Exploratory Factor Analysis, Confirmatory Factor Analysis, And Structural Equation Modeling For ...

136 Pages·2008·0.65 MB·English

by  Bengt Muthen

Mplus Short Courses
Topic 1
Exploratory Factor Analysis, Confirmatory
Factor Analysis, And Structural Equation
Modeling For Continuous Outcomes
Linda K. Muthén
Bengt Muthén
Copyright © 2008 Muthén & Muthén
www.statmodel.com
1
Table Of Contents
General Latent Variable Modeling Framework 4
Regression Analysis 14
Path Analysis 26
Indirect Effects 35
Measurement Errors And Multiple Indicators Of Latent Variables 39
Factor Analysis 47
Exploratory Factor Analysis 58
Confirmatory Factor Analysis 106
Technical Aspects Of Maximum-Likelihood Estimation And Testing 120
EFA In A CFA Framework 132
Simple Structure CFA 145
Special Factor Analysis Models 154
Measurement Invariance And Population Heterogeneity 163
CFA With Covariates (MIMIC) 168
Multiple Group Analysis 196
Structural Equation Modeling (SEM) 223
Model Identification 234
Formative Indicators 238
Latent Variable Interactions 246
Monte Carlo Simulations 248
Model Constraints 254
Model Test 257
References 260 2
1
Mplus Background
• Inefficient dissemination of statistical methods:
– Many good methods contributions from biostatistics,
psychometrics, etc are underutilized in practice
• Fragmented presentation of methods:
– Technical descriptions in many different journals
– Many different pieces of limited software
• Mplus: Integration of methods in one framework
– Easy to use: Simple, non-technical language, graphics
– Powerful: General modeling capabilities
• Mplus versions
– V1: November 1998
– V2: February 2001
– V3: March 2004
– V4: February 2006
– V5: November 2007
• Mplus team: Linda & Bengt Muthén, Thuy Nguyen,
Tihomir Asparouhov, Michelle Conn, Jean Maninger
3
Statistical Analysis With Latent Variables
A General Modeling Framework
Statistical Concepts Captured By Latent Variables
Continuous Latent Variables Categorical Latent Variables
• Measurement errors • Latent classes
• Factors • Clusters
• Random effects • Finite mixtures
• Frailties, liabilities • Missing data
• Variance components
• Missing data
4
2
Statistical Analysis With Latent Variables
A General Modeling Framework (Continued)
Models That Use Latent Variables
Continuous Latent Variables Categorical Latent Variables
• Factor analysis models • Latent class models
• Structural equation models • Mixture models
• Growth curve models • Discrete-time survival models
• Multilevel models • Missing data models
Mplus integrates the statistical concepts captured by
latent variables into a general modeling framework that
includes not only all of the models listed above but also
combinations and extensions of these models.
5
General Latent Variable Modeling Framework
• Observed variables
x background variables (no model structure)
y continuous and censored outcome variables
u categorical (dichotomous, ordinal, nominal) and
count outcome variables
• Latent variables
f continuous variables
–interactions among f’s
c categorical variables
–multiple c’s 6
3
General Latent Variable Modeling Framework
7
General Latent Variable Modeling Framework
8
4
General Latent Variable Modeling Framework
9
General Latent Variable Modeling Framework
• Observed variables
x background variables (no model structure)
y continuous and censored outcome variables
u categorical (dichotomous, ordinal, nominal) and
count outcome variables
• Latent variables
f continuous variables
–interactions among f’s
c categorical variables
–multiple c’s 10
5
Mplus
Several programs in one
• Exploratory factor analysis
• Structural equation modeling
• Item response theory analysis
• Latent class analysis
• Latent transition analysis
• Survival analysis
• Growth modeling
• Multilevel analysis
• Complex survey data analysis
• Monte Carlo simulation
Fully integrated in the general latent variable framework
11
Overview
Single-Level Analysis
Cross-Sectional Longitudinal
Continuous Observed Day 1 Day 2
And Latent Variables Regression Analysis Growth Analysis
Path Analysis
Exploratory Factor Analysis
Confirmatory Factor Analysis
Structural Equation Modeling
Adding Categorical Day 3 Day 4
Observed And Latent Regression Analysis Latent Transition Analysis
Variables Path Analysis Latent Class Growth Analysis
Exploratory Factor Analysis Growth Analysis
Confirmatory Factor Analysis
Growth Mixture Modeling
Structural Equation Modeling
Discrete-Time Survival
Latent Class Analysis
Mixture Analysis
Factor Mixture Analysis
Missing Data Analysis
Structural Equation Mixture
Modeling
12
6
Overview (Continued)
Multilevel Analysis
Cross-Sectional Longitudinal
Continuous Observed Day 5 Day 5
And Latent Variables Regression Analysis Growth Analysis
Path Analysis
Exploratory Factor Analysis
Confirmatory Factor Analysis
Structural Equation Modeling
Adding Categorical Day 5 Day 5
Observed And Latent Latent Class Analysis Growth Mixture Modeling
Variables Factor Mixture Analysis
13
Regression Analysis
14
7
LSAY Math Regression
100
95
90
85
80
75
70
ATH10 6605
M 55
50
45
40
35
30
25
20
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
MATH7
15
Regression Analysis
Regression model:
y = α + β x + ε , (1)
i i i
E(ε | x) = E(ε) = E(ε) = 0 (x and ε uncorrelated), (2)
i i i
V(ε | x) = V(ε) = V(ε) (constant variance). (3)
i i i
For inference and ML estimation, we also assume ε normal.
The model implies
E(y| x) = α + β x (conditional expectation function)
V(y| x) = V(ε) (homoscedasticity)
16
8
Regression Analysis (Continued)
b

=
x

|
y a ε

=
x E(y| x) = α + βx

|
ε
x
x= a x= b
17
Regression Analysis (Continued)
Population formulas:
y = α + β x + ε , (1)
i i i
E(y) = E(α) + E(β x) + E(ε)
= α + βE(x) (2)
V(y) = V(α) + V(β x) + V(ε)
= β2 V(x) + V(ε) (3)
Cov(y, x) = E[y –E(y)] [x –E(x)] = βV(x) (4)
R2 = β2 V(x) / (β2V(x) + V(ε)) (5)
SD(x)
Stdyx β= β (6)
SD(y)
18
9
Regression Analysis (Continued)
The model has 3 parameters: α, β, and V(ε)
Note: E(x) and V(x) are not model parameters
Formulas for ML and OLS parameter estimates based on a
random sample
βˆ = s /s
yx xx
αˆ = y−βˆ x
Vˆ(ε)= s −βˆ2s
yy xx
Prediction
yˆ =αˆ +βˆ x
i i
19
Regression Analysis (Continued)
x 0/1 dummy variable (e.g. gender), x continuous variable
1 2
y = α + β x + β x + ε
i 1 1i 2 2i i
E(y| x = 0, x ) = α + β x
1 2 2 2
E(y| x = 1, x ) = α + β + β x
1 2 1 2 2
intercept
E(y| x , x ) β > 0
1 2 1
x = 1
1
x = 0
1
α+ β
1 α
x
2
Analogous to ANCOVA
20
10