G1, like F*, fails to compensate for
the effect of model complexity. Consider a sequence of *nested* models,
where the models with more degrees of freedom are special cases of those
with fewer degrees of freedom. (See Steiger, Shapiro, and Browne, 1985,
for a discussion of the statistical properties of *Chi-square* tests
with nested models.) For a nested sequence of models, the more complex
models (i.e., those with more free parameters and fewer degrees of freedom)
will always have G1 coefficients as low or lower than
those which are less complex.

Goodness of fit, as measured by G1,
improves more or less inevitably as more parameters are added. The adjusted
population *gamma* index G2 attempts to compensate for this
tendency.

Just as G1 is computed by subtracting a ratio
of sums of squares from 1, G2 is obtained by subtracting a corresponding
ratio of mean squares from 1. Let *p** =* p*(*p* + 1)/2.
Let **s** be a p*x1
vector of non-duplicated elements of the population reproduced covariance
matrix **S**(**q**), as in Equation 99**, **for a model with *n* degrees of freedom, and **e** a corresponding vector of
residuals. Then G2
is

(110)

Consistent estimates and confidence intervals for G1 may thus be converted into corresponding quantities for G2 by applying Equation 110.