Confirmatory factor analyses (CFA) are widely used in the organizational literature. As a result, understanding how to properly conduct these analyses, report the results, and interpret their implications is critically important for advancing organizational research. The goal of this paper is to summarize the complexities of CFA models and, therefore, to provide a resource for journal reviewers and researchers who are using CFA in their research. The topics covered in this paper include the estimation process, power analyses, model fit, and model modifications, among other things. In addition, this paper concludes with a checklist that summarizes the key points that are discussed and can be used to evaluate future studies that incorporate CFA.
AppelbaumM.CooperH.KlineR. B.Mayo-WilsonE.NezuA. M.RaoS. M. (2018). Journal article reporting standards for quantitative research in psychology: The APA publications and communications board task. American Psychologist, 73(1), 3-25. https://doi.org/10.1037/amp0000191
BakerT.PollockT. G. 2007. Making the marriage work: the benefits of strategy’s takeover of entrepreneurship for strategic organization. Strategic Organization, 5(3), 297-312. https://doi.org/10.1177/1476127007079957
4.
BandalosD. L.FinneyS. J. (2019). Factor analysis: Exploratory and confirmatory. In HancockG. R.StapletonL. M.MuellerR. O. (Eds.), The reviewer’s guide to quantitative methods in the social sciences (2nd ed., pp. 98–122). Routledge.
BollenK. A. (1989). Structural equations with latent variables. Wiley.
8.
BollenK. A.HardenJ. J.RayS.ZaviscaJ. (2014). BIC And alternative Bayesian information criteria in the selection of structural equation models. Structural Equation Modeling: A Multidisciplinary Journal, 21(1), 1-19. https://doi.org/10.1080/10705511.2014.856691
9.
BoscoF. A.AguinisH.FieldJ. G.PierceC.A.DaltonD. R. (2016). HARKing’s threat to organizational research: Evidence from primary and meta-analytic sources. Personnel Psychology, 69(3), 709-750. https://doi.org/10.1111/peps.12111
10.
CheungG. W.RensvoldR. B. (1999). Testing factorial invariance across groups: A reconceptualization and proposed new method. Journal of Management, 25(1), 1–27.
11.
ChenF.BollenK. A.PaxtonP.CurranP. J.KirbyJ. B. (2001). Improper solutions in structural equation models: Causes, consequences, and strategies. Sociological Methods & Research, 29(4), 468-508. https://doi.org/10.1177/0049124101029004003
12.
ChouC. P.BentlerP. M. (1993). Invariant standardized estimated parameter change for model modification in covariance structure analysis. Multivariate Behavioral Research, 28(1), 97-110. https://doi.org/10.1207/s15327906mbr2801_6
13.
CohenJ. (1988). Statistical power analysis for the behavioral sciences. Lawrence Erlbaum.
14.
ColeD. A.CieslaJ. A.SteigerJ. H. (2007). The insidious effects of failing to include design-driven correlated residuals in latent-variable covariance structure analysis. Psychological Methods, 12(4), 381-398. https://doi.org/10.1037/1082-989X.12.4.381
15.
CollinsL. M.SchaferJ. L.KamC. M. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological Methods, 6(4), 330-351. https://doi.org/10.1037/1082-989X.6.4.330
16.
CortinaJ. M.GreenJ. P.KeelerK. R.VandenbergR. J. (2017). Degrees of freedom in SEM: Are we testing the models that we claim to test?Organizational Research Methods, 20(3), 350-378. https://doi.org/10.1177/1094428116676345
17.
DunbarK. N.FuselsangJ. (2005). Causal thinking in science: How scientists and students interpret the unexpected. In GormanM. E.TweneyR. D.GoodingD. C.KincannonA. P. (Eds), Scientific and technical thinking (pp. 57-80). Lawrence Erlbaum Associates.
EndersC. K. (2006). Analyzing structural equation models with missing data. In HancockG. R.MuellerR. O. (Eds.), Structural equation modeling: A second course (pp. 315-344). Information Age Publishing.
20.
FanX.SivoS. A. (2007). Sensitivity of fit indices to model misspecification and model types. Multivariate Behavioral Research, 42(3), 509-529. https://doi.org/10.1080/00273170701382864
21.
FinneyS. J.DiStefanoC. (2006). Non-normal and categorical data in structural equation modeling. In HancockG. R.MuellerR. D. (Eds.), Structural equation modeling: A second course (pp. 269-314). Information Age Publishing.
