1. G89.2247 Lecture 51 Estimation details Testing Fit Fit indices Arguing for models

2. G89.2247 Lecture 52 A simulated example EQS makes it easy to simulate an example  Y2Y2 Y1Y1 X2X2       X1X1 

3. G89.2247 Lecture 53 EQS Syntax

4. G89.2247 Lecture 54 Estimating the Recursive Example Specify parameters to be estimated in model (  Compute sample variance/covariance S Construct expected representation of variance covariance under model [  Choose estimates that make representation as close as possible according to a fitting function

5. G89.2247 Lecture 55 Form of  The form of  can be worked out with expectation operators

6. G89.2247 Lecture 56 Fitting Functions ML minimizes ULS minimizes GLS minimizes ADF minimizes a weighted least squares criterion, but with a weight different than GLS

7. G89.2247 Lecture 57 Estimation Example using Excel

8. G89.2247 Lecture 58 Measures of Fit: Chi Square If model is not saturated, and If residuals of Y have multivariate normal distribution  ML*(N-1) and GLS*(N-1) have large sample chi squared distributions  Degrees of freedom given by difference in number of parameters in model compared to saturated model

9. G89.2247 Lecture 59 Chi Square Test Issues Appeal of Chi Square Test  Makes model fit appear confirmatory  Can reject an ill fitting model  Can compare nested models  Can calculate power Problems with Chi Square Test  Global test will reject a good model if data are not multivariate normal  Usual issues of significance testing

10. G89.2247 Lecture 510 Global Fit Measures A host of measures have been developed to provide an index of fit  Many scaled between 0 and 1 Analogous to R-square in regression Began to be used as test of model fit oConventional cut point: >.90  Some focus on residuals Conventional cut point <.05  All measures look at global rather than specific fit

11. G89.2247 Lecture 511 Hu and Bentler (1995) Found that fit measures were not robust  Different answers for different sample sizes  Affected by distribution and independence of latent variables  Gave different answers depending on which fitting algorithm is used. When ML is used, N>250, and latent variables are independent, H&B recommend  NNFI, Bollen89(IFI), CFI, MacDonald (MFI)  Best for comparing models rather than for rigid test >.90

12. G89.2247 Lecture 512 Details of Fit Statistics Measures based on Relative Fit  Define a “strawman” model that is point of comparison Usually model that assumes that all variables are uncorrelated with each other Compute LR test statistic, T B on df B  Construct credible “target” model Compute LR test statistic, T T on df T

13. G89.2247 Lecture 513 Type I Comparative Index Bentler and Bonett NFI  “Normed Fit Index”  Falls in range 0,1. Interpreted as amount of available discrepancy accounted by target model  Apparently is affected by N, and estimation method when sample size is small

14. G89.2247 Lecture 514 Type II Comparative Index: (refined by central chi square) Tucker-Lewis (NNFI) Reaches 1.0 if T T equals its expected value Not normed in range 0,1 Seems to be less affected by N, and relatively stable across GLS and ML  Hu and Bentler recommend it

15. G89.2247 Lecture 515 Type III Comparative Index (refined by noncentral chi square) The Comparative Fit Index (CFI) CFI=BFI (if BFI≥0) CFI=0 (otherwise) Appears more stable for small samples (n>250) and relatively stable from ML to GLS when latent variables are independent.  Hu and Bentler recommend

16. G89.2247 Lecture 516 Absolute Fit Measures: GFI and AGFI Jöreskog suggested two measures based on multivariate variance expressions AGFI adjusts for parsimony given r=(p+q) variables Hu and Bentler report that GFI and AGFI accept many models when N is large, and they do not perform well when latent variables are associated.

17. G89.2247 Lecture 517 Other Absolute Fit Measures Akaike AIC=T T +2f  Where T T is the ML chi square and f is the number of parameters estimated by the model  Models with smaller AIC are preferred Root Mean Square Error of Approximation  where F is the minimized fitting function Look for values less than.05 See Browne and Cudeck(1993)

18. G89.2247 Lecture 518 Error of Approximation vs. Errors of Estimation Browne, M. W. and Cudeck, R. (1993) Alternative ways of assessing model fit. In Bollen, K.A. & Long, J.S. (eds) Testing structural equation models. Newbury Park, CA: Sage. When measurement models are considered along with structural models, almost all SEM models are misspecified to some extent  We should ask about that extent  We should distinguish between how close the model approximates the true covariance structure and how fuzzy is our estimate of the model.  They find support for RMSEA as an index

19. G89.2247 Lecture 519 Arguing for Models People report many fit statistics and claim that a model fits if values exceed.90. Be skeptical about global fit statistics – The fit can be great in one part of model and awful in another Most impressive is a series of models, with one fitting better than another If model is developed with a data set, then the fit statistics should be calculated with a replication data set.