where r is the number of free parameters. A definition in terms of chi-square also exists and gives the same results as discussed in connection with BIC below.
Bengt O. Muthen posted on Tuesday, August 31, 2010 - 2:36 pm Comparing models using the formula "chi2-df (ln(N))" is the same as using the Mplus BIC = -2logL + p*ln(N), where p is the number of parameters. Note that chi2 = -2(logL_a - logL_b), where a is a model nested within b. In the usual SEM case b is the totally unrestricted model called H1. Note also that df = p_b - p_a, where p is the number of parameters. So when you look at the difference between the BIC of two models using the formula chi2-df (ln(N)) there is a canceling out of the terms -2logL_b and of the terms p_b*ln(N). This means that BIC differences are the same for both formulas.