Let's work our way up the right-hand side of the "SEM Pyramid of Success," examining how the Pearson correlation gives rise to exploratory factor analysis (EFA).
Starting out with a fairly large set of variables (usually single items), EFA will arrange the variables into subsets, where the variables within each subset are strongly correlated with each other. These subsets are organized along axes (the plural of "axis," not the "axe" like a hatchet).
You could have a one-factor (one-dimensional) solution, in which case all the variables will be capable of being located along a single line (e.g., across from left to right, with low scores to the left and high scores to the right). Or there could be a two-factor (two-dimensional) solution, where the axes are across and up-and-down. Three-factor (three-dimensional) solutions are harder to describe verbally, so let's look at a
picture. These examples hold only as long as the axes are
orthogonal (at 90-degree angles) to each other (which denotes completely
uncorrelated factors), an issue to which we'll return. Solutions can also exceed three factors, but we cannot visualize four spatial dimensions (at least I can't).
In conducting factor analyses with a program such as SPSS, there are
three main steps, at each of which a decision has to be made:
(1) One must first decide what extraction method to use (i.e., how to "pull out" the dimensions). The two best-known approaches are Principal Axis Factoring (PAF; also known as common factor analysis) and Principal Components Analysis (PCA). There's only one difference, computationally, between PAF and PCA, as described in this document, yet some authors portray the two techniques as being very different (further, PCA is technically not a form of factor analysis, but many researchers treat it as such).
(ADDED 9/11/18). This EFA tutorial from Columbia University's Mailman School of Public Health provides an intuitive illustration of the distinction between PAF and PCA. As shown in Figure 4 of the document, there are three potential sources of variation on a variable (or, more loosely, three reasons why someone obtains his or her total score on a variable). Let's use an example from the music-liking items in our SPSS practice dataset. Each of the 11 items lists a music style (e.g., big band, bluegrass, classical, jazz) and asks the respondent how much he/she likes it. Let's look specifically at liking for classical music. Someone's liking score for classical music will emerge from some combination of: (a) his or her liking of musical in general (corresponding to "common variance" in Figure 4 of the Columbia document); (b) reasons the person likes classical music that don't pertain to other musical styles such as jazz, blues, etc. (e.g., he or she studied great composers in European history, corresponding to unique or "specific variance" in Figure 4); and (c) any kind of random measurement error such as the person misunderstanding the survey item or accidentally selecting an unintended answer choice (corresponding to "error variance" in Figure 4). As shown by the faint red and purple ovals in Figure 4, PCA seeks to explain variance from all three boxes, whereas PAF only seeks to explain common variance (the first box). Hence, PAF begins with R-squares from a series of multiple-regression analyses predicting liking for each style of music, one at a time, from the remaining styles of music. Doing so reveals the amount of variance common to the music styles. On the other hand, PCA takes it as a given that it is "trying" to explain 100% of the variance in all of the variables.
(2) Second, one must decide how many factors to retain. There is no absolute, definitive answer to this question. There are various tests, including the Kaiser Criterion (how many factors or components have eigenvalues greater than or equal to 1.00) and Scree Test (an "elbow curve," where one looks for drop-off in the sizes of the eigenvalues).
The book
Does Measurement Measure Up?, by John Henshaw, addresses the indeterminacy of factor analysis in the context of intelligence testing as follows: "Statistical analyses of intelligence test data... have been performed for a long time. Given the same set of data, one can make a convincing, statistically sound argument for a single, overriding intelligence (sometimes called the g factor) or an equally sound argument for multiple intelligences. In Frames of Mind,
Howard Gardner argues that 'when it comes to the interpretation of intelligence testing, we are faced with an issue of taste or preference rather than one on which scientific closure is likely to be reached' " (p. 95).
(3) The axes from the original solution will not necessarily come close to sets of data points (loosely speaking, it's like the best-fitting line in a correlational plot). The axes can be rotated to put them into better alignment with the data points. The third decision, therefore, involves the choice of rotation method. Two classes of rotation methods are orthogonal (as described above) and oblique (in which the axes are free to intersect at other than 90-degree angles, which allows the factors to be correlated with each other). Mathworks has a web document on factor rotation, including a nice color-coded depiction of orthogonal and oblique rotation. (As of January 2015, the graphics do not show up in the Mathworks document; however, I had previously saved a copy of the factor-rotation diagram, which I reproduce below.)
