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We will illustrate path analysis with a model that has two exogenous variables (ses and female) and two endogenous variables (read and write). We begin with a just identified model, illustrated below.
We use Type = meanstructure so that we get an intercept with model. The Model command block uses the keyword ON to indicate that the model regresses write on read and female. The Output Standardized was included to obtain standardized regression coefficients.
Title:
Path analysis -- just identified model
Data:
File is hsb2.dat ;
Variable:
Names are
id female race ses schtyp prog read write math science socst ;
Usevariables are
female ses read write ;
Model:
ses WITH female ; ! means ses is correlated WITH female
read ON ses female ;
write ON ses read female ;
Output:
Standardized ;
Here is the regression output from Mplus.
INPUT READING TERMINATED NORMALLY
Path analysis -- just identified model
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 200
Number of dependent variables 2
Number of independent variables 2
Number of continuous latent variables 0
Observed dependent variables
Continuous
READ WRITE
Observed independent variables
FEMALE SES
Estimator ML
Information matrix EXPECTED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
hsb2.dat
Input data format FREE
THE MODEL ESTIMATION TERMINATED NORMALLY
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 135.440
Degrees of Freedom 5
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.000
Loglikelihood
H0 Value -1775.717
H1 Value -1775.717
Information Criteria
Number of Free Parameters 10
Akaike (AIC) 3571.434
Bayesian (BIC) 3604.417
Sample-Size Adjusted BIC 3572.736
(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. Std StdYX
READ ON
SES 4.122 0.964 4.274 4.122 0.291
FEMALE -0.342 1.399 -0.245 -0.342 -0.017
WRITE ON
SES 0.930 0.727 1.280 0.930 0.071
FEMALE 5.635 1.009 5.583 5.635 0.297
READ 0.547 0.051 10.727 0.547 0.592
SES WITH
FEMALE -0.045 0.026 -1.754 -0.045 -0.125
Variances
FEMALE 0.248 0.025 10.000 0.248 1.000
SES 0.522 0.052 10.000 0.522 1.000
Residual Variances
READ 95.578 9.558 10.000 95.578 0.914
WRITE 49.710 4.971 10.000 49.710 0.556
R-SQUARE
Observed
Variable R-Square
READ 0.086
WRITE 0.444
Beginning Time: 07:49:24
Ending Time: 07:49:24
Elapsed Time: 00:00:00
One of the appealing aspects of path models is the ability to assess indirect effects of one variable upon another and the total effect of a variable upon another. (The total effect is the combination of the direct effect and indirect effects). This example shows the total, total indirect, and direct effects of ses on write). We obtain these effect by adding these statements to our model
Model indirect: write ind ses;
Here is an illusration of the indirect effect of ses on write
Here is the entire program
Title:
Path analysis -- with indirect effects.
Data:
File is hsb2.dat ;
Variable:
Names are
id female race ses schtyp prog read write math science socst ;
Usevariables are
female ses read write ;
Model:
ses WITH female ;
read ON ses female ;
write ON ses read female ;
Model indirect:
write ind ses;
Output:
Standardized ;
Here are the results
INPUT READING TERMINATED NORMALLY
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 200
Number of dependent variables 2
Number of independent variables 2
Number of continuous latent variables 0
Observed dependent variables
Continuous
READ WRITE
Observed independent variables
FEMALE SES
Estimator ML
Information matrix EXPECTED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
hsb2.dat
Input data format FREE
THE MODEL ESTIMATION TERMINATED NORMALLY
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 135.440
Degrees of Freedom 5
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.000
Loglikelihood
H0 Value -1775.717
H1 Value -1775.717
Information Criteria
Number of Free Parameters 10
Akaike (AIC) 3571.434
Bayesian (BIC) 3604.417
Sample-Size Adjusted BIC 3572.736
(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. Std StdYX
READ ON
SES 4.122 0.964 4.274 4.122 0.291
FEMALE -0.342 1.399 -0.245 -0.342 -0.017
WRITE ON
SES 0.930 0.727 1.280 0.930 0.071
READ 0.547 0.051 10.727 0.547 0.592
FEMALE 5.635 1.009 5.583 5.635 0.297
SES WITH
FEMALE -0.045 0.026 -1.754 -0.045 -0.125
Variances
FEMALE 0.248 0.025 10.000 0.248 1.000
SES 0.522 0.052 10.000 0.522 1.000
Residual Variances
READ 95.578 9.558 10.000 95.578 0.914
WRITE 49.710 4.971 10.000 49.710 0.556
R-SQUARE
Observed
Variable R-Square
READ 0.086
WRITE 0.444
TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS
Estimates S.E. Est./S.E. Std StdYX
Effects from SES to WRITE
Total 3.185 0.873 3.648 3.185 0.243
Total indirect 2.255 0.568 3.971 2.255 0.172
Specific indirect
WRITE
READ
SES 2.255 0.568 3.971 2.255 0.172
Direct
WRITE
SES 0.930 0.727 1.280 0.930 0.071
The above example was a bit overly simple since there was only one possible indirect effect. Often you might have multiple indirect effects. With Mplus you can assess both the breakdown of the total and indirect effects with ind but you can also specify particular indirect routes with via. Say that we put in a preposterous path that female is regressed on ses. Then we can look at the multiple paths leading to write from ses with write ind ses; and can test the specific path of ses to female to read to write with write via read female ses;. The diagram of these indirect effects is shown below.
