nach oben

Journal of the Academy of Marketing Science

Erschienen in:

17.09.2018 | Methodological Paper

Flexible cutoff values for fit indices in the evaluation of structural equation models

verfasst von: Thomas Niemand, Robert Mai

Erschienen in: Journal of the Academy of Marketing Science | Ausgabe 6/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

Researchers often struggle when applying ‘golden rules of thumb’ to evaluate structural equation models. This paper questions the notion of universal thresholds and calls for adjusted orientation points that account for sample size, factor loadings, the number of latent variables and indicators, as well as data (non-)normality. This research explores the need for flexible cutoffs and their accuracy in single- and two-index strategies. Study 1 reveals that many indices are biased; thus, rigid cutoffs can become imprecise. Flexible cutoff values are shown to compensate for the unique distorting patterns and prove to be particularly beneficial for moderate misspecification. Study 2 sheds further light on this ‘gray’ area of misspecification and disentangles the different sources of misspecification. Study 3 finally investigates the performance of flexible cutoffs for non-normal data. Having substantiated higher performance for flexible reference values, this paper provides to managers an easy-to-use tool that facilitates the determination of adequate cutoffs.

Vorheriger Artikel I thought it was over, but now it is back: customer reactions to ex post time extensions of sales promotions

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

The definitions of Type I and Type II errors are based on Hu and Bentler (1999, p. 5) and Marsh et al. (2004, pp. 335-336) (see the Web Appendix for a more detailed explanation).

Note that Hu and Bentler’s (1999) recommendations are often ‘inappropriately promoted’ (Marsh et al. 2004, p. 322) in textbooks or guidelines, citing single-index cutoffs while clearly promoting two-index cutoffs.

Pairwise manipulation is necessary to ensure model identification and comparability (i.e., positive and constant degrees of freedom across all combinations).

For example, for two five-indicator constructs, the initial (a1-a5 on A) and final specification (b1-b5 on A) are identical, forming the level MM = 0, because the meaning of the two constructs is reversed. For the same reason, the second (b1, a2-a5) and fourth (b1-b4, a5) specifications are equal (MM = 1), and only the third (a1-a3, b4-b5) specification entails the highest possible degree of misspecification (MM = 2).

It should be noted that robust and scaled versions of the χ² statistic and χ² based fit indices are readily available in modern statistical software (e.g., LISREL, lavaan). Users of this software should check whether these corrections are automatically applied or can be applied when estimating fit indices (if non-normality may be an issue).

Anderson, J. C., & Gerbing, D. W. (1984). The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis. Psychometrika, 49(2), 155–173.CrossRef

Babakus, E., & Boller, G. W. (1992). An empirical assessment of the SERVQUAL scale. Journal of Business Research, 24(3), 253–268.CrossRef

Bagozzi, R. P., & Yi, Y. (1988). On the evaluation of structural equation models. Journal of the Academy of Marketing Science, 16(1), 74–94.CrossRef

Bagozzi, R. P., & Yi, Y. (1989). On the use of structural equation models in experimental designs. Journal of Marketing Research, 26(3), 271–284.CrossRef

Bandalos, D. L. (2002). The effects of item parceling on goodness-of-fit and parameter estimate bias in structural equation modeling. Structural Equation Modeling, 9(1), 78–102.CrossRef

Bearden, W. O., Sharma, S., & Teel, J. E. (1982). Sample size effects on chi square and other statistics used in evaluating causal models. Journal of Marketing Research, 19(4), 425–430.CrossRef

Beauducel, A., & Wittmann, W. W. (2005). Simulation study on fit indexes in CFA based on data with slightly distorted simple structure. Structural Equation Modeling, 12(1), 41–75.CrossRef

Bentler, P. M. (1990). Comparative fit indexes in structural models. Psychological Bulletin, 107(2), 238–246.CrossRef

Bentler, P. M. (1995). EQS structural equations program manual. Encino: Multivariate Software.

Bentler, P. M. (2007). On tests and indices for evaluating structural models. Personality and Individual Differences, 42(5), 825–829.CrossRef

Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88(3), 588–606.CrossRef

Bollen, K. A., Harden, J. J., Ray, S., & Zavisca, J. (2014). BIC and alternative Bayesian information criteria in the selection of structural equation models. Structural Equation Modeling, 21(1), 1–19.CrossRef

Bone, P. F., Sharma, S., & Shimp, T. A. (1989). A bootstrap procedure for evaluating goodness-of-fit indices of structural equation and confirmatory factor models. Journal of Marketing Research, 26(1), 105–111.CrossRef

Boomsma, A. (1985). Nonconvergence, improper solutions, and starting values in LISREL maximum likelihood estimation. Psychometrika, 50(2), 229–242.CrossRef

Breivik, E., & Olsson, U. H. (2001). Adding variables to improve fit: The effect of model size on fit assessment in LISREL. In S. DuToit, R. Cudeck, & D. Sorbom (Eds.), Structural equation modeling: Present and future (pp. 169–194). Lincolnwood: Scientific Software International, Inc..

