Evaluation of structural equation models Hans Baumgartner Penn State University

Evaluating structural equation models 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]

Evaluating structural equation models 11 22 33 44 55 66 77 88 x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 11 22

x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 11 22 11 22

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

Evaluating structural equation models

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

Evaluating structural equation models 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

Evaluating structural equation models 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]

Evaluating structural equation models

known - random population covariance matrix best fit of the model to  0 for a given discrepancy function unknown - fixed best fit of the model to S for a given discrepancy function error of approximation (an unknown constant) error of estimation (an unknown random variable) overall error (an unknown random variable) Types of error in covariance structure modeling

Evaluating structural equation models Incremental fit indices GF t, BF t =value of some stand-alone goodness- or badness-of-fit index for the target model; GF n, BF n =value of the stand-alone index for the null model; E(GF t ), E(BF t ) =expected value of GF t or BF t assuming that the target model is true; type I indices: type II indices:

Evaluating structural equation models 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

Evaluating structural equation models Direct, indirect, and total effects inconveniences rewards encumbrances AactBIB inconveniences rewards encumbrances BI B.24 inconveniences rewards encumbrances AactBIB direct indirect total

Evaluating structural equation models 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]

Evaluating structural equation models Decision True state of nature Accept H 0 H 0 trueH 0 false Reject H 0 Correct decision Correct decision Type I error (  Type II error ( 

Evaluating structural equation models test statistic power non- significant low high

Evaluating structural equation models Model comparisons saturated structural model (M s ) null structural model (M n ) target model (M t ) next most likely unconstrained model (M u ) next most likely constrained model (M c ) lowest  2 lowest df highest  2 highest df

Evaluating structural equation models