Review calculation of GFI

[M-E-Rademaker]

Currently we implement the GFI as: 1 - trace[(S - Sigma_hat)^2] / trace[S^2]. This is the GFI for the ULS fitting function according to Jöreskog (1982) - Recent development in structural equation modeling.

Schermelleh-Engel (2003) define the GFI as: 1 - F_t / F_n where

• F_t : fit function of target/postulated model (e.g., calculateDML())
• F_n: fit function of the null model (e.g. the fit function with model-implied matrix set to the identity matrix).

[M-E-Rademaker]

Problem

Jöreskog & Sörbom (1993) - LISREL 8 define the GFI as a measure that measures "how much better the model fits as compared to no model at all." (p. 122). In a formula:

GFI = 1 - F(S, Sigma_hat) / F(S, Sigma(0))

They dont say what Sigma(0) is supposed to mean, however, Mulaik (1989) essentially assumes that what Jöreskog and Sörbom had in mind when "inventing" the GFI was essentially a R^2 analog. So in general

GFI = 1 - tr[(W^-1/2(S - Sigma_hat)W^-1/2)^2] / tr[(W^-1/2 S W^{-1/2})^2]

where W^{-1/2} depends on the fitting function to compute the GFI for. So for ULS:

GFI = 1 - tr[(S - Sigma_hat))^2] / tr[(S)^2]

since W^-1/2 = I. And for ML

GFI = 1 - tr[(\Sigma_hat^-1 S - I))^2] / tr[(Sigma_hat^-1 S)^2]

since W^-1/2 = Sigma_hat^-1/2

So there is no "the" GFI.