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Structural Equation Modeling

Structural Equation Modeling. Mgmt 291 Lecture 3 – CFA and Hybrid Models Oct. 12, 2009. Measurement is Everything. Nothing can be done with wrong or unreliable measurements. “Measurement is Everything”.

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Structural Equation Modeling

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  1. Structural Equation Modeling Mgmt 291 Lecture 3 – CFA and Hybrid Models Oct. 12, 2009

  2. Measurement is Everything • Nothing can be done with wrong or unreliable measurements. • “Measurement is Everything”. • In research presentation or paper submission, measurement is the part being challenged the most.

  3. Latent Variables are everywhere in research • “The true power of SEM comes from latent variable modelling “ • “Variables in psychology and other social science are rarely (never?) measured directly” • the effects of the variable are measured • Intelligence, self-esteem, depression • Reaction time, diagnostic skill • Democracy, Socio-Economical Status • Legitimacy, Management Skill • (soul, angels, … - hypothetical construct)

  4. Beyond Validity and Reliability:Between concept and indicators • Validity: Measures what it intends to measure. • Reliability: Consistency • Precision • repeatability

  5. Latent Variable Latent Vars and Observed Indicators Indicator1 Indicator3 • What to be studied is: L1 L2 X1.1 X2.1 X3.1 X4.1 X5.1 X1.2 X2.2 X3.2 X4.2 X5.2 E 1.1 E 2.1 E 3.1 E 4.1 E 5.1 E 1.2 E 2.2 E 3.2 E 4.2 E 5.2 Latent vs. Observed

  6. Exploratory Factor Analysis SPSS For Data Reduction Factor analysis GIGO

  7. Confirmative Factor Analysis • 1) Equations & Diagrams: model representation • 2) Identification & Estimation • 3) Errors and Evaluation: assumptions & fit indexes • 4) Explanation

  8. 1) Equations & Diagrams: model representation • X 1.1 = Ø1 L1 + e 1.1 • X 2.1 = Ø2 L1 + e 2.1 • X 3.1 = Ø3 L1 + e 3.1 • X 4.1 = Ø4 L1 + e 4.1 • Loadings - Ø1 … • X ~ similar to endogenous variables • L ~ similar to exogenous variables

  9. More on Equations X = L+ e Error Measured True Score Relationship between Measured <–> true score Observed <–> latent variable Indicator <–> construct or factor Unique factor

  10. Diagram representation e1 Research Presentation Assignment Report Knowing SEM e2 X 1.1 ~ X 5.1 load on L 1 Classroom Participation e3 Co-vary L1 L2 X1.1 X2.1 X3.1 X4.1 X5.1 X1.2 X2.2 X3.2 X4.2 X5.2 E4.1 E5.1 E1.2 E2.2 E3.2 E4.2 E5.2 E1.1 E2.1 E3.1

  11. Types of Measurement Models • Uni-dimensional (each indicator loads only on one factor, error terms independent from each other) • Multi-dimensional • Single-factor • Multifactors L1 L2 X1.1 X2.1 X3.1 X4.1 X2.2 X3.2 X4.2

  12. Non-recursive Type Income Education Occupation ? Socio-economical Status

  13. 2) Identification and Estimation • Parameters <= Observations • Scale for every factor • Single factor & >= 3 indicators • 2 or more factors & 2 or more indicators per factor • Less than 2 indicators for one or more factors --- ??? Not an issue As recursive In literature, 3 indicators or 2 with 2 correlated factors or sample size > 200

  14. a) How to scale the latent variable • 1) fix variances as a constant • 2) fix one loading as 1

  15. b) How to count • # parameters = # loadings + vars & co-vas of factors + vars & co-vas of errors • # obs = v(v+1)/2 ~ number of observed variables

  16. Examples E1 E2 E1 E2 E3 X2 X1 X1 X2 X3 1.0 1.0 A A E2 E3 E1 E4 X1 X2 X3 X4 1.0 1.0 A B 4, 6, 9

  17. Identification of EFA GIGO ?

  18. Estimation Methods • ML – most often used • Generalized least squared • Un-weighted least squared

  19. 3) Errors and Evaluation: Assumptions • Multivariate normality

  20. Fit Indices • All the fit indices for path analysis applied to CFA • Chi squared / df < 3 • GFI (Goodness Fit Index), AGFI close to 1 • SRMR (Standardized Root Mean Squared Residual) close to 0

