Let’s look at an example. This example is based on example 5.8 from the Mplus
User’s Guide. (Note: the Mplus User’s Guide, as well as all files needed to
run the examples can be downloaded from the
Mplus website.)
data: file is ex5.8.dat;
variable:
names are y1-y6 x1-x3;
usevariables are y1-y6;
model:
f1 by y1-y3;
f2 by y4-y6;
output: tech1;
TESTS OF MODEL FIT
Chi-Square Test of Model Fit
Value 9.257
Degrees of Freedom 8
P-Value 0.3211
Chi-Square Test of Model Fit for the Baseline Model
Value 3295.491
Degrees of Freedom 15
P-Value 0.0000
CFI/TLI
CFI 1.000
TLI 0.999
Loglikelihood
H0 Value -4435.790
H1 Value -4431.161
Information Criteria
Number of Free Parameters 19
Akaike (AIC) 8909.580
Bayesian (BIC) 8989.657
Sample-Size Adjusted BIC 8929.350
(n* = (n + 2) / 24)
RMSEA (Root Mean Square Error Of Approximation)
Estimate 0.018
90 Percent C.I. 0.000 0.057
Probability RMSEA <= .05 0.897
SRMR (Standardized Root Mean Square Residual)
Value 0.007
MODEL RESULTS
Two-Tailed
Estimate S.E. Est./S.E. P-Value
F1 BY
Y1 1.000 0.000 999.000 999.000
Y2 1.035 0.030 34.332 0.000
Y3 1.025 0.030 34.549 0.000
F2 BY
Y4 1.000 0.000 999.000 999.000
Y5 0.972 0.026 37.948 0.000
Y6 0.969 0.028 35.122 0.000
F2 WITH
F1 2.333 0.180 12.930 0.000
Intercepts
Y1 2.066 0.080 25.719 0.000
Y2 2.088 0.082 25.361 0.000
Y3 2.088 0.081 25.714 0.000
Y4 1.663 0.085 19.606 0.000
Y5 1.623 0.081 20.012 0.000
Y6 1.596 0.083 19.255 0.000
Variances
F1 2.688 0.204 13.173 0.000
F2 3.071 0.228 13.485 0.000
Residual Variances
Y1 0.539 0.048 11.219 0.000
Y2 0.508 0.048 10.545 0.000
Y3 0.472 0.046 10.230 0.000
Y4 0.524 0.049 10.812 0.000
Y5 0.387 0.040 9.573 0.000
Y6 0.550 0.048 11.405 0.000
There are 6 manifest variables used in the model. So the total number of
parameters that one can estimate is 7*6/2 + 6 = 27, coming from the variance-covariance structure of y1 – y6
and their means. By default, the loading for y1 and
y4 are fixed at 1. The parameters estimated by default are: the means of y1
– y6 (6 parameters), the residual variances of y1 – y6 (6 parameters),
loadings for f1 and f2 (2 each, so 4 total) and the variance-covariance structure for f1
and f2 (3 parameters, 2 variances and 1 covariance). So all together there
are 6 + 6 + 2*2 + 3 = 19 parameters in the model. If you want to know
exactly what parameters are being model, you can use the "tech1" option for the output statement.
Since there are total of 27 degrees of freedom and we have estimated
19 parameters, the degrees of freedom for this model is 27-18 = 8. This information is
in the first part of the output labeled as "TESTS OF MODEL FIT":
Chi-Square Test of Model FitValue 9.257 Degrees of Freedom 8 P-Value 0.3211
Now where does this 9.257 come from? This is 2 times the difference in
the log likelihood given by Mplus: 2*(4435.790 – 4431.161) = 9.258
Loglikelihood
H0 Value -4435.790
H1 Value -4431.161
H0 is the current model and H1 is the saturated model, where the
variance-covariance of y1-y6 and their means are parameters to be modeled,
resulting in a model with zero degree of freedom.
The "Baseline model" referred by the following section of the output is
the mean structure model, where there are 12 parameters, 6 means and 6
variances. This leads to (27-12) = 15 degrees of freedom.
Chi-Square Test of Model Fit for the Baseline Model
Value 3295.491
Degrees of Freedom 15
P-Value 0.0000