Confirmatory Factor Analysis CFA

Confirmatory Factor Analysis CFA PowerPoint PPT Presentation


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Confirmatory Factor Analysis. a priori measurement model is specified and testedfactor loadings direct relations between observed and latent variables are modelederror terms are the leftovers"estimate the variance of these thingsfactor variances/covariancestypically we are interested in the standardized covariances between factorsfactor correlations.

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Confirmatory Factor Analysis CFA

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1. Confirmatory Factor Analysis (CFA) CFA is used when strong theory and/or when a strong empirical base is available Specify relations a priori number of factors relations among factors (i.e., correlated vs. uncorrelated) variables specified as fixed or free on a respective factor(s)

2. Confirmatory Factor Analysis a priori measurement model is specified and tested factor loadings direct relations between observed and latent variables are modeled error terms are the “leftovers” estimate the variance of these things factor variances/covariances typically we are interested in the standardized covariances between factors factor correlations

3. Two-factor correlated CFA model

4. The Process Specifying the pattern matrix: * = parameter we will estimated (free) 0 = parameter noted estimated (fixed)

5. The Process Model specification via the Bentler-Weeks model all variables in a model are categorized as IVs or DVs DV = variable with unidirectional arrows aiming at it want to explain the variance in these variables with other variables (IVs) e.g., our eight observed variables IV = no unidirectional arrows aiming at it but there can be an unanalyzed association mathematically we estimate the error variances from slide 3 as well

6. The Process you write an equation for each DV S1 = ?(Academic Self-Esteem) + e1 S2 = ?(Academic Self-Esteem) + e2 ..... S5 = ?(Relationship Self-Esteem) + e5 S6 = ?(Relationship Self-Esteem) + e6 .... ? represents a regression coefficient (factor loading) e represents error (or residual), this path is not directly estimated (fixed) predetermined by the factor loading

7. The Process core parameters in CFA are these factor loadings and variances/covariances for IVS notice that the latter are not directly specified in these equations e.g., covariance between Academic and Relationship Self-Esteem they are a function of the equations that you see via some complicated matrix algebra however, you must specify them to solve the equations software does this for us

8. The Process Estimating the model using primarily maximum likelihood estimation produces a fit function Determination of model fit done at two levels overall model fit individual parameter fit parameters = generally factor loadings in CFA but include factor covariances (correlations) if specified

9. The Process Overall model fit (Goodness of fit) tells us if the model should be accepted or rejected if model is accepted, interpret model parameters if model is rejected, do not interpret model parameters Determining goodness of fit test statistic ?2 provides a statistical test of fit ?2 = ( fit function ) ( N – 1) we want this to be nonsignificant

10. The Process types of descriptive indices absolute fit indices indexes the amount of variance/covariance accounted for by a model goodness of fit index (GFI) and adjusted GFI want values > .90 root mean square residual (RMSR) average size of residuals generated by a model want standardized values < .05 if model is good

11. The Process comparative fit indices (CFI) compare target model to a baseline model baseline model = null or independence model null model = specifies no factors CFI values > .90 are good, .93 better, .95 great parsimony adjusted fit indices adjusts fit by weighting values by the number of parameters estimated root mean square error of approximation (RMSEA) is best values less than .08 are good, .05 are better

12. The Process Fit of individual parameters we have statistical tests for each factor loading and each factor co(variance) evaluate the critical ratios (CR) these are distributed as z-values What if my model and/or individual parameters do not fit? report that and stop, or go to the model modification phase the LaGrange Multiplier test the Wald test

13. Practical Issues Identification also needed to mathematically solve the equations based largely on degrees of freedom (df) for the model df = nonredundant elements in ? - parameters estimated elements in ? = # variances & covariances this equals p (p+1) / 2, where p = # observed variables parameters estimated count up factor loadings, factor covariances, and IV variances estimated

14. Identification of a one-factor model e.g., 4 MVs ? 4 (4 + 1) /2 = 10 variances/covariances e.g., 4 factor loadings, 4 error variances e.g., df = 10 – 8 = 2

15. Practical Issues over-identified (the ideal) positive df = more information than parameters to estimate can determine overall model fit under-identified too many parameters, not enough information model cannot be estimated just-identified (df = 0) parameters to be estimated = amount of information no overall model fit, but you can interpret parameter estimates

16. Practical Issues EQS will present unstandardized factor loadings and factor covariances remember, the analyses are based on ? however, we generally interpret standardized solutions this makes factor loadings range (roughly) between 1 and –1 and makes factor covariances into factor correlations

17. The Structural Model Testing the directional relations among latent variables This is just path analysis with latent variables Latent variables are developed through confirmatory factor analysis (CFA) General modeling process is the same as with CFA

18. Structural Model Equations

19. Comparing nested models

20. Comparing nested models

21. Comparing nested models we can statistically compare models 1 and 2 model 1: does not have the direct effect model 2: does have the direct effect both models have the mediated or indirect effect model 1, then, is nested within model 2, and thus they can be statistically compared we do this using the ?2 difference test (? ?2 )

22. ?2 Difference Test Statistically compares nested models nested = lower-order models that contain a subset of the parameters from a target higher-order model e.g., model 1 is nested within model 2 (target) ??2 = ? 2nested - ? 2target; ?df = dfnested – dftarget notice that the nested model will always have worse overall model fit (higher ?2 ) and more degrees of freedom because we are estimating fewer things ??2 and ?df will always be positive because of this

23. ?2 Difference Test if ? ?2 is not significant... there is no difference between models the simpler or more parsimonious model fits "better" if ? ?2 is significant... target model fits better and...

24. Comparing nonnested models This is a direct comparison between models that have at least a subset of variables that differ We typically use other descriptive fit indices for these purposes Akaike Information Criterion (AIC) Bayesian Information Criterion (BIC) Expected Cross-Validation Index (ECVI) For all of the above indices, the model with the smaller index value is the better-fitting model

25. Setting the Scale (Metric) for an Endogenous Latent Variable We need to do this to mathematically solve the equations of the model Two options: fix variance of latent variable to 1 (standardize) fix a factor loading for each LV to 1 For an endogenous LV, you can only use the second option we want to predict the variance of the endogenous LV setting this value to 1 does not allow for this possibility

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