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.

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:

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)]

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.

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.

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.

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 |