G Lecture 51 Estimation details Testing Fit Fit indices Arguing for models

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

G Lecture 53 EQS Syntax

G 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

G Lecture 55 Form of  The form of  can be worked out with expectation operators

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

G Lecture 57 Estimation Example using Excel

G 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

G 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

G 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

G 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

G 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

G 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

G 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

G 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

G 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.

G 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)

G 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

G 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.