Assume you
have scores on a number of variables (most often 10-100 variables). The purpose
of factor analysis is to discover simple patterns in the pattern of relationships
among the variables. In particular, it seeks to discover if the observed
variables can be explained largely or entirely in terms of a much smaller
number of variables called ** factors**.

*Factor Analysis or
Principal Components Analysis*

If your purpose is to reduce the information in many variables into a
set of weighted linear combinations of those variables, use *Principal Components Analysis (PCA)*,
which does not differentiate between common and unique variance.

If your purpose is to identify the latent variables which are
contributing to the common variance in a set of measured variables, use *Factor Analysis (FA),* which will attempt
to exclude unique variance from the analysis.

*Exploratory or
Confirmatory Factor Analysis*

If you wish to restrict the number of factors extracted to a particular
number and specify particular patterns of relationship between measured
variables and common factors, and this is done a priori (before seeing the
data), then the confirmatory procedure is for you. If you have no such well
specified a priori restrictions, then use the exploratory procedure.

Exploratory
factor analysis (EFA) is often recommended when researchers have no hypotheses
about the nature of the underlying 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.

*Decide the
Number of factors** *

If more factors extracted, it is likely to lead to a solution where the
major factors are well estimated by the obtained loadings but where there are
also additional poorly defined factors.

If too few factors extracted, it is likely to lead to factors that are
poorly estimated (poor correspondence between the structure of the true factors
and that of the estimated factors).

One can use scree test, then select the factors based on Kaiser’s
run (eigenvalue >1) or parallel analysis, or else.

*Choosing Factor Extraction
Method*

Maximum Likelihood (ML) extraction allows computation of assorted
indices of goodness-of-fit (of data to the model) and the testing of the
significance of loadings and correlations between factors, but *requires the assumption of multivariate
normality. *

We can first examine the distributions of the measured variables for
normality. Unless there are severe problems (|skew| > 2, kurtosis > 7),
we can use Maximum Likelihood extraction. If there are severe problems, we can
also try to correct the problems by transforming variables.

Principal Factors (PF) methods have no distributional assumptions, but
it is not preferred.

The extraction method will produce *factor
loadings* for every item on every extracted factor. Researchers hope their
results will show what is called simple structure, with *most items having a large loading on one factor but small loadings on
other factors.*

*Choosing a
Rotation Method*

Once an initial solution is obtained, the *loadings *are rotated. *Rotation is a way of maximizing high
loadings and minimizing low loadings so that the simplest possible structure is
achieved*. There are two basic types of rotation: *orthogonal and oblique.*

·
*Orthogonal*
means the factors are assumed to be uncorrelated with one another.

Common algoriths for
orthogonal rotation: *varimax**, quartamax, equamax*.

·
*Oblique*
rotation derives factor loadings based on the assumption that the factors are
correlated, and this is probably most likely the case for most measures. So,
oblique rotation gives the correlation between the factors in addition to the
loadings.

Common
algorithms for oblique rotation: *oblimin**, promax, direct quartimin.*

Empower gives *“varimax” and “promax”* for you
to choose.

*Scores*

To produce scores for each observation, you can choose
"regression” or “Bartlett” method. “Regression” gives Thompson’s scores,
“Bartlett” gives Bartlett’s weighted least-squares scores.

Below is the **sample
input window**

Below is the **sample
output ** of the above model:

Maximum likelihood factor analysis

Call:

factanal(x = tmp.xx,
factors = 3, scores = "regression", rotation = "promax")

Uniquenesses:

RATING COMPLAINTS PRIVILEGES
LEARNING RAISES CRITICAL
ADVANCE

0.227 0.080 0.639
0.005 0.239 0.771
0.299

Loadings:

Factor1 Factor2 Factor3

RATING 0.872 -0.140 0.122

COMPLAINTS 0.998

PRIVILEGES 0.408 0.185 0.117

LEARNING 0.148 0.132
0.839

RAISES 0.402
0.570

CRITICAL 0.499 -0.231

ADVANCE -0.304
0.829 0.265

Factor1 Factor2 Factor3

SS loadings 2.206
1.333 0.863

Proportion Var 0.315
0.190 0.123

Cumulative Var 0.315
0.505 0.629

Test of the hypothesis that 3
factors are sufficient.

The chi square statistic is 2.06 on
3 degrees of freedom.

The p-value is 0.56

**Empower also
output following loading plot.**