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

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.CrossRef

Cheung, G. W., & Rensvold, R. B. (2001). The effects of model parsimony and sampling error on the fit of structural equation models. Organizational Research Methods, 4(3), 236–264.CrossRef

Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale: Lawrence Erlbaum.

Curran, P. J., West, S. G., & Finch, J. F. (1996). The robustness of test statistics to nonnormality and specification error in confirmatory factor analysis. Psychological Methods, 1(1), 16–29.CrossRef

Curran, P. J., Bollen, K. A., Paxton, P., Kirby, J., & Chen, F. (2002). The noncentral chi-square distribution in misspecified structural equation models: Finite sample results from a Monte Carlo simulation. Multivariate Behavioral Research, 37(1), 1–36.CrossRef

Enders, C. K. (2001). The impact of nonnormality on full information maximum-likelihood estimation for structural equation models with missing data. Psychological Methods, 6(4), 352–370.CrossRef

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, 12(3), 343–367.CrossRef

Fan, X., & Sivo, S. A. (2007). Sensitivity of fit indices to model misspecification and model types. Multivariate Behavioral Research, 42(3), 509–529.CrossRef

Fouladi, R. T. (2000). Performance of modified test statistics in covariance and correlation structure analysis under conditions of multivariate nonnormality. Structural Equation Modeling, 7(3), 356–410.CrossRef

Gagne, P., & Hancock, G. R. (2006). Measurement model quality, sample size, and solution propriety in confirmatory factor models. Multivariate Behavioral Research, 41(1), 65–83.CrossRef

Gupta, S., Hanssens, D., Hauser, J. R., Lehmann, D., & Schmitt, B. (2014). Introduction to theory and practice in marketing conference special section of marketing science. Marketing Science, 33(1), 1–5.CrossRef

Hau, K. T., & Marsh, H. W. (2004). The use of item parcels in structural equation modelling: Non-normal data and small sample sizes. British Journal of Mathematical and Statistical Psychology, 57(2), 327–351.CrossRef

Hayduk, L., Cummings, G., Boadu, K., Pazderka-Robinson, H., & Boulianne, S. (2007). Testing! Testing! One, two, three–testing the theory in structural equation models! Personality and Individual Differences, 42(5), 841–850.CrossRef

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(3), 319–336.CrossRef

Herzog, W., Boomsma, A., & Reinecke, S. (2007). The model-size effect on traditional and modified tests of covariance structures. Structural Equation Modeling, 14(3), 361–390.CrossRef

Hu, L., & Bentler, P. M. (1998). Fit indices in covariance structure modeling: Sensitivity to underparameterized model misspecification. Psychological Methods, 3(4), 424–453.CrossRef

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.CrossRef

Hulland, J., Baumgartner, H., & Smith, K. M. (2017). Marketing survey research best practices: Evidence and recommendations from a review of JAMS articles. Journal of the Academy of Marketing Science, 46(1), 92–108.CrossRef

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365.

Jackson, D. L. (2007). The effect of the number of observations per parameter in misspecified confirmatory factor analytic models. Structural Equation Modeling, 14(1), 48–76.CrossRef

Jackson, D. L., Gillaspy, J. A. J., & Purc-Stephenson, R. (2009). Reporting practices in confirmatory factor analysis: An overview and some recommendations. Psychological Methods, 14(1), 6–23.CrossRef

Jöreskog, K. G., & Sörbom, D. (1981). LISREL V: Analysis of linear structural relationships by maximum likelihood and least squares methods. University of Uppsala, Department of Statistics.

Jöreskog, K. G., & Sörbom, D. (1982). Recent developments in structural equation modeling. Journal of Marketing Research, 19(4), 404–416.CrossRef

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(3), 333–351.CrossRef

Kenny, D. A., Kaniskan, B., & McCoach, D. B. (2015). The performance of RMSEA in models with small degrees of freedom. Sociological Methods & Research, 44(3), 486–507.CrossRef

Kumar, V., Sharma, A., & Gupta, S. (2017). Accessing the influence of strategic marketing research on generating impact: Moderating roles of models, journals, and estimation approaches. Journal of the Academy of Marketing Science, 45(2), 164–185.CrossRef

La Du, T. J., & Tanaka, J. S. (1989). Influence of sample size, estimation method, and model specification on goodness-of-fit assessments in structural equation models. Journal of Applied Psychology, 74(4), 625–635.CrossRef

Lehmann, D. R. (2014). Introduction to the special issue on theory and practice in marketing. Journal of Marketing Research, 51(6), 645–646.CrossRef

Lehmann, D. R., McAlister, L., & Staelin, R. (2013). Sophistication in research in marketing. Journal of Marketing, 75(4), 155–165.CrossRef

Lombardi, L., & Pastore, M. (2012). Sensitivity of fit indices to fake perturbation of ordinal data: A sample by replacement approach. Multivariate Behavioral Research, 47(4), 519–546.CrossRef

