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Performing Factor Analysis/Principal
Components on computers
JMP has a nice interactive spinning plot, only does VARIMAX rotation
Uses of factor analysis
Types of structures
How many factors?
A market researcher has collected spending information from 100 households. The variables measured are
Uses of Factor Analysis
Y1j, = a11F1j
Y2j, = a21F1j + a22F2j + U2j,
Y3j, = a31F1j + a32F2j + U3j,
Y4j, = a41F1j + a42F2j + U4j,
Y5j, = a51F1j + a52F2j + U5j.
These equations form the factor structure. The coefficients in the equations, aij, are factor structure coefficients. (If the Yij had been represented in their standardized forms Zij = (Yij, - µi)÷si then these coefficients would be called the factor loadings.)
Assumptions of Factor Analysis
The purpose of factor rotation is to find a factor loadings pattern that is easier to interpret. Recall the factor indeterminance problem, which says that the factor structure is not unique. There is an infinite range of possible factor structures that will fit the observed data equally well. Factor indeterminance gives us the freedom to impose an additional criterion for the factor structure. We not only want it to provide a good fit, but we want it to be easy to interpret. Several additional criteria are available, each having its own factor rotation method.
Orthagonal rotation methods are methods that will preserve the zero correlation between the factors.
VARIMAX: One of the most commonly used methods is the VARIMAX rotation. This seeks to maximize the separation of the factor loadings in a column of the factor pattern; that is, for each factor, it tries to say that the factor is mostly correlated with a group of observed Y variables and uncorrelated witht the other Y variables. JMP uses VARIMAX.
QUARTIMAX: This method does the opposite of VARIMAX--it seeks to maximize the separation between the rows of the loadings pattern. It tries to say that a given observed variable Y is mostly correlated with a small group of factors and uncorrelated with the rest.
EQUIMAX: This method is a compromise between VARIMAX and QUARTIMAX.
There are several variants on each of these methods.
Oblique rotation methods will allow the factors to be correlated in the process of increasing the separation of the factor loadings. Two common methods are
characteristic root: See eigenvalue
eigenvalue: (Also called characteristic root or latent root) A measure of the strength of a correlation among a group of variables. The number of eigenvalues for a dataset equals the number of variables in the data set though there is no way to relate an eigenvalue to a single variable. The sum of all eigenvalues must equal the number of variables in the dataset.
latent root: See eigenvalue
scree plot: A graph of the eigenvalues (on the vertical axis) against the rank of the eigenvalue (on the horizontal axis). Used for determining the number of factors to use. (The term is sometimes erroneously capitalized. The graph takes its name not from a person, but from geology. Scree is the rocky debris at the base of a cliff.)
Type P, S, or T factor analsys: Don't worry about them.
Type R factor analysis: A factor analysis which seeks to find similar variables. The most common use of factor analysis.
Type Q factor analysis: A factor analysis which seeks to find similar observations. Similar to cluster analysis.
last updated April 27, 2001, by Jim Frederick
copyright 2001 James R. Frederick