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.

Population Noncentrality Parameter. This index directly estimates the population noncentrality parameter. We present this here primarily because it can be used to compute other noncentrality-based indices, including some we have not presented because they do not lead to confidence interval estimates.

Steiger-Lind RMSEA Index. This index corrects the population noncentrality by compensating for model parsimony. All other things being equal, more parsimonious models (those with fewer parameters) tend not to fit as well as less parsimonious models. Consequently, an index of fit that fails to compensate for the number of parameters in the model can be somewhat misleading. The Steiger-Lind index compensates for model parsimony by dividing the estimate of the population noncentrality parameter by the degrees of freedom. This ratio, in a sense, represents a "mean square badness-of-fit." Taking the square root leads to a measure that is analogous, in a sense, to a root mean square measure of model misfit.

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).

McDonald's Index of Noncentrality. McDonald proposed this index of noncentrality in a 1989 article in the Journal of Classification. The index represents one approach to transforming the population noncentrality index F* into the range from 0 to 1. The index does not compensate for model parsimony, and the rationale for the exponential transformation it uses is primarily pragmatic.

The index may be expressed as

e-F*/2

Good fit is indicated by values above .95.

Population Gamma Index. This index was proposed by Steiger (1989) as an extension of the rationale for the GFI given by Joreskog and Sorbom (1984).  These latter two authors proposed their index as strictly a descriptive (sample based) statistic. However, Tanaka and Huba (1989) showed that the GFI and AGFI could be justified on the basis of a "coefficient of determination" rationale. Steiger (1989) noted that this rationale could be extended to the population as well as the sample, and developed the asymptotic sampling theory of the statistic (this result was also derived independently by Maiti and Mukherjee, 1990). The population gamma index is an estimate of the "population GFI," the value of the GFI that would be obtained if we could analyze the population covariance matrix σ.

For this index, good fit is indicated by values above .95.

Adjusted Population Gamma Index. This index is to the Joreskog and Sorbom (1984) AGFI as the Population Gamma Index is to the GFI. It is, basically, an estimate of the population GFI corrected for model parsimony. Good fit is indicated by values above .95.