Single factor structure

From InterSciWiki

1 Factor Analysis
2 Single-factor criteria
3 Multiple Measures
4 Radial factors
5 Learn more about factor analysis
6 Links

[edit] Factor Analysis

Also known as principal component analysis (PCA). The idea here is to use the SCCS data file with Spss commands /Analyze/Data Reduction and select appropriate variables under a single topic to test the idea of a single factor structure. You may want to look at <PCA> or at <factor analysis advantages> on Wikipedia. Note that if you continued to add one or more additional variables, Spss /Analyze/Data Reduction would begin to identify multiple factors.

[edit] Single-factor criteria

The basic idea here, for our concern, is to input a series of measures that you think are not only related to one another, but that either (a) all measure the same thing or (b) measure some common composite idea. In the first case - (a) - Spss /Data Reduction should return only a single component of correlated variation. If you get two or more components there is not a single factor. In the second case if you do get a single factor then you have a strong composite variables.

Added Oct 21 2007 thanks to Tanya Nielsen: What is the test of a single factor?

The first number in the Eigenvalues column of the Total Variance table should be close to 3, and considerably greater than one.
The second number should be less than one.
The ambiguous case is where the second number is 1 or just a tiny amount greater (e.g. 1.05 max), and the first is 2 or more times greater.
An eigenvalue of 1 or less is an indicator that this and subsequent (first, second, third, etc) are all random.
Also, each of the numbers in the "Component" table should be greater than 0.6 (the maximum is 1).

For more on the single-factor model, see http://eclectic.ss.uci.edu/~drwhite/pub/Reliability1990e.pdf page 119:

With k variables, the first factor should have well over 1/k of the variance
... and second factor under 1/k (Schuessler 1971: 129)
The variance accounted for by the first factor is at least three times larger than that of the second (Romney 1989: 189; Lord 1980: 21).

[edit] Multiple Measures

What is the purpose of multiple measures of the same thing or concept? Multiple measures, combined into a single "data reduction" factor, give a more realiable measure of the concept than a single measure. Single factor measures, then, as composite variables, are capable of overcoming random measurement errors which cancel each other out, and thus of giving better prediction than single measures, provided that there really are "dependent variables" that are related to the central concept being measured.

[edit] Radial factors

Once you find a single factor, however, you can enter additional variables that measure something quite different, but still related to the main factor, and you can see how the main factor may predict or be correlated with a wide variety of other variables. Once "main" factor may be quite powerful in helping to predict a great many other variables. A set of core "radial" factors may, in different combinations (see <[multiple regression]>), predict a much larger set of dependent variables. Hence single factors and radial factors may provide a parsimonious set of measures that predict a wide variety of related phenomena. This is the power of the radial idea: for a more elaborate view on the sources and implications of this approach, see the <Radex theory of complex interactions>.

Note that if you simply continued to try and add one or more additional variables to a single factor, Spss /Data Reduction would begin to identify multiple factors, and new variables might continue to fail the single factor test.