22.
GagneP.HancockG. R. (2006). Measurement model quality, sample size, and solution propriety in confirmatory factor models. Multivariate Behavioral Research, 41(1), 65–83.
23.
GerbingD. W.AndersonJ. C. (1984). On the meaning of within-factor correlated measurement errors. Journal of Consumer Research, 11(1), 572-580. https://doi.org/10.1086/208993
24.
GrahamJ. W. (2003). Adding missing-data-relevant variables to FIML-based structural equation models. Structural Equation Modeling, 10(1), 80-100. https://doi.org/10.1207/S15328007SEM1001_4
GrahamJ. W.CoffmanD. L. (2012). Structural equation modeling with missing data. In HoyleR. H. (Ed.), Handbook of structural equation modeling (pp. 277-295). The Guilford Press.
27.
GrahamJ. W.OlchowskiA. E.GilreathT. D. (2007). How many imputations are really needed? Some practical clarifications of multiple imputation theory. Prevention Science, 8(3), 206-213. https://doi.org/10.1007/s11121-007-0070-9
28.
GreenwaldA. G. (2012). There is nothing so theoretical as a good method. Perspectives on Psychological Science, 7(2), 99-108. https://doi.org/10.1177/1745691611434210
29.
HambrickD. C. (2007). The field of management’s devotion to theory: Too much of a good thing?Academy of Management Journal, 50(6), 1346-1352. https://doi.org/10.5465/amj.2007.28166119
30.
HancockG. R.MuellerR. O. (2011). The reliability paradox in assessing structural relations within covariance structure models. Educational & Psychological Measurement, 71(2), 306-324. https://doi.org/10.1177/0013164410384856
31.
HaydukL.CummingsG.BoaduK.Pazderka-RobinsonH.BoulianneS. (2007). Testing! testing! one, two, three–testing the theory in structural equation models!. Personality and Individual Differences, 42(5), 841-850. https://doi.org/10.1016/j.paid.2006.10.001
32.
HenselerJ.DijkstraT. K.SarstedtM.RingleC. M.DiamantopoulosA.StraubD. W., Ketchen, D. J. Jr., Hair, J. F., Hult, G. T. M., & Calantone, R. J. (2014). Common beliefs and reality about PLS: Comments on Rönkkö and Evermann (2013). Organizational Research Methods, 17(2), 182–209. https://doi.org/10.1177/1094428114526928
33.
HerzogW.BoomsmaA. (2009). Small-sample robust estimators of noncentrality-based and incremental model fit. Structural Equation Modeling, 16(1), 1-27. https://doi.org/10.1080/10705510802561279
34.
HollenbeckJ. R.WrightP. M. (2017). Harking, sharking, and tharking: Making the case for post hoc analysis of scientific data. Journal of Management, 43(1), 5-18. https://doi.org/10.1177/0149206316679487
35.
HopwoodC. J.DonnellanM. B. (2010). How should the internal structure of personality inventories be evaluated?. Personality and Social Psychology Review, 14(3), 332–346.
36.
HuL.BentlerP. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6(1), 1–55 https://doi.org/10.1080/10705519909540118
37.
JorgensenT. D.PornprasertmanitS.SchoemannA. M.RosseelY. (2018). Semtools: Useful tools for structural equation modeling. R package version 0.5–1. Retrieved fromhttps://CRAN.R-project.org/package=semTools
KiersH. A. L.TakaneYten BergeJ. M. F. (1996). The analysis of multitrait-multimethod matrices via constrained components analysis. Psychometrika, 61(4), 601-628. https://doi.org/10.1007/BF02294039
40.
KlineR. B. (2016). Principles and practice of structural equation modeling (4th ed.). Guilford Press.
41.
LandisR. S.EdwardsB. D.CortinaJ. M. (2009). On the practice of allowing correlated residuals among indicators in structural equation models. In LanceC. E.VandenbergR. J. (Eds.), Statistical and methodological myths and urban legends: doctrine, verity and fable in the organizational and social sciences (pp. 193-215). Routledge/Psychology Press.
42.
LiF.DuncanT. E.HarmerP.AcockA.StoolmillerM. (1998). Analyzing measurement models of latent variables through multilevel confirmatory factor analysis and hierarchical linear modeling approaches. Structural Equation Modeling: A Multidisciplinary Journal, 5(3), 294–306.
