ExploratoryFactor Analysis: Purpose
Exploratory factor analysis (EFA) isgenerally used to discover the factor structure of a measure and to examine itsinternal reliability.
EFA is oftenrecommended when researchers have no hypotheses about the nature of theunderlying factor structure of their measure.
Exploratory factor analysis has three basic decision points:
(1) decide the number of factors, (2)choosing an extraction method, (3) choosing a rotation method.
ExploratoryFactor Analysis: Deciding the number of factors
The most common approach to decidingthe number of factors is to generate a scree plot.
The scree plot is a two dimensional graphwith factors on the x-axis and eigenvalues on the y-axis.
Eigenvalues are produced by a process called principalcomponents analysis (PCA) and represent the variance accounted for by eachunderlying factor.
They are notrepresented by percentages but scores that total to the number of items.
A 12-item scale will theoretically have 12possible underlying factors, each factor will have an eigenvalue that indicatesthe amount of variation in the items accounted for by each factor.
If a the first factor has an eigenvalue of3.0, it accounts for 25% of the variance (3/12=.25).
The total of all the eigenvalues will be 12if there are 12 items, so some factors will have smaller eigenvalues.
They are typically arranged in a scree plotin descending order
From the scree plot you can see thatthe first couple of factors account for most of the variance, then theremaining factors all have small eigenvalues.
The term “scree” is taken from the word for the rubble at the bottom ofa mountain.
A researcher might examinethis plot and decide there are 2 underlying factors and the remainder offactors are just “scree” or error variation.
So, this approach to selecting the number of factors involves a certainamount of subjective judgment.
Another approach is called theKaiser-Guttman rule and simply states that the number of factors are equal tothe number of factors with eigenvalues greater than 1.0.
I tend to recommend the scree plot approachbecause the Kaiser-Guttman approach seems to produce many factors.
ExploratoryFactor Analysis: Factor Extraction
Once the number of factors are decidedthe researcher runs another factor analysis to get the loadings for each of thefactors.
To do this, one has to decidedwhich mathematical solution to use to find the loadings.
There are about five basic extraction methods(1) PCA, which is the default in most packages.
PCA assumes there is no measurement error and is considered not to be atrue exploratory factor analysis;
(2)
maximum likelihood (a.k.a.canonical factoring); (3) alpha factoring, (4) image factoring, (5) principalaxis factoring with iterated communalities (a.k.a. least squares).
Without getting into the details ofeach of these, I think the best evidence supports the use of principal axisfactoring and maximum likelihood approaches.
I typically use the former.
Gorsuch (1989) recommends the latter if only a few iterations areperformed (not really possible in most packages).
Snook and Gorsuch (1989) show that PCA cangive poor estimates of the population loadings in small samples.
With larger samples, most approaches willhave similar results.
The extraction method will producefactor loadings for every item on every extracted factor.
Researchers hope their results will showwhat is called simple structure, with most items having a large loadingon one factor but small loadings on other factors.
ExploratoryFactor Analysis: Rotation
Once an initial solution is obtained,the loadings are rotated.
Rotation is a way of maximizing high loadings and minimizing lowloadings so that the simplest possible structure is achieved.
There are two basic types of rotation: orthogonal and oblique.
Orthogonal means the factors are assumed tobe uncorrelated with one another.
Thisis the default setting in all statistical packages but is rarely a logicalassumption about factors in the social sciences.
Not all researchers using EFA realize thatorthogonal rotations imply the assumption that they probably would notconsciously make.
Oblique rotationderives factor loadings based on the assumption that the factors arecorrelated, and this is probably most likely the case for most measures.
So, oblique rotation gives the correlationbetween the factors in addition to the loadings.
Here are some common algorithms fororthogonal and oblique rotation:
I am not an expert on the advantagesand disadvantages of each of these rotation algorithms, and they reportedlyproduce fairly similar results under most circumstances (although orthogonaland oblique rotations will be rather different). I tend to use promax rotationbecause it is known to be relatively efficient at achieving simple obliquestructure.
References andFurther Readings
Kim, J.-O., &Mueller, C.W. (1978). Introduction tofactor analysis: What
it is and how todo it.
Newbury Park: Sage.
Snook, S.C., &Gorsuch, R.L.
(1989).
Principal component analysis versus commonfactor analysis:
A Monte Carlostudy. Psychological Bulletin, 106,148-154.
Tabachnick, B.G.,& Fidell, L.S.
(2000). Using multivariate statistics
(4th Ed.).
New York:
Harper-Collins.
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