Page 299 - DMGT404 RESEARCH_METHODOLOGY

Unit 13: Multivariate Analysis

possible pairs of predictor variables that would give the same predictions, which is simplest to Notes
use, in the logic of minimizing the number of predictor variables needed in the typical regression?
The pair of predictor variables maximising some measure of minimalism could be said to have
simple structure. In this example involving grades, you might be able to predict grades in some
courses correctly from just a verbal test score, and predict grades in other courses accurately
from just a math score. If so, then you would have achieved a “simpler structure” in your
predictions than if you had used both tests for each and every predictions.

Simple Structure in Factor Analysis

The points of the preceding section are relevant when the predictor variables are factors. Think
of the m factors F as a set of independent or predictor variables, and imagine of the p observed
variables X as a set of dependent or criterion variables. Think a set of p multiple regressions,
each predicting one of the variables from all m factors. The standardized coefficients in this set of
regressions structure a p x m matrix called the factor loading matrix. If we replaced the original
factors by a set of linear functions of those factors, we would get just the same predictions as
before, but the factor loading matrix would be different. So we can ask which, of the many
possible sets of linear functions we might use, produces the simplest factor loading matrix.
Specially we will define simplicity as the number of zeros or near-zero entries in the factor
loading matrix—the more zeros, the simpler the structure. Rotation does not alter matrix C or
U at all, but does transform the factor loading matrix.
In the intense case of simple structure, each X-variable will have merely one large entry, so that
all the others can be ignored. But that would be a simpler structure than you would usually
expect to achieve; after all, in the real world each variable isn’t in general affected by only one
other variable. You then name the factors subjectively, based on an examination of their loadings.

In common factor analysis the procedure of rotation is in fact somewhat more abstract that I
have implied here, since you don’t actually know the individual scores of cases on factors.
However, the statistics for a multiple regression that is mainly relevant here—the multiple
correlation and the standardized regression slopes—can all be calculated just from the correlations
of the variables and factors involved. So we can base the calculations for rotation to simple
structure on just those correlations, devoid of using any individual scores.
A rotation which necessitates the factors to remain uncorrelated is an orthogonal rotation, while
others are oblique rotations. Oblique rotations regularly achieve greater simple structure, though
at the cost that you have to also consider the matrix of factor intercorrelations when interpreting
results. Manuals are usually clear which is which, but if there is ever any ambiguity, a simple
rule is that if there is any capability to print out a matrix of factor correlations, then the rotation
is oblique, as no such capacity is needed for orthogonal rotations.

Self Assessment

Fill in the blanks:
9. When the objective is to summarise information from a large set of variables into fewer
factors, ....................... analysis is used.
10. Correspondence analysis is a ....................... technique.
11. In a typical correspondence analysis, a cross-tabulation table of frequencies is first
.......................

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