43.
LinL. C.HuangP. H.WengL. J. (2017). Selecting path models in SEM: A comparison of model selection criteria. Structural Equation Modeling: A Multidisciplinary Journal, 24(6), 855-869. https://doi.org/10.1080/10705511.2017.1363652
44.
LittleR. J. (1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association, 83(404), 1198-1202. https://doi.org/10.1080/01621459.1988.10478722
45.
LittleJ. A.RubinD. B. (2020). Statistical analyses with missing data (3rd Edition). Wiley.
46.
LittleT. D.SlegersD. W.CardN. A. (2006). A non-arbitrary method of identifying and scaling latent variables in SEM and MACS models. Structural Equation Modeling, 13(1), 59-72. https://doi.org/10.1207/s15328007sem1301_3
MacCallumR. C.BrowneM. W.SugawaraH. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological Methods, 1(2), 130-149. https://doi.org/10.1037/1082-989X.1.2.130
49.
MacCallumR. C.RoznowskiM.NecowitzL. B. (1992). Model modifications in covariance structure analysis: The problem of capitalization on chance. Psychological Bulletin, 111(3), 490-504. https://doi.org/10.1037/0033-2909.111.3.490
50.
McIntoshC. N.EdwardsJ. R.AntonakisJ. (2014). Reflections on partial least squares path modeling. Organizational Research Methods, 17(2), 210-251. https://doi.org/10.1177/1094428114529165
51.
McNeishD. (2020). Should we use F-tests for model fit instead of chi-square in overidentified structural equation models?Organizational Research Methods, 23(3), 487-510. https://doi.org/10.1177/1094428118809495
52.
McNeishD.AnJ.HancockG. R. (2018). The thorny relation between measurement quality and fit index cutoffs in latent variable models. Journal of Personality Assessment, 100(1), 43-52. https://doi.org/10.1080/00223891.2017.1281286
53.
MeadeA. W.JohnsonE. C.BraddyP. W. (2008). Power and sensitivity of alternative fit indices in tests of measurement invariance. Journal of Applied Psychology, 93(3), 568-592. https://doi.org/10.1037/0021-9010.93.3.568
54.
MoshagenM. (2012). The model size effect in SEM: Inflated goodness-of-fit statistics are due to the size of the covariance matrix. Structural Equation Modeling: A Multidisciplinary Journal, 19(1), 86-98. https://doi.org/10.1080/10705511.2012.634724
55.
MurphyK. R.AguinisH. (2019). HARKing: How badly can cherry-picking and question trolling produce bias in published results?Journal of Business and Psychology, 34(1), 1-17. https://doi.org/10.1007/s10869-017-9524-7
56.
Muthén, L. K., & Muthén, B. O. (2017). Mplus user’s guide (8th ed). Muthén & Muthén.
57.
NewmanD. A. (2009). Missing data techniques and low response rates: The role of systematic nonresponse parameters. In LanceC. EVandenbergR. J. (Eds.). Statistical and methodological myths and urban legends (pp. 7-36). Routledge.
NyeC. D.AllemandM.GoslingS. D.PotterJ.RobertsB. W. (2016). Personality trait differences between young and middle-aged adults: Measurement artifacts or actual trends?Journal of Personality, 84(4), 473-492. https://doi.org/10.1111/jopy.12173
60.
NyeC. D.DrasgowF. (2011). Assessing goodness of fit: simple rules of thumb simply do not work. Organizational Research Methods, 14(3), 548-570. https://doi.org/10.1177/1094428110368562
PodsakoffP. M.MacKenzieS. B.LeeJ. Y.PodsakoffN. P. (2003). Common method biases in behavioral research: A critical review of the literature and recommended remedies. Journal of Applied Psychology, 88(5), 879-903. https://doi.org/10.1037/0021-9010.88.5.879
63.
PreacherK. J.CoffmanD. L. (2006, May). Computing power and minimum sample size for RMSEA [Computer software].
64.
RindskopfD. (1984). Structural equation models: Empirical identification, heywood cases, and related problems. Sociological Methods & Research, 13(1), 109-119. https://doi.org/10.1177/0049124184013001004
RubinD. B. (1976). Inference and missing data. Biometrika, 63, 581–592.
67.
SarisW. E.SatorraA. (1993). Power evaluations in structural equation models. In BollenK. A.LongJ. S. (Eds.), Testing structural equation models (pp. 181-204). Sage.
