. https://doi.org/10.3389/fpsyg.2015.01715
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Savalei, V. (2021). Improving fit indices in structural equation modeling with categorical data. Multivariate Behavioral Research, 56, 390–407.
   PubMed Web of Science ®Google Scholar
• Savalei, V., & Yuan, K. H. (2009). On the model-based bootstrap with missing data: Obtaining a p-value for a test of exact fit. Multivariate Behavioral Research, 44, 741–763. https://doi.org/10.1080/00273170903333590
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Shi, D., & Maydeu-Olivares, A. (2020). The effect of estimation methods on SEM fit indices. Educational and Psychological Measurement, 80, 421–445. https://doi.org/10.1177/0013164419885164
   PubMed Web of Science ®Google Scholar
• Shi, D., DiStefano, C., Maydeu-Olivares, A., & Lee, T. (2022). Evaluating SEM model fit with small degrees of freedom. Multivariate Behavioral Research, 57, 179–207. https://doi.org/10.1080/00273171.2020.1868965
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Shi, D., Maydeu-Olivares, A., & DiStefano, C. (2018). The relationship between the standardized root mean square residual and model misspecification in factor analysis models. Multivariate Behavioral Research, 53, 676–694. https://doi.org/10.1080/00273171.2018.1476221
   PubMed Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Spielberg, S. (Director). (1993). Jurassic Park [Film]. Universal Pictures.
   Google Scholar
• Steiger, J. H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25, 173–180. https://doi.org/10.1207/s15327906mbr2502_4
   PubMed Web of Science ®Google Scholar
• Steiger, J. H. (2007). Understanding the limitations of global fit assessment in structural equation modeling. Personality and Individual Differences, 42, 893–898. https://doi.org/10.1016/j.paid.2006.09.017
   Web of Science ®Google Scholar
• Steiger, J. H., & Lind, J. M. (1980). Statistically based tests for the number of common factors. Paper Presented at the Annual Meeting of the Psychometric Society, Iowa City, IA.
   Google Scholar
• 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
   Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Tucker, L. R., Koopman, R. F., & Linn, R. L. (1969). Evaluation of factor analytic research procedures by means of simulated correlation matrices. Psychometrika, 34, 421–459. https://doi.org/10.1007/BF02290601
   Web of Science ®Google Scholar
• Ullman, J. B., & Bentler, P. M. (2012). Structural equation modeling. In J. A. Schinka, W. F. Velicer, & I. B. Weiner (Eds.), Handbook of psychology: Research methods in psychology (2nd ed., pp. 661–690). Wiley.
   Google Scholar
• Vale, C. D., & Maurelli, V. A. (1983). Simulating multivariate nonnormal distributions. Psychometrika, 48, 465–471. https://doi.org/10.1007/BF02293687
   Web of Science ®Google Scholar
• van Harmelen, A.-L., Gibson, J. L., St Clair, M. C., Owens, M., Brodbeck, J., Dunn, V., Lewis, G., Croudace, T., Jones, P. B., Kievit, R. A., & Goodyer, I. M. (2016). Friendships and family support reduce subsequent depressive symptoms in at-risk adolescents. PLOS One, 11, e0153715. https://doi.org/10.1371/journal.pone.0153715
   PubMed Web of Science ®Google Scholar
• Van Noorden, R., Maher, B., & Nuzzo, R. (2014). The top 100 papers. Nature, 514, 550–553. https://doi.org/10.1038/514550a
   PubMed Web of Science ®Google Scholar
• van Tilburg, W. A. P., & van Tilburg, L. J. A. (2023). Impossible hypotheses and effect-size limits. Advances in Methods and Practices in Psychological Science, 6. https://doi.org/10.1177/25152459231197605
   PubMedGoogle Scholar
• 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.
   Google Scholar
• Willmott, C. J., & Matsuura, K. (2005). Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Climate Research, 30, 79–82. https://doi.org/10.3354/cr030079
   Web of Science ®Google Scholar
• Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: Current approaches and future directions. Psychological Methods, 12, 58–79. https://doi.org/10.1037/1082-989X.12.1.58
   PubMed Web of Science ®Google Scholar
• Wolf, M. G., & McNeish, D. (2021). Dynamic model fit (version 1.1.0). [Software]. www.dynamicfit.app
   Google Scholar
• 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
   PubMed Web of Science ®Google Scholar
• Wu, H., & Browne, M. W. (2015). Quantifying adventitious error in a covariance structure as a random effect. Psychometrika, 80, 571–600. https://doi.org/10.1007/s11336-015-9451-3
   PubMed Web of Science ®Google Scholar
• Xia, Y., & Yang, Y. (2018). The influence of number of categories and threshold values on fit indices in structural equation modeling with ordered categorical data. Multivariate Behavioral Research, 53, 731–755. https://doi.org/10.1080/00273171.2018.1480346
   PubMed Web of Science ®Google Scholar
• Xia, Y., & Yang, Y. (2019). RMSEA, CFI, and TLI in structural equation modeling with ordered categorical data: The story they tell depends on the estimation methods. Behavior Research Methods, 51, 409–428. https://doi.org/10.3758/s13428-018-1055-2
   PubMed Web of Science ®Google Scholar
• Yuan, K. H., Chan, W., Marcoulides, G. A., & Bentler, P. M. (2016). Assessing structural equation models by equivalence testing with adjusted fit indexes. Structural Equation Modeling, 23, 319–330. https://doi.org/10.1080/10705511.2015.1065414
   Web of Science ®Google Scholar
• 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
   Web of Science ®Google Scholar
• Zhang, W., Leng, C., & Tang, C. Y. (2015). A joint modelling approach for longitudinal studies. Journal of the Royal Statistical Society Series B, 77, 219–238. https://doi.org/10.1111/rssb.12065
   Google Scholar
• Zhang, X., & Savalei, V. (2020). Examining the effect of missing data on RMSEA and CFI under normal theory full-information maximum likelihood. Structural Equation Modeling, 27, 219–239. https://doi.org/10.1080/10705511.2019.1642111
   Web of Science ®Google Scholar
• 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
   Google Scholar