From Mathworks, Factor Analysis
***
The particular combination of Principal Components Analysis for extraction, the Kaiser Criterion to determine the number of factors, and orthogonal rotation (specifically one called Varimax) is known as the "
Little Jiffy" routine
, presumably because it works quickly. I've always been a Little Jiffy guy myself (and have written a song about it, below), but in recent years, Little Jiffy has been criticized, both collectively and in terms of its individual steps.
An article by K.J. Preacher and R.C. MacCallum (2003) entitled "
Repairing Tom Swift’s Electric Factor Analysis Machine" (explanation of
"Tom Swift" reference) gives the following pieces of advice (shown in italics, with my comments inserted in between):
Three recommendations are made regarding the use of exploratory techniques like EFA and PCA. First, it is strongly recommended that PCA be avoided unless the researcher is specifically interested in data reduction... If the researcher wishes to identify factors that account for correlations among [measured variables], it is generally more appropriate to use EFA than PCA...
Another article we'll discuss (Russell, 2002,
Personality and Social Psychology Bulletin) concurs that PAF is preferable to PCA, although it acknowledges that the solutions produced by the two extraction techniques are sometimes very similar. Also, data reduction (i.e., wanting to present results in terms of, say, three factor-based subscales instead of 30 original items) seems to be a respectable goal, for which PCA appears appropriate.
Second, it is recommended that a combination of criteria be used to determine the appropriate number of factors to retain... Use of the Kaiser criterion as the sole decision rule should be avoided altogether, although this criterion may be used as one piece of information in conjunction with other means of determining the number of factors to retain.
I concur with this, and Russell's recommendation seems consistent with this.
Third, it is recommended that the mechanical use of orthogonal varimax rotation be avoided... The use of orthogonal rotation methods, in general, is rarely defensible because factors are rarely if ever uncorrelated in empirical studies. Rather, researchers should use oblique rotation methods.
As we'll see, Russell has some interesting suggestions in this area.
One final area, discussed in the Russell article, concerns how to create subscales or indices based on your factor analysis. Knowing that certain items align well with a particular factor (i.e., having high
factor loadings), we can either multiply each item by its factor loading before summing the items (hypothetically, e.g., [.35 X Item 1] + [.42 X Item 2] + [.50 X Item 3].......) or just add the items up with equal (or "unit") weighting (Item 1 + Item 2 + Item 3). Russell recommends the latter. It should be noted that, if one obtains a varimax-rotated solution, the newly created subscales will only have zero correlation (independence or orthogonality) with each other if the items are weighted by exact factor scores in creating the subscales.
I have created a
diagram to explicate factor scoring in greater detail.
Here's another perspective on the issue of
factor scores (see the heading "Factor Scores/Scale Scores" when the new page opens).
Here's the Little Jiffy song.
Little Jiffy
Lyrics by Alan Reifman
(May be sung to the tune of “
Desperado,” Frey/Henley)
“Little Jiffy,” you know your status is iffy,
Some top statisticians, think that you’re no good,
You are so simple, the users just take the defaults,
And thus halts the process, of finding structure,
(Bridge)
You are a three-stage, procedure,
For making, your data concise,
And upon some simple guidelines, you do rest,
But for each step, in the routine,
Little Jiffy’s not so precise,
And the experts say, other choices are best…
For extraction, you use Principal Components,
While all your opponents, advocate P-A-F,
You use Kaiser’s test, to tell the number of factors,
While all your detractors, support the Scree test,
On your behalf, some researchers claim,
Components and factors, yield almost the same,
But computers give, several more options today,
With “Little Jiffy,” you don’t have to stay,
You can experiment, with different ways…
Varimax is used, to implement your rotation,
There’s no correlation, among your axes,
If one goes oblique, like critics urge that you ought to,
The items you brought, ooh (yes, the items that you brought, ooh),
The items you brought, ooh... (Pause)
Will fall close to the lines…
***
Finally, here are some references for students wishing to pursue EFA in greater detail.
Conway,
J.M., & Huffcutt, A.I. (2003). A review and evaluation of
exploratory factor analysis practices in organizational research.
Organizational Research Methods, 6, 147-168.
Henson,
R.K., & Roberts, J.K. (2006). Use of exploratory factor
analysis in published research: Common errors and some comment on
improved practice.
Educational and Psychological Measurement, 66, 393-416.
