It's Greek to me
Β - beta
η - eta
ξ - xi
ζ - zeta
Γ - gamma
Λ - lambda (upper case)
λ - lambda (lower case)
δ - delta
ε - epsilon
Structural Equation Modeling
Structural Equation Model
Notation
Measurement Model
Notation
Example with Full Notation
Example: Just Identified Model
Output: Just Identified Model
This is the same example as used in the path analysis unit with variables: x1-ξ1-ses, x2-ξ2-iq, y1-η1-am and y2-η2-gpa.
Mplus ESTIMATES
Mplus VERSION 2.02
INPUT INSTRUCTIONS
TITLE:
path analysis;
DATA:
FILE IS ..\data\ped.dat;
VARIABLE:
NAMES ARE ses iq am gpa;
USEVAR = ses iq am gpa;
ANALYSIS: TYPE=meanstructure;
MODEL:
iq on ses;
am on ses iq;
gpa on am ses iq;
OUTPUT: sampstat residual;
Correlations
IQ AM GPA SES
________ ________ ________ ________
IQ 1.000
AM 0.160 1.000
GPA 0.570 0.500 1.000
SES 0.300 0.410 0.330 1.000
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 0.000
Degrees of Freedom 0
P-Value 0.0000
Chi-Square Test of Model Fit for the Baseline Model
Value 289.885
Degrees of Freedom 6
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.000
Loglikelihood
H0 Value -1555.780
H1 Value -1555.780
Information Criteria
Number of Free Parameters 12
Akaike (AIC) 3135.561
Bayesian (BIC) 3180.006
Sample-Size Adjusted BIC 3141.949
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.000
90 Percent C.I. 0.000 0.000
Probability RMSEA <= .05 0.000
SRMR (Standardized Root Mean Square Residual)
Value 0.000
MODEL RESULTS
Estimates S.E. Est./S.E.
IQ ON
SES 0.300 0.055 5.447
AM ON
SES 0.398 0.055 7.213
IQ 0.041 0.055 0.737
GPA ON
AM 0.416 0.045 9.256
SES 0.009 0.047 0.198
IQ 0.501 0.043 11.647
Residual Variances
IQ 0.907 0.074 12.247
AM 0.828 0.068 12.247
GPA 0.502 0.041 12.247
Intercepts
IQ 0.000 0.055 0.000
AM 0.000 0.053 0.000
GPA 0.000 0.041 0.000
RESIDUAL OUTPUT
ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED)
Model Estimated Covariances/Correlations/Residual Correlations
IQ AM GPA SES
________ ________ ________ ________
IQ 0.997
AM 0.159 0.997
GPA 0.568 0.498 0.997
SES 0.299 0.409 0.329 0.997
LISREL ESTIMATES
BETA
ETA1 ETA2
EQ 1 1.000 0.000
EQ 2 -0.416 1.000
GAMMA
KSI1 KSI2
EQ 1 0.398 0.041
EQ 2 0.009 0.501
PHI
KSI1 KSI2
KSI1 1.000
KSI2 0.300 1.000
PSI
EQ1 EQ2
1 0.830 0.504
TEST OF GOODNESS OF FIT
CHI SQUARE WITH 0 DF = 0.00
PROBABILITY LEVEL = 1.000
Example: Overidentified Model
Output: Overidentified Model
Mplus ESTIMATES
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 0.582
Degrees of Freedom 2
P-Value 0.7469
Chi-Square Test of Model Fit for the Baseline Model
Value 289.885
Degrees of Freedom 6
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.015
Loglikelihood
H0 Value -1556.071
H1 Value -1555.780
Information Criteria
Number of Free Parameters 10
Akaike (AIC) 3132.143
Bayesian (BIC) 3169.181
Sample-Size Adjusted BIC 3137.467
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.000
90 Percent C.I. 0.000 0.079
Probability RMSEA <= .05 0.868
SRMR (Standardized Root Mean Square Residual)
Value 0.013
MODEL RESULTS
Estimates S.E. Est./S.E.
IQ ON
SES 0.300 0.055 5.447
AM ON
SES 0.410 0.053 7.786
GPA ON
AM 0.420 0.041 10.162
IQ 0.503 0.041 12.180
Residual Variances
IQ 0.907 0.074 12.247
AM 0.829 0.068 12.247
GPA 0.502 0.041 12.247
Intercepts
IQ 0.000 0.055 0.000
AM 0.000 0.053 0.000
GPA 0.000 0.041 0.000
RESIDUAL OUTPUT
ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED)
Model Estimated Covariances/Correlations/Residual Correlations
IQ AM GPA SES
________ ________ ________ ________
IQ 0.997
AM 0.123 0.997
GPA 0.553 0.480 0.981
SES 0.299 0.409 0.322 0.997
LISREL ESTIMATES
BETA
ETA1 ETA2
EQ 1 1.000 0.000
EQ 2 -0.420 1.000
GAMMA
KSI1 KSI2
EQ 1 0.410 0.000
EQ 2 0.000 0.503
PHI
KSI1 KSI2
KSI1 1.000
KSI2 0.300 1.000
PSI
EQ1 EQ2
1 0.832 0.504
TEST OF GOODNESS OF FIT
CHI SQUARE WITH 2 DF = 0.1921
PROBABILITY LEVEL = 0.9084