These commands are shown in bold as part of the program below.
Title:
Multiple indirect paths
Data:
File is hsb2.dat ;
Variable:
Names are
id female race ses schtyp prog read write math science socst;
Usevariables are
female ses read write ;
Model:
female on ses ;
read ON ses female ;
write ON ses read female ;
Model indirect:
write ind ses;
write via read female ses;
Output:
Standardized ;
And here is the output
INPUT READING TERMINATED NORMALLY
Multiple indirect paths
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 200
Number of dependent variables 3
Number of independent variables 1
Number of continuous latent variables 0
Observed dependent variables
Continuous
FEMALE READ WRITE
Observed independent variables
SES
Estimator ML
Information matrix EXPECTED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
hsb2.dat
Input data format FREE
THE MODEL ESTIMATION TERMINATED NORMALLY
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 138.590
Degrees of Freedom 6
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.000
Loglikelihood
H0 Value -1775.717
H1 Value -1775.717
Information Criteria
Number of Free Parameters 9
Akaike (AIC) 3569.434
Bayesian (BIC) 3599.119
Sample-Size Adjusted BIC 3570.606
(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. Std StdYX
FEMALE ON
SES -0.086 0.048 -1.782 -0.086 -0.125
READ ON
SES 4.123 0.964 4.275 4.123 0.291
FEMALE -0.343 1.399 -0.245 -0.343 -0.017
WRITE ON
SES 0.930 0.727 1.280 0.930 0.071
READ 0.547 0.051 10.726 0.547 0.592
FEMALE 5.635 1.009 5.584 5.635 0.297
Residual Variances
FEMALE 0.244 0.024 10.000 0.244 0.984
READ 95.565 9.557 10.000 95.565 0.914
WRITE 49.705 4.971 10.000 49.705 0.556
R-SQUARE
Observed
Variable R-Square
FEMALE 0.016
READ 0.086
WRITE 0.444
TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS
Estimates S.E. Est./S.E. Std StdYX
Effects from SES to WRITE
Total 2.715 0.905 3.000 2.715 0.208
Total indirect 1.786 0.629 2.841 1.786 0.136
Specific indirect
WRITE
FEMALE
SES -0.486 0.286 -1.698 -0.486 -0.037
WRITE
READ
SES 2.255 0.568 3.971 2.255 0.172
WRITE
READ
FEMALE
SES 0.016 0.067 0.242 0.016 0.001
Direct
WRITE
SES 0.930 0.727 1.280 0.930 0.071
Effects from SES to WRITE via READ FEMALE
Sum of indirect 0.016 0.067 0.242 0.016 0.001
Specific indirect
WRITE
READ
FEMALE
SES 0.016 0.067 0.242 0.016 0.001
This is an example of an overidentified model. It has fewer paths than the just identified model and hence we can test the fit of the model.
Title:
Path analysis -- over identified model
Data:
File is hsb2.dat ;
Variable:
Names are
id female race ses schtyp prog read write math science socst ;
Usevariables are
female ses read write ;
Model:
ses WITH female ;
read ON ses ;
write ON read female ;
Output:
Standardized ;
Here is the censored regression output from Mplus.
INPUT READING TERMINATED NORMALLY
Path analysis -- over identified model
SUMMARY OF ANALYSIS
Number of groups 1
Number of observations 200
Number of dependent variables 2
Number of independent variables 2
Number of continuous latent variables 0
Observed dependent variables
Continuous
READ WRITE
Observed independent variables
FEMALE SES
Estimator ML
Information matrix EXPECTED
Maximum number of iterations 1000
Convergence criterion 0.500D-04
Maximum number of steepest descent iterations 20
Input data file(s)
hsb2.dat
Input data format FREE
THE MODEL ESTIMATION TERMINATED NORMALLY
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 1.690
Degrees of Freedom 2
P-Value 0.4259
Chi-Square Test of Model Fit for the Baseline Model
Value 135.440
Degrees of Freedom 5
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 1.006
Loglikelihood
H0 Value -1776.562
H1 Value -1775.717
Information Criteria
Number of Free Parameters 8
Akaike (AIC) 3569.124
Bayesian (BIC) 3595.511
Sample-Size Adjusted BIC 3570.166
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.000
90 Percent C.I. 0.000 0.133
Probability RMSEA <= .05 0.587
SRMR (Standardized Root Mean Square Residual)
Value 0.021
MODEL RESULTS
Estimates S.E. Est./S.E. Std StdYX
READ ON
SES 4.152 0.957 4.339 4.152 0.293
WRITE ON
FEMALE 5.487 1.006 5.455 5.487 0.288
READ 0.566 0.049 11.554 0.566 0.610
SES WITH
FEMALE -0.045 0.026 -1.754 -0.045 -0.125
Variances
FEMALE 0.248 0.025 10.000 0.248 1.000
SES 0.522 0.052 10.000 0.522 1.000
Residual Variances
READ 95.601 9.560 10.000 95.601 0.914
WRITE 50.113 5.011 10.000 50.113 0.557
R-SQUARE
Observed
Variable R-Square
READ 0.086
WRITE 0.443
Beginning Time: 08:01:29
Ending Time: 08:01:29
Elapsed Time: 00:00:00
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