Single Sample Goodness of Fit Indices
Statistica provides a variety of single sample goodness-of-fit indices for evaluating the SEPATH model you have created. You can access these indices by clicking the Other Single Sample Indices button on the Advanced tab of the Structural Equation Modeling Results dialog box. This button activates a spreadsheet with a sampling of some of the better known single sample indices of fit, and some related measures.
Joreskog AGFI. Values above .95 indicate good fit. This index is, like the GFI, a negatively biased estimate of its population equivalent. As with the GFI, we give this index primarily because of its historical popularity. The Adjusted Population Gamma index is a superior realization of the same rationale.
Ak = FML,k + (2nk/N+1)
where
nk | degrees of freedom for the k'th model |
FML,k | the maximum likelihood discrepancy function for the k'th model |
N | sample size |
Sk = FML,k + [nkln(N)/N-1]
where
nk | degrees of freedom for the k'th model |
FML,k | the maximum likelihood discrepancy function for the k'th model |
N | sample size |
FML[Sn,Sk(q)]
In this case, F is the maximum likelihood discrepancy function, S is the covariance matrix calculated on a cross-validation sample.
σk(q)
is the reproduced covariance matrix obtained by fitting model k to the original calibration sample. In general, better models will have smaller cross-validation indices.
The drawback of the original procedure is that it requires two samples, i.e., the calibration sample for fitting the models, and the cross-validation sample. The new measure estimates the original cross-validation index from a single sample.
The measure is:
Ck = FML[Sn,Sk(q)] + 2fk/(N-p-2)
where
N | sample size |
p | the number of manifest variables |
fk | the number of free parameters for the k'th model |
Bk = (F0 - Fk)/F0
where
F0 | the discrepancy function for the "Null Model" |
Fk | the discrepancy function for the k'th model |
This index approaches 1 in value as fit becomes perfect. However, it does not compensate for model parsimony.
BBNk = [(c02/n0) - (ck2/nk)]/[(c02/n0)-1]
where
c02 | chi-square for the "Null Model" |
ck2 | chi-square for the k'th model |
n0 | degrees of freedom for the "Null Model" |
nk | degrees of freedom for the k'th model |
1 - (t-hatk/t-hat0)
where
t-hatk | estimated non-centrality parameter for the k'th model |
t-hat0 | estimated non-centrality parameter for the "Null Model" |
πk = (nk/n0)Bk
where
n0 | degrees of freedom for the "Null Model" |
nk | degrees of freedom for the k'th model |
Bk | Bentler-Bonnet normed fit index |
rk = [(F0/n0) - (Fk/nk)]/(F0/n0)
where
F0 | discrepancy function for the "Null Model" |
Fk | discrepancy function for the k'th model |
n0 | degrees of freedom for the "Null Model" |
nk | degrees of freedom for the k'th model |
Dk = (F0 - Fk)/(F0 - nk/N)
where
F0 | discrepancy function for the "Null Model" |
Fk | discrepancy function for the k'th model |
nk | degrees of freedom for the k'th model |