MacCallum, R. C., Widaman, K. F., Preacher, K. J., & Hong, S. (2001). Sample size in factor analysis: The role of model error. Multivariate Behavioral Research, 36(4), 611–637.CrossRef

Marsh, H. W., Balla, J. R., & McDonald, R. P. (1988). Goodness-of-fit indexes in confirmatory factor analysis: The effect of sample size. Psychological Bulletin, 103(3), 391–410.CrossRef

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(2), 181–220.CrossRef

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(3), 320–341.CrossRef

Miao, C. F., & Wang, G. (2017). Effects of work–family interface conflicts on salesperson behaviors: A double-edged sword. Journal of the Academy of Marketing Science, 45(5), 762–783.CrossRef

Mishra, S., Umesh, U., & Stem Jr., D. E. (1993). Antecedents of the attraction effect: An information-processing approach. Journal of Marketing Research, 30(3), 331–349.CrossRef

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(1), 86–98.CrossRef

Muthén, B., & Asparouhov, T. (2012). Bayesian structural equation modeling: A more flexible representation of substantive theory. Psychological Methods, 17(3), 313–335.CrossRef

Muthén, B., & Kaplan, D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38(2), 171–189.CrossRef

Nasser, F., & Wisenbaker, J. (2003). A Monte Carlo study investigating the impact of item parceling on measures of fit in confirmatory factor analysis. Educational and Psychological Measurement, 63(5), 729–757.CrossRef

Nye, C. D., & Drasgow, F. (2011). Assessing goodness of fit: Simple rules of thumb simply do not work. Organizational Research Methods, 14(3), 548–570.CrossRef

Olsson, U. H., Foss, T., Troye, S. V., & Howell, R. D. (2000). The performance of ML, GLS, and WLS estimation in structural equation modeling under conditions of misspecification and nonnormality. Structural Equation Modeling, 7(4), 557–595.CrossRef

Parasuraman, A., Zeithaml, V. A., & Berry, L. L. (1988). Servqual: A multiple-item scale for measuring consumer perc. Journal of Retailing, 64(1), 12–40.

Podsakoff, P. M., MacKenzie, S. B., Lee, J.-Y., & Podsakoff, N. 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.CrossRef

Satorra, A., & Bentler, P. M. (2001). A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66(4), 507–514.CrossRef

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.CrossRef

Schermelleh-Engel, K., Moosbrugger, H., & Müller, H. (2003). Evaluating the fit of structural equation models: Tests of significance and descriptive goodness-of-fit measures. Methods of Psychological Research Online, 8(2), 23–74.

Sedlmeier, P., & Gigerenzer, G. (1989). Do studies of statistical power have an effect on the power of studies? Psychological Bulletin, 105(2), 309–316.CrossRef

Sharma, S., Mukherjee, S., Kumar, A., & Dillon, W. R. (2005). A simulation study to investigate the use of cutoff values for assessing model fit in covariance structure models. Journal of Business Research, 58(7), 935–943.CrossRef

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. Journal of Experimental Education, 74(3), 267–288.CrossRef

Sleep, S., Bharadwaj, S., & Lam, S. K. (2015). Walking a tightrope: The joint impact of customer and within-firm boundary spanning activities on perceived customer satisfaction and team performance. Journal of the Academy of Marketing Science, 43(4), 472–489.CrossRef

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

Tucker, L. R., & Lewis, C. (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38(1), 1–10.CrossRef

West, S. G., Taylor, A. B., & Wu, W. (2012). Model fit and model selection in structural equation modeling. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (Vol. 1, pp. 209–231). New York: Guilford Press.

Ximénez, C. (2009). Recovery of weak factor loadings in confirmatory factor analysis under conditions of model misspecification. Behavior Research Methods, 41(4), 1038–1052.CrossRef

Zboja, J. J., Clark, R. A., & Haytko, D. L. (2016). An offer you can’t refuse: Consumer perceptions of sales pressure. Journal of the Academy of Marketing Science, 44(6), 806–821.CrossRef

Titel: Flexible cutoff values for fit indices in the evaluation of structural equation models
verfasst von: Thomas Niemand
Robert Mai
Publikationsdatum: 17.09.2018
Verlag: Springer US
Erschienen in: Journal of the Academy of Marketing Science / Ausgabe 6/2018
Print ISSN: 0092-0703
Elektronische ISSN: 1552-7824
DOI: https://doi.org/10.1007/s11747-018-0602-9

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 6/2018

Modeling the effects of dynamic group influence on shopper zone choice, purchase conversion, and spending

I thought it was over, but now it is back: customer reactions to ex post time extensions of sales promotions

Understanding the quality–quantity conundrum of customer referral programs: effects of contribution margin, extraversion, and opinion leadership

Institutions and opportunism in buyer–supplier exchanges: the moderated mediating effects of contractual and relational governance

Correction to: selling, general, and administrative expense (SGA)-based metrics in marketing: conceptual and measurement challenges

Selling, general, and administrative expense (SGA)-based metrics in marketing: conceptual and measurement challenges