Noncentrality-Based, Goodness-of-Fit Indices
Statistica provides a variety of noncentrality-based, goodness-of-fit indices for evaluating the SEPATH model you have created. You can access these indices by clicking the Noncentrality-Based Indices button on the Advanced tab of the Structural Equation Modeling Results dialog box.
These indices are all based on the idea, first proposed by Steiger and Lind (1980), of basing goodness-of-fit assessment on an estimation of the population noncentrality parameter. Steiger, Shapiro, and Browne (1985) proved that, under some realistic simplifying assumptions, the Chi-square test statistic has a distribution which is asymptotically non-central Chi-square, with a non-centrality parameter equal to N times the population discrepancy function. The population discrepancy function is that value of the discrepancy function that you would obtain if (a) you actually knew the population covariance matrix S, and (b) you were to analyze it as if it were sample data. It is a rather natural index of "badness-of-fit." The population discrepancy function can be estimated, with a confidence interval, from sample data. So can other fit indices that are functions of the population discrepancy function. The following estimates are computed, along with a 90% confidence interval.
The philosophy behind "noncentrality interval estimation" (Steiger, 1990) represents a change of emphasis in assessing model fit. Instead of testing the hypothesis that the fit is perfect, we ask the questions (a) "How bad is the fit of our model to our Statistical population?" and (b) "How accurately have we determined population badness-of-fit from our sample data."
The indices presented here allow us to assess both questions, because they allow confidence interval assessment as well as the more traditional point estimates. As a result, they reward high sample size, and high power, with a narrower confidence interval expressing high precision of estimate.
If F* is the population badness-of-fit, and n the degrees of freedom, the Steiger-Lind index can be written as
sqrt(F*/n)
In general, values of the RMSEA index below .05 indicate good fit, and values below .01 indicate outstanding fit. In general, the RMSEA index tends to produce the same conclusions about population fit as the Adjusted Population Gamma Index (see below).
The index may be expressed as
e-F*/2
Good fit is indicated by values above .95.
For this index, good fit is indicated by values above .95.