68.
SarisW. E.SatorraA.SörbomD. (1987). The detection and correction of specification errors in structural equation models. In CloggC. C. (Ed.), Sociological methodology (pp. 105-129). Jossey-Bass.
69.
SatorraA.BentlerP.M. (2010). Ensuring positiveness of the scaled difference chi-square test statistic. Psychometrika, 75(2), 243-248. https://doi.org/10.1007/s11336-009-9135-y
70.
Schermelleh-EngelK.MoosbruggerH.MullerH. (2003). Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures. Methods of Psychological Research, 8(2), 23-74.
SchumakerR. E.LomaxR. G. (2016). A beginner’s guide to structural equation modeling (4th ed.). Routledge.
73.
SilviaE. S. M.MacCallumR. C. (1988). Some factors affecting the success of specification searches in covariance structure modeling. Multivariate Behavioral Research, 23(3), 297-326. https://doi.org/10.1207/s15327906mbr2303_2
74.
SomarajuA. V., NyeC. DOlenickJ. (in press). A review of measurement equivalence in organizational research: what’s old, what’s new, and what’s next?Organizational Research Methods. https://doi.org/10.1177/10944281211056524
75.
StapletonL. M.McNeishD. M.YangJ. S. (2016). Multilevel and single-level models for measured and latent variables when data are clustered. Educational Psychologist, 51(3–4), 317-330. https://doi.org/10.1080/00461520.2016.1207178
76.
StarkS.ChernyshenkoO. S.DrasgowF. (2006). Detecting differential item functioning with confirmatory factor analysis and item response theory: Toward a unified strategy. Journal of Applied Psychology, 91(6), 1292-1306. https://doi.org/10.1037/0021-9010.91.6.1292
77.
SteigerJ. H. (1990). Structural model evaluation and modification: An interval estimation approach. Multivariate Behavioral Research, 25(2), 173-180. https://doi.org/10.1207/s15327906mbr2502_4
78.
SuR.TayL.LiaoH. Y.ZhangQ.RoundsJ. (2019). Toward a dimensional model of vocational interests. Journal of Applied Psychology, 104(5), 690–714.
79.
TanakaJ. S. (1987). “How big is big enough?": Sample size and goodness of fit in structural equation models with latent variables. Child Development, 58(1), 134-146. https://doi.org/10.2307/1130296
80.
TayL.WooS. E.VermuntJ. K. (2014). A conceptual and methodological framework for psychometric isomorphism: Validation of multilevel construct measures. Organizational Research Methods, 17(1), 77-106. https://doi.org/10.1177/1094428113517008
81.
ThompsonB. (2000). Ten commandments of structural equation modeling. In GrimmL. G.YarnoldP. R. (Eds.), Reading and understanding MORE multivariate statistics (pp. 261-283). American Psychological Association.
82.
VandenbergR. J.GrelleD. M. (2009). Alternative model specifications in structural equation modeling: Fact, fictions, and truth. In LanceC. E.VandenbergR. J. (Eds.), Statistical and methodological myths and urban legends: doctrine, verity and fable in the organizational and social sciences (pp. 165-191). Routledge/Psychology Press.
83.
VandenbergR. J.LanceC. E. (2000). A review and synthesis of the measurement invariance literature: Suggestions, practices, and recommendations for organizational research. Organizational Research Methods, 3(1), 4-70. https://doi.org/10.1177/109442810031002
84.
VandenbergR. J.MorelliN. A. (2016). A contemporary update on testing for measurement equivalence and invariance. In MeyerJ. P. (Ed.), The handbook of employee commitment (pp. 449–461). Edward Elgar.
85.
WangY. A., & RhemtullaM. (2021). Power analysis for parameter estimation in structural equation modeling: A discussion and tutorial. Advances in Methods and Practices in Psychological Science, 4(1), 1-17. https://doi.org/10.1177/2515245920918253
86.
YuanK.-H.BentlerP. M.ZhangW. (2005). The effect of skewness and kurtosis on mean and covariance structure analysis: The univariate case and its multivariate implication. Sociological Methods and Research, 34(2), 240-258. https://doi.org/10.1177/0049124105280200
87.
ZhangG.PreacherK. J. (2015). Factor rotation and standard errors in exploratory factor analysis. Journal of Educational and Behavioral Statistics, 40(6), 579-603.
