Evaluation of structural equation models. Hans Baumgartner Penn State University

Transcription

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 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

4 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

5 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 and Appendix F on how to do bootstrapping]

6 Model estimation and testing Measurement model factor loadings, factor (co)variances, and error variances reliabilities and discriminant validity [see Appendix G on how to compute composite reliability in LISREL] Latent variable model structural coefficients and equation disturbances direct, indirect, and total effects [see Appendix H] explained variation in endogenous constructs

7 Model estimation and testing Power [see Appendix I] Model modification and model comparison [see Appendix J] Measurement model Latent variable model Model-based residual analysis Cross-validation Model equivalence and near equivalence [see Appendix K] Latent variable scores [see Appendix L]

8 Appendix A: Reflective vs. formative measurement models focal construct focal construct focal construct focal construct

9 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

10 Appendix C: Consumer Behavior Consumer Behavior Attitudes A ad as a mediator of advertising effectiveness: Four structural specifications (MacKenzie et al. 1986) Affect transfer hypothesis Dual mediation hypothesis C ad A ad C ad A ad C b A b BI C b A b BI Reciprocal mediation hypothesis Independent influences hypothesis C ad A ad C ad A ad C b A b BI C b A b BI

11 Appendix D: Overall fit indices Stand-alone fit indices Incremental fit indices χ 2 test and variations Noncentralitybased measures Information theory-based measures Others Type I indices Type II indices minimum fit function χ 2 (C1) normal theory WLS χ 2 (C2) S-B scaled χ 2 (C3) χ 2 corrected for nonnormality (C4) NCP Rescaled NCP (t) RMSEA MC AIC SBC CIC ECVI (S)RMR GFI PGFI AGFI Gamma hat CN NFI RFI [χ 2 or f] [χ 2 /df] CFI [χ 2 -df] TLI [(χ 2 -df)/df] IFI TLI χ 2 /df minimum fit function f Scaled LR

12 Types of error in covariance structure modeling best fit of the model to S for a given discrepancy function Σˆ known - random ~ Σ 0 unknown - fixed error of approximation (an unknown constant) Σ 0 unknown - fixed best fit of the model to Σ 0 for a given discrepancy function population covariance matrix

13 Incremental fit indices type I indices: GF t GF GF n t or BF n BF BF t n type II indices: GF GF t n E( GF ) GF t n or BF n BF n BF t E( BF ) t 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;

14 direct indirect Appendix H: Direct, indirect, and total effects inconveniences rewards encumbrances inconveniences rewards encumbrances Aact BI B -.15 BI B inconveniences rewards encumbrances Aact BI B

15 Appendix I: True state of nature H 0 true H 0 false Accept H 0 Correct decision Type II error (β) Decision Reject H 0 Type I error (α) Correct decision

16 low power high nonsignificant test statistic significant

17 Appendix J: Model comparisons saturated structural model (M s ) next most likely unconstrained model (M u ) lowest χ 2 lowest df target model (M t ) next most likely constrained model (M c ) null structural model (M n ) highest χ 2 highest df

18 Appendix K: Model equivalence η 3 η 3 η 1 η 1 η 4 η 4 η 2 η 2 η 5 η 5 η 3 η 3 η 1 η 1 η 4 η 4 η 2 η 2 η 5 η 5