About Factor Analysis

WinCross’ Factor Analysis module performs a standard R-Factor Analysis on a set of items. Using varimax rotation, the user is allowed to preset the number of factors or to determine the number of factors based on a size-of-eigenvalues criterion (the default being the set of eigenvalues that are greater than 1.0). This module also produces factor scores for each respondent for each factor as determined by the factor analysis. These factor scores can themselves be saved for further analysis. In addition, this module determines the factor for which the respondent has the highest factor score, and assigns the respondent to the segment, also knows as a factor group, corresponding to that factor. The factor group assignments can also be saved, so that they can be used by the WinCross user in crosstabulations.

Considerations

How large does a factor loading have to be to consider that variable as a defining part of that factor? A general rule of thumb is that if the absolute value of the standardized loading is greater than 0.3, the variable is relevant for the particular factor. Another rule-of-thumb terms loadings as "weak" if less than 0.4, "strong" if more than 0.6, and otherwise as "moderate." These rules are arbitrary, and do not constitute a test of significance. The meaning of the factor loading magnitudes varies by research context. (For instance, loadings of 0.45 might be considered "high" for dichotomous items but for Likert scales a 0.6 might be required to be considered "high.")

In their article in the 1967 Psychological Bulletin, Cliff and Hamburger have suggested the reciprocal of the square root of the sample size as an approximate standard error for the factor loading. Thus, for a sample of 100 the standard error is 0.1, and so one can dub a pair of factor loadings as "significantly different" if they differ by 0.2. (Cudeck, R. & O'Dell, L. L. (1994). Applications of standard error estimates in unrestricted factor analysis: Significance tests for factor loadings and correlations. Psychological Bulletin, 115, 475-487 present a more rigorous, but computationally difficult to implement, method for significance testing of rotated loadings.)

WinCross has supplemented these rules of thumb with additional rules that govern the designation of "importance" to a factor loading, and hence to its labeling:

1. If the factor loading is greater than 0.3 in absolute value, we label it with a *.
2. The largest loading across the factors is labeled with an XX. This determination overrides the previous labeling.
3. If the factor loading is at least 0.2 higher than the next higher loading, we label it with ****. This latter overrides both of the above.

Usually the largest factor loading satisfies (3), and there are no other factor loadings that exceed 0.3 in absolute value. In that case the largest factor loading is labeled by a **** and no other loading receives a label.

Following are three examples in which this is not the case.

• In the first instance (item 5), though the highest loading exceeds the next highest by 0.2, and so is labeled by a ****, there are two others loadings that exceed 0.3, and so they are labeled by a *.
• In the second instance (item 12), the highest loading does not exceed the next highest loading by 0.2, and so it is labeled by an XX, and the next highest loading, being greater than 0.3, is labeled by a *.
• In the third instance (item 74), the highest loading does not exceed the next highest loading by 0.2, and so it is labeled by an XX, and the next highest loading, being smaller than 0.3, gets no label.

The table presents the items, their factor loadings on 20 factors, and the importance labels.

Related topics:

Factor Analysis - Variable data

Factor Analysis - ASCII data