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Confirmatory Factor Analysis

Confirmatory Factor Analysis. Intro. Factor Analysis. Exploratory Principle components Rotations Confirmatory Split sample Structural equations. Structural Equation Approach. Structural equation or covariance structure models. Components. Latent variables (endogenous)

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Confirmatory Factor Analysis

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  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.

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