# Factor Analysis

Factor analysis attempts to identify underlying variables, or **factors**,
that explain the pattern of correlations within a set of observed
variables. Factor analysis is often used in data reduction to identify
a small number of factors that explain most of the variance that is
observed in a much larger number of manifest variables. Factor analysis
can also be used to generate hypotheses regarding causal mechanisms
or to screen variables for subsequent analysis (for example, to identify
collinearity prior to performing a linear regression analysis).

The factor analysis procedure offers a high degree of flexibility:

- Seven methods of factor extraction are available.
- Five methods of rotation are available, including direct oblimin and promax for nonorthogonal rotations.
- Three methods of computing factor scores are available, and scores can be saved as variables for further analysis.

**Example.** What underlying attitudes lead people to respond
to the questions on a political survey as they do? Examining the correlations
among the survey items reveals that there is significant overlap among
various subgroups of items--questions about taxes tend to correlate
with each other, questions about military issues correlate with each
other, and so on. With factor analysis, you can investigate the number
of underlying factors and, in many cases, identify what the factors
represent conceptually. Additionally, you can compute factor scores
for each respondent, which can then be used in subsequent analyses.
For example, you might build a logistic regression model to predict
voting behavior based on factor scores.

**Statistics.** For each variable: number of valid cases, mean,
and standard deviation. For each factor analysis: correlation matrix
of variables, including significance levels, determinant, and inverse;
reproduced correlation matrix, including anti-image; initial solution
(communalities, eigenvalues, and percentage of variance explained);
Kaiser-Meyer-Olkin measure of sampling adequacy and Bartlett's test
of sphericity; unrotated solution, including factor loadings, communalities,
and eigenvalues; and rotated solution, including rotated pattern matrix
and transformation matrix. For oblique rotations: rotated pattern
and structure matrices; factor score coefficient matrix and factor
covariance matrix. Plots: scree plot of eigenvalues and loading plot
of first two or three factors.

Factor Analysis Data Considerations

**Data.** The variables should be quantitative at the interval or ratio level.
Categorical data (such as religion or country of origin) are not suitable
for factor analysis. Data for which Pearson correlation coefficients
can sensibly be calculated should be suitable for factor analysis.

**Assumptions. **The data should have a bivariate normal distribution
for each pair of variables, and observations should be independent.
The factor analysis model specifies that variables are determined
by common factors (the factors estimated by the model) and unique
factors (which do not overlap between observed variables); the computed
estimates are based on the assumption that all unique factors are
uncorrelated with each other and with the common factors.

To Obtain a Factor Analysis

This feature requires the Statistics Base option.

- From the menus choose:
- Select the variables for the factor analysis.

This procedure pastes FACTOR command syntax.