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**Confirmatory Factor Analysis**

Intro

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**Factor Analysis Exploratory Confirmatory Principle components**

Rotations Confirmatory Split sample Structural equations

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**Structural Equation Approach**

Structural equation or covariance structure models

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**Components Latent variables (endogenous)**

Manifest variables (exogenous) Residual variables Covariances Influences

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**Path Diagrams (components)**

Observed Variable E Residual or Error Latent Variable Influence Path Covariance between exogenous variables or errors

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**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

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**Regression All variables are manifest One error term**

All covariances allowed among independent variables

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**Two Factor Confirmatory Path Model**

V1 V2 V3 V4 V5 V6 E1 E1 E1 E1 E1 E1

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**Confirmatory Model F1 and F2 correlated (oblique)**

Components of F1 and F2 are separate indicator variables

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**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

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**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)

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**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

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**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

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**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.

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