Factor Analysis

Factor analysis is a procedure for reducing scotes on many variables (e.g. tests) to scores on a smaller number of "factors."

The identification of factors is based on the correlations among scores.

Here is a "factor matrix" that summarizes a factor analysis of subjects' scores on six WAIS subtests:

 

 

Subtest

 

Loading on Factor 1

 

Loading on Factor 2

 

Communality

 

Information

.65

.20

.4625

 

Vocabulary

.70

.09

.4981

 

Similarities

.60

30

.4500

 

Picture Completion

.20

.60

.4000

 

Block Design

.15

.55

.3250

 

Object Assembly

-.25

.50

.3125

 

Extracted variance (eigenvalue)

1.40

.80

2.2

 

Percentage variance

23.33%

13.33%

36.66%

 

Some Importantant Terminology

Factor loading:

This is the correlation between a given test and a given factor. Like any correlation, it will range from -1.0 to +1.0, and it can be squared to determine the proportion of variability acounted for by this factor.

Communality (common variance)

This is the overall proportion of variance attributable to the factors

(e.g., for Picture Completion, communality = .20*.02 + .60*.60 = .4000)

Extracted variance (eigenvalue)

This is a measure of the amount of variance in all the tests that is accounted for by the factor (it is a sum of squares).

The eigenvalue is often converted to percentages:

(eigenvalue x 100)/number of tests

 

When a factor has a large eigenvalue, we assume this is because the factor represents some trait or characteristic common to the tests.

Factor Rotation

In order to better interpret the factors, a procedure called rotation is often used.

Rotation involves re-distributing the tests’ commonalities so that a clearer pattern of loadings emerges. The aim is to find an arrangement in which tests load high on one factor and low on others.

The factors can be viewed as axes that define a space in which the tests are displayed. Rotation moves the factors until the most simple alignment with the tests is found. Rotation can be orthogonal or oblique: in the former the factors are kept "at right angles" to each other (i.e. they remain uncorrelated); in the latter they are allowed to become correlated.

Although the factor loadings of each subtest are changed by rotation, their communality and the factors’ eigenvalues are unchanged:

Factor loadings after rotation:

Subtest

Loadings on Factor 1'

Loadings on Factor 2'

communality

Information

.68

.01

.4625

Vocabulary

.70

.09

.4981

Similarities

.66

.12

.4500

Picture Completion

.05

.63

.4000

Block Design

.01

.57

.3250

Object Assembly

.10

.55

.3125

Extracted variance (eigenvalues)

1.40

.80

2.2

Percentage variance

23.33%

13.33%

36.66%

 

Notice that the subtests have loadings on the rotated factors that tend to be higher or lower than they were on the unrotated factors. The rotated factors "line up" better with the tests.