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Confirmatory Factor Analysis

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Presentation on theme: "Confirmatory Factor Analysis"— Presentation transcript:

1 Confirmatory Factor Analysis

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 Model
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 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

displays the residual correlation matrix and the associated partial correlation matrix. The diagonal elements of the residual correlation matrix are the unique variances.

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