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 GFI. Values above .95 indicate good fit. This index is a negatively biased estimate of the population GFI, so it tends to produce a slightly pessimistic view of the quality of population fit. We give this index primarily because of its historical popularity. The Population Gamma index is a superior realization of the same rationale.

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.

Akaike Information Criterion. This criterion is useful primarily for deciding which of several nested models provides the best approximation to the data. When trying to decide between several nested models, choose the one with the smallest Akaike criterion. The criterion is defined, for the k'th model, as:

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

Schwarz's Bayesian Criterion. This criterion, like the Akaike, is used for deciding among several models in a nested sequence. When deciding among several nested models, choose the one with the smallest Schwarz criterion value. The criterion is defined, for the k'th model, as

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

Browne-Cudeck Cross Validation Index. Browne and Cudeck (1989) proposed a single sample cross-validation index as a follow-up to their earlier (Cudeck & Browne,1983) paper on cross-validation. Cudeck and Browne had proposed a cross-validation index which, for the k'th model in a set of competing models is of the form

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

Independence Model Chi-square and df. These are the Chi-square goodness-of-fit statistic, and associated degrees of freedom, for the hypothesis that the population covariances are all zero. Under the assumption of multivariate normality, this hypothesis can only be true if the variables are all independent. The "Independence Model" is used as the "Null Model" in several comparative fit indices.

Bentler-Bonett (1980) Normed Fit Index. One of the most important and original fit indices, the Bentler-Bonett index measures the relative decrease in the discrepancy function caused by switching from a "Null Model" or baseline model, to a more complex model.  It is defined as:

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.

Bentler-Bonett Non-Normed Fit Index. This comparative index takes into account model parsimony.  It may be written as

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

Bentler Comparative Fit Index. This comparative index estimates the relative decrease in population noncentrality obtained by changing from the "Null Model" to the k'th model.  The index may be computed as:

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"

James-Mulaik-Brett Parsimonious Fit Index. This index was one of the earliest (along with the Steiger-Lind index) to compensate for model parsimony. Basically, it operates by rescaling the Bentler-Bonnet Normed fit index to compensate for model parsimony. The formula for the index is:

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

Bollen's Rho. This comparative fit index computes the relative reduction in the discrepancy function per degree of freedom when moving from the "Null Model" to the k'th model. It is computed as:

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

Bollen's Delta. This index is similar in form to the Bentler-Bonnet index, but rewards simpler models (those with higher degrees of freedom). It is computed as:

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