The material below began as notes I copied from the blackboard in Karl Jöreskog's factor analysis course at Princeton in 1970/71. Over the years I've added topics that appeared as Frequently Asked Questions in consulting and teaching.
This document outlines the phases of a factor analytic study and a number of the practical questions and issues that need to be addressed. As an outline, it does not go into much detail. Instead, you should consult one or more of these sources:
PRIORS
statement in
PROC FACTOR
).
PROC STANDARD
to set the means in each group to zero before computing correlations.)
Alternatively, code the group variable with dummy variables and examine
the correlations with the factor variables.
Rather than regarding c^{2}; as a formal test statistic, one should regard it as a badness of fit measure in the sense that large c^{2}; values correspond to bad fit and small values correspond to good fit. From this perspective, the statistical problem is not one of testing a given hypothesis (which may be considered false a priori), but rather one of fitting the model to the data to decide whether the fit is adequate or not. With greater N you can extract more statistically significant factors.
The c^{2}; measure is sensitive to sample size and very sensitive to departures from multivariate normality. Large sample sizes and departure from normality tend to increase c^{2}; over and above what can be expected due to misspecification of the model. (See CFA below for alternative measures).
A more reasonable way to use the c^{2} value is to compare the differences in c^{2}; to the differences in degrees of freedom as more factors are added to the model. A large drop in c^{2}; compared to the difference in d.f. indicates that the addition of one more factor represents a real improvement. A drop in c^{2}; close to the difference in d.f. indicates that the improvement in fit is obtained by 'capitalizing on chance', and the added factor may not have real significance or meaning.
 (1) 

The vague basis for the .3.4 rule of thumb appears to be this: With N=100, the minimum significant correlation at p< .05 is about 0.2. Doubling this gives 0.4. By this rule of thumb, interpreting a structure correlation of 0.3 as significant would require N>175.
Note that with very large sample sizes, loadings so small as to be uninterpretable may still be significant. This may be another reason for the popularity of 0.3 as an absolute minimum.

Note: For a LISREL factor analysis model to be identified, it is necessary to fix at least one loading on each factor to a nonzero value (e.g., 1.0) in order to fix the measurement scale of that factor.
PROC CALIS
calls these
"Lagrange multiplier" tests. This index is the expected decrease
in c^{2}; if a single constraint in the hypothesis is relaxed, and all
estimated parameters are held fixed at their estimated values.
Each modification index is a c^{2} with 1 df, and the parameter with
the largest index will improve fit maximally. Relaxing parameters based on
the modification index is only recommended when the parameter(s) freed
make sense from a substantive point of view.
Similarly, the tvalues for each free parameter provide a test of the
hypothesis that the parameter equals 0. PROC CALIS
provides
a Wald test statistic as well, which is a 1 df c^{2}; value.
Both statistics evaluate whether a restriction (setting the parameter = 0)
can be imposed on the estimated model.
For example if H _{1} and H _{2} both specify the same factor pattern, but H _{1} fixes the factor correlations, F = I, while H _{2} allows factor correlations to be free, the c^{2}; difference is attributable to the correlations among the factors.

Goodness of fit index (GFI): A measure of the relative amount of variances and covariances accounted for by the model, and an adjusted goodness of fit value (AGFI), adjusted for degrees of freedom. Both measures are between 0 and 1, where 1=perfect fit. Unlike the c^{2}; , Jöreskog & Sorbom (1984) claim that both the GFI and AGFI index are independent of sample size and relatively robust against departure from normality. Their distributional properties are unknown, however, so there is no significance test associated with them.
It should be emphasized that the measures c^{2}, GFI, and AGFI are measures of the overall fit of the model to the data and do not express the quality of the model by any other criteria. For example, it can happen that the overall fit of the model is very good, but one or more relationships in the model is poorly determined (as indicated by the squared multiple correlations), or vice versa. Furthermore, if any of the overall measures indicates that the model does not fit well, that fact does not tell what is wrong with the model. Diagnosing what part of the model is wrong can be done by inspecting the normalized residuals (which correlations are not well fit?) and/or the modification indices (which fixed parameters might be relaxed?).
Of all available software, only AMOS provides these model comparison statistics automatically when you fit a series of models. My CALISCMP macro provides similar model comparison statistics for a set of models fit using PROC CALIS.
Good luck!
© 1995 Michael Friendly
Author: Michael Friendly