Noncentrality-Based Indices of Fit - Adjusted Population Gamma Index

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.