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”. In research presentation or paper submission, measurement is the part being challenged the most.

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)

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

Latent Vars and Observed Indicators What to be studied is: L1L1 X 3.1 X 4.1 X 5.1 X 2.1 X 1.1 L2L2 X 3.2 X 4.2 X 5.2 X 2.2 X 1.2 Latent Variable Indicator3Indicator1 vs. Latent Observed E 1.1E 2.1E 3.1E 4.1E 5.1E 1.2E 2.2E 3.2E 4.2E 5.2

Exploratory Factor Analysis SPSS For Data Reduction Factor analysis GIGO

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

1) Equations & Diagrams: model representation X 1.1 = Ø 1 L1 + e 1.1 X 2.1 = Ø 2 L1 + e 2.1 X 3.1 = Ø 3 L1 + e 3.1 X 4.1 = Ø 4 L1 + e 4.1 Loadings - Ø 1 … X ~ similar to endogenous variables L ~ similar to exogenous variables

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

Diagram representation L1L1 X 3.1 X 4.1 X 5.1 X 2.1 X 1.1 L2L2 X 3.2 X 4.2 X 5.2 X 2.2 X 1.2 E 1.1 E 4.1 E 5.1 E 3.1 E 2.1 E 1.2 E 2.2 E 3.2 E 4.2 E 5.2 Knowing SEM Research Presentation Assignment Report Classroom Participation e1 e2 e3 Co-vary X 1.1 ~ X 5.1 load on L 1

Types of Measurement Models Uni-dimensional (each indicator loads only on one factor, error terms independent from each other) Multi-dimensional Single-factor Multifactors L1L1 X 3.1 X 4.1 X 2.1 X 1.1 L2L2 X 3.2 X 4.2 X 2.2

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

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

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

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

Examples A A A X1 X2 X3X2X1 X4X3X1X2 B 1.0 E1E1 E2E2 E1E1 E4E4 E3E3 E2E2 E1E1 E2E2 E3E3 4, 6, 9

Identification of EFA GIGO ?

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

3) Errors and Evaluation: Assumptions Multivariate normality

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

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

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

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

Example: Results R 2 Chi Square Chi-square = Df = 19 ~ 2-factor model For one factor (df=20) IndicatorR2R2 Hand.25 Number.65 Word.65 Gestalt.25 Triangle.52 Matrix.43 Spatial.35 Photo.61

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

Example: Explanation Standardized & Un-standardized coefficients & variances (8.71 / = 8.71 / =.75).5 2 = IndicatorSD Hand3.4 Number2.4 Word2.9 Gestalt2.7 Triangle2.7 Matrix4.2 Spatial2.8 Photo3 Hand Movements Number Recall Sequential 8.71 (.75) 2.01 (.34) 1.0 (.50) 1.15 (.81)

Hybrid Models – Combination of Measurement and Structure Models

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 – e Y Error Terms for Endogenous Obs - e X

Diagram representation K X E Y eXeX eYeY 1, LY 2, LX 3, BE 4, GA 5, PH 6, PS 7, TE 8, TD eEeE

More on Diagram representation K X E1 Y1 eXeX eYeY 1, LY 2, LX 3, BE 4, GA 5, PH 6, PS 7, TE 8, TD E2 Y2Y3Y4 eYeY eYeY eYeY e E1 e E2

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

Model representation K X E Y eXeX eYeY 1, LY (NY X NE) 2, LX (NX X NK) 3, BE 4, GA 5, PH 6, PS 7, TE (NY X NY) 8, TD eEeE

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

Estimation Methods ML again

3) Errors & Model Evaluation Fit Indexes Chi-squares

4) Explanation path coefficients and loadings

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

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

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)

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