Determine the degrees of freedom as well as the free parameters in an SEM

Question

I need to determine both the

• number of free parameters to estimate and

• degrees of freedom

of a structural equation model. I know how to calculate these values by hand. However, when I use the R package lavaan, I get a different number. I calculate 59 parameters to estimate.

library(lavaan)
modelo1 <- '
    ST ~ A*PA + B*MT+C*INF

    PA=~CP127_1+CP127_2+CP127_3+CP127_4+CP127_5+CP127_6
    MT=~CP128_1+CP128_2+CP128_3+CP128_4+CP128_5+CP128_6
    INF=~CP138_1+CP138_2+CP138_3+CP138_4+CP138_5+CP138_6+CP138_7+CP138_8
    ST=~CP139_1+CP139_2+CP139_3+CP139_4+CP139_5+CP139_6'

ajuste1 <- sem(modelo1,data=dataof,ordered =TRUE)
summary(fit.measures=TRUE,ajuste1)

---

lavaan (0.5-23.1097) converged normally after  65 iterations

Number of observations                           416

Estimator                                       DWLS      Robust
Minimum Function Test Statistic              375.660     555.434
Degrees of freedom                               293         293
P-value (Chi-square)                           0.001       0.000
Scaling correction factor                                  0.977
Shift parameter                                          170.951
for simple second-order correction (Mplus variant)

Answer 1

The total number of elements in your initial covariance matrix is k(k+1)/2 where k = the number of variables in the matrix. In this case, that gives you (26*27)/2 = 351. In terms of the number of parameters that you have to estimate in the model, you have 26 observed variable variances, 22 loadings, 4 latent variable variances, 3 regression paths, and 3 latent variable covariances. Add those up and you have 58 not 59 free parameters to estimate in the model. 351-58 = 293 degrees of freedom.