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Evaluation of structural equation models

Evaluation of structural equation models. Hans Baumgartner Penn State University. Issues related to the initial specification of theoretical models of interest. Model specification: Measurement model: EFA vs. CFA reflective vs. formative indicators [see Appendix A]

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Evaluation of structural equation models

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  1. Evaluation of structural equation models Hans Baumgartner Penn State University

  2. Issues related to the initial specification of theoretical models of interest • Model specification: • Measurement model: • EFA vs. CFA • reflective vs. formative indicators [see Appendix A] • number of indicators per construct [see Appendix B] • total aggregation model • partial aggregation model • total disaggregation model • Latent variable model: • recursive vs. nonrecursive models • alternatives to the target model [see Appendix C for an example]

  3. x1 x2 x1 x2 x3 x4 x5 x6 x7 x8 d1 d2 d3 d4 d5 d6 d7 d8

  4. z1 z2 x1 x2 x1 x2 x3 x4 x5 x6 x7 x8

  5. Criteria for distinguishing between reflective and formative indicator models • Are the indicators manifestations of the underlying construct or defining characteristics of it? • Are the indicators conceptually interchangeable? • Are the indicators expected to covary? • Are all of the indicators expected to have the same antecedents and/or consequences? Based on MacKenzie, Podsakoff and Jarvis, JAP 2005, pp. 710-730.

  6. Issues related to the initial specification of theoretical models of interest • Model misspecification • omission/inclusion of (ir)relevant variables • omission/inclusion of (ir)relevant relationships • misspecification of the functional form of relationships • Model identification • Sample size • Statistical assumptions

  7. Data screening • Inspection of the raw data • detection of coding errors • recoding of variables • treatment of missing values • Outlier detection • Assessment of normality • Measures of association • regular vs. specialized measures • covariances vs. correlations • non-positive definite input matrices

  8. Model estimation and testing • Model estimation • Estimation problems • nonconvergence or convergence to a local optimum • improper solutions • problems with standard errors • empirical underidentification • Overall fit assessment [see Appendix D] • Local fit measures [see Appendix E on how to obtain robust standard errors]

  9. Types of error in covariance structure modeling best fit of the model to S for a given discrepancy function error of estimation (an unknown random variable) overall error (an unknown random variable) known - random error of approximation (an unknown constant) unknown - fixed unknown - fixed best fit of the model to S0 for a given discrepancy function population covariance matrix

  10. Incremental fit indices • type I indices: GFt, BFt = value of some stand-alone goodness- or badness-of-fit index for the target model; GFn, BFn = value of the stand-alone index for the null model; E(GFt), E(BFt) = expected value of GFt or BFt assuming that the target model is true; • type II indices:

  11. Model estimation and testing • Measurement model • factor loadings, factor (co)variances, and error variances • reliabilities and discriminant validity • Latent variable model • structural coefficients and equation disturbances • direct, indirect, and total effects [see Appendix F] • explained variation in endogenous constructs

  12. Direct, indirect, and total effects -.28 inconveniences .44 1.10 .49 direct rewards Aact BI B encumbrances -.05 inconveniences -.15 -.31 .48 .24 BI rewards B indirect -.05 encumbrances -.03 inconveniences -.15 -.28 -.31 .48 .24 .44 rewards Aact BI B total -.05 -.05 encumbrances -.03

  13. Model estimation and testing • Power [see Appendix G] • Model modification and model comparison [see Appendix H] • Measurement model • Latent variable model • Model-based residual analysis • Cross-validation • Model equivalence and near equivalence [see Appendix I] • Latent variable scores [see Appendix J]

  14. True state of nature H0 true H0 false Correct decision Type II error (b) Accept H0 Decision Type I error (a) Correct decision Reject H0

  15. power low high non- significant test statistic significant

  16. Model comparisons lowest c2 saturated structural model (Ms) lowest df next most likely unconstrained model (Mu) target model (Mt) next most likely constrained model (Mc) highest c2 null structural model (Mn) highest df

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