  21. 4) Explanation: Factor loadings • Un-standardized coefficients • (similar to regression coefficients) • Standardized coefficients

  22. R 2 • Proportion of explained variances • (1 – measurement error variance / observed variances) • 1-R 2 ~ proportion of unique variances

  23. Example: The Model Representation Hand Movements Number Recall Word Order Gestalt Closure Spatial Memo Matrix Analogies Photo Series Triangles 1 1 Sequential Simultaneous

  24. Example: Results • R2 • Chi Square • Chi-square = 38.13 • Df = 19 ~ 2-factor model • For one factor • 104.90 (df=20)

  25. Example: Diagram to Rep Results 3.50 (.39) 3.44 (.47) 5.13 (.56) 10.05 (.65) 2.01 (.34) 2.92 (.34) 5.45 (.75) 8.71 (.75) Hand Movements Number Recall Word Order Gestalt Closure Spatial Memo Matrix Analogies Photo Series Triangles 1.21 (.66) 1.45 (.73) 2.03 (.59) 1.39 (.81) 1.0 (.50) 1.15 (.81) 1.0 (.50) 1.73 (.78) Sequential Simultaneous Standardized coefficients inside parenthesis

  26. Example: Explanation 2.01 (.34) 8.71 (.75) • Standardized & Un-standardized coefficients & variances • (8.71 / 3.4 2 = 8.71 / 11.56 = .75) • .5 2 = 1 - .75 Hand Movements Number Recall 1.0 (.50) 1.15 (.81) Sequential

  27. Hybrid Models– Combination of Measurement and Structure Models

  28. 1) Equations and Diagram: Model representation of Hybrid Model • 6 Types of Terms • Observed Exogenous - X • Observed Endogenous - Y • Latent Exogenous - K • Latent Endogenous - E • Error Terms for Exogenous Obs – eY • Error Terms for Endogenous Obs - eX

  29. Diagram representation eE 6, PS 5, PH 3, BE 4, GA E K 1, LY 2, LX Y X 8, TD 7, TE eX eY

  30. More on Diagram representation eE1 eE2 6, PS 5, PH 3, BE 4, GA E1 K E2 2, LX 1, LY Y1 Y2 Y3 Y4 X 8, TD 7, TE eX eY eY eY eY

  31. Model Representation • NY = # observed endogenous • NX = # observed exogenous • NE = # latent endogenous • NK = # latent exogenous

  32. Model representation eE 6, PS 5, PH 3, BE 4, GA E K 1, LY (NY X NE) 2, LX (NX X NK) Y X 8, TD 7, TE (NY X NY) eX eY

  33. 2) Identification and Estimation • Number of parameters <(p+q)(p+q+1)/2 • Two-Step Rule - Measurement Model Identification - Structural Model Identification

  34. Estimation Methods • ML again

  35. 3) Errors & Model Evaluation • Fit Indexes • Chi-squares

  36. 4) Explanation • path coefficients • and loadings

  37. Example: Model e e e e e e Parental Psychopathology Low Family SES Extroversion Reading Arithmetic Spelling Scholastic Achievement Classroom Adjustment Emotional Stability Familial Risk Cognitive Ability Scholastic Motivation Harmony e Memory Verbal Visual- Spatial e e e e e

  38. Example: Identification e e e e e e Parental Psychopathology Low Family SES Extroversion Reading Arithmetic Spelling Scholastic Achievement Classroom Adjustment Emotional Stability Familial Risk Cognitive Ability Scholastic Motivation Harmony e Memory Verbal Visual- Spatial e e e e D e D D Scholastic Achievement Classroom Adjustment Familial Risk Cognitive Ability

  39. Example: Errors & Fix Indexes for Evaluation • Better chi square/df for 3-factor measurement model (cognitive & scholar merger) (2.05 vs. 3.92) • (also GFI and SRMR better) • Good chi square/df for hybrid model • (2.05)

  40. Example: results explanation e e e e e e Parental Psychopathology Low Family SES Extroversion Reading Arithmetic Spelling Scholastic Achievement Classroom Adjustment Emotional Stability Familial Risk Cognitive Ability Scholastic Motivation Harmony e Memory Verbal Visual- Spatial e e e e e

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