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