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Note. The first-order and second-order factors used SMRs plus 2 iterations as communality estimates for the ECFA using principal factors extraction and Promax rotation. The correlations of g (i.e., the general factor) with the scales was by extension analysis (Gorsuch, 1997).

Note. The first-order and second-order factors used SMRs plus 2 iterations as communality estimates for the ECFA using principal factors extraction and Promax rotation. The correlations of g (i.e., the general factor) with the scales was by extension analysis (Gorsuch, 1997).

If one suspects that there is a general factor and CA or ECFA is used, that general factor will usually be found if and only if a higher-order analysis is computed from unrestricted rotation.

Item analysis is probably the most common situation in which a rotation restricted to orthogonality is misleading. The author of a scale includes items that each measure the underlying characteristic; then a total score is computed by adding the items together. So the author is assuming that there is a general factor—that is, one that loads all of the items. What happens when the scale is factored? Because factor analysis is a sensitive tool, it will take into account the almost universal fact that some items will correlate more highly with each other than with the rest of the items. There are generally several subsets of items that correlate slightly higher among themselves than with the other items because they have the same distributions or use similar words. Then several factors will be found. These factors may, for example, be one for the easy items, one for the medium-difficulty items, and one for the hard items. None of these factors will be a general factor because, as in Table 6.4, the general factor is found in the correlations among the factors. Varimax, however, never allows such correlations to occur. The decision to restrict item analysis rotation to orthogonality is a decision with major implications. It is far better to use Promax, an unrestricted rotation, and see whether a general factor happens to occur among the factors.

An instructive example can be drawn from the factor analyses of the Beck Depression Inventory (BDI). Chan (Gorsuch & Chan, 1991) ran analyses in Chinese and U.S. samples, and computed the relationships of previous U.S. and Canadian factor analyses to her factors. The table clearly showed that (a) primary factors did not replicate, whether within or across countries; (b) all primary factors correlated highly; and (c) the second-order depression factor replicated both within and across countries. That general factor is the same as the total score. The prior studies missed this fact because they only provided first-order analyses, and the erroneous conclusion from those would have been that there were no replicable factors. Chan showed the correct conclusion to be that there is one factor in the BDI, just as the author designed it.

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