1. Confirmatory Factor Analysis Intro

2. Factor Analysis • Exploratory • Principle components • Rotations • Confirmatory • Split sample • Structural equations

3. Structural Equation Approach • Structural equation or covariance structure models

4. Components • Latent variables (endogenous) • Manifest variables (exogenous) • Residual variables • Covariances • Influences

5. Path Diagrams (components) Observed Variable E1 Residual or Error Latent Variable Influence Path Covariance between exogenous variables or errors

6. Path Diagram for Multiple Regressiony = a0 + a1*x1 +a2*x2 + a3*x3 + a4*x4 + e1 X1 X2 Y E1 X3 X4

7. Regression • All variables are manifest • One error term • All covariances allowed among independent variables

8. Two Factor Confirmatory Path Model F1 F2 V1 V2 V3 V4 V5 V6 E1 E1 E1 E1 E1 E1

9. Confirmatory Model • F1 and F2 correlated (oblique) • Components of F1 and F2 are separate indicator variables

10. Y = v + e1 X = u + e2 X’ = u + e3 X, Y & X’ are manifest U, V are latent e1, e2, e3 are residual/errors e1, e2, e3 independent with mean = 0 e2, e3, u uncorrelated e1, v uncorrelated Example

11. Example Covariance

12. FACTOR Model Specification • You can specify the FACTOR statement to compute factor loadings F and unique variances U of an exploratory or confirmatory first-order factor (or component) analysis. By default, the factor correlation matrix P is an identity matrix. C = FF’ + U,    U = diag C= data covariance matrix

13. First-order Confirmatory Factor Analysis • For a first-order confirmatory factor analysis, you can use MATRIX statements to define elements in the matrices F, P, and U of the more general model C = FPF' + U,     P = P' ,     U = diag • factor loadings F • unique variances U • factor correlation matrix P • data covariance matrix C

14. PROC FACTOR • RESIDUALS / RES • displays the residual correlation matrix and the associated partial correlation matrix. The diagonal elements of the residual correlation matrix are the unique variances.