1. Confirmatory Factor Analysis
Intro

2. Factor Analysis Exploratory Confirmatory 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
E Residual or Error
Latent Variable
Influence Path
Covariance between exogenous variables or errors

6. Path Diagram for Multiple Regression y = a0 + a1. x1 +a2. x2 + a3
Path Diagram for Multiple Regression y = 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 ModelV1 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. Example 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

11. Example Covariance Y X X’ Var(Y)= Var(v) + Var(e1) Cov(XY) = Cov(uv)
Var(X) =
Var(u) + Var(e2)
Cov(X’Y) =
Cov(X/X) =
Var(u)
Var(X’) =
Var(u) + Var(e3)

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.