Confirmatory Factor Analysis Intro

Factor Analysis Exploratory Confirmatory Principle components Rotations Confirmatory Split sample Structural equations

Structural Equation Approach Structural equation or covariance structure models

Components Latent variables (endogenous) Manifest variables (exogenous) Residual variables Covariances Influences

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

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

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

Two Factor Confirmatory Path Model V1 V2 V3 V4 V5 V6 E1 E1 E1 E1 E1 E1

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

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

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)

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

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

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.