**Factor Analysis **** with Maximum
Likelihood Extraction in SPSS.**
Before we begin with the analysis; let's take a moment to address and
hopefully clarify one of the most confusing and misarticulated issues in
statistical teaching and practice literature.
**First, Principal Components Analysis (PCA)** is a variable reduction
technique which maximizes the amount of variance accounted for in the observed
variables by a smaller group of variables called COMPONENTS. As an example,
consider the following situation. Let's say, we have 500 questions on a survey
we designed to measure persistence. We want to reduce the number of questions so
that it does not take someone 3 hours to complete the survey. It would be
appropriate to use PCA to reduce the number of questions by identifying and
removing redundant questions. For instance, if question 122 and question 356 are
virtually identical (i.e. they ask the exact same thing but in different ways),
then one of them is not necessary. The PCA process allows us to reduce the
number of questions or variables down to their PRINCIPAL COMPONENTS.
PCA is commonly, but very confusingly, called exploratory factor analysis (EFA).
The use of the word *factor* in EFA is inappropriate and confusing because
we are really interested in COMPONENTS, not factors. This issue is made more
confusing by some software packages (e.g. PASW/SPSS & SAS) which list or use PCA
under the heading factor analysis.
**Second, Factor Analysis (FA)** is typically used to confirm the latent
factor structure for a group of measured variables. Latent factors are
unobserved variables which typically can not be directly measured; but, they are
assumed to *cause* the scores we observe on the measured or indicator
variables. FA is a model based technique. It is concerned with modeling the
relationships between measured variables, latent factors, and error.
As stated in O'Rourke, Hatcher, and Stepanski (2005): "Both (PCA & FA) are
methods that can be used to identify groups of observed variables that tend to
hang together empirically. Both procedures can also be performed with the SAS
FACTOR procedure and they generally tend to provide similar results.
Nonetheless, there are some important conceptual differences between principal
component analysis and factor analysis that should be understood at the outset.
Perhaps the most important deals with the assumption of an underlying causal
structure. Factor analysis assumes that the covariation in the observed
variables is due to the presence of one or more latent variables (factors) that
exert causal influence on these observed variables" (p. 436).
Final thoughts. Both PCA and FA can be used as exploratory analysis. But; PCA
is predominantly used in an exploratory fashion and almost never used in a
confirmatory fashion. FA can be used in an exploratory fashion, but most of the
time it is used in a confirmatory fashion because it is concerned with modeling
factor structure. The choice of which is used should be driven by the goals of
the analyst. If you are interested in reducing the observed variables down to
their principal components while maximizing the variance accounted for in the
variables by the components, then you should be using PCA. If you are concerned
with modeling the latent factors (and their relationships) which cause the
scores on your observed variables, then you should be using FA.
O'Rourke, N., Hatcher, L., & Stepanski, E.J. (2005). A step-by-step approach
to using SAS for univariate and multivariate statistics, Second Edition. Cary,
NC: SAS Institute Inc.
**Factor Analysis**
The following covers a few of the SPSS procedures for conducting factor
analysis with maximum likelihood extraction. For the duration of this tutorial
we will be using the
ExampleData4.sav file.
**FA 1**. Begin by clicking on Analyze, Dimension Reduction,
Factor...
Next, highlight all the variables of interest (y1 - y15) and use the top
arrow button to move them to the Variables: box. Then click the Descriptives
button and select the following. Then click the Continue button.
Next, click on the Extraction button. In the Method drop-down menu, choose
Maximum likelihood. Then select Unrotated factor solution and Scree plot. Notice
the extraction is based on factors with eigenvalues greater than 1 (by default).
There are a number of perspectives on determining the number of factors to
extract and what criteria to use for extraction. Originally, eigenvalues greater
than 1 was generally accepted. However, more recently
Zwick and Velicer
(1986) have suggested, Horn’s (1965) parallel analysis tends to be more precise
in determining the number of reliable components or factors. Unfortunately,
Parallel Analysis is not available in SPSS. Therefore, a review of the parallel
analysis engine (Patil,
Singh, Mishra, & Donavan, 2007) is strongly
recommended. Next,
click the Continue button.
Next, click on Scores and select Save as Variables. This will create new
variables (1 per extracted factor) which will allow us to evaluate which type of
rotation strategy is appropriate in subsequent factor analysis.
Next, click the OK button.
The output should be similar to that displayed below.
The
Descriptive Statistics table simply provides mean, standard deviation, and
number of observations for each variable included in the analysis.
The Correlation Matrix table provides correlation coefficients and p-values
for each pair of variables included in the analysis. A close inspection of these
correlations can offer insights into the factor structure.
The
next table is used as to test assumptions; essentially, the Kaiser-Meyer-Olking
(KMO) statistic should be greater than 0.600 and the Bartlett's test should be
significant (e.g. *p *< .05). KMO is used for assessing sampling adequacy
and evaluates the correlations and partial correlations to determine if the data
are likely to coalesce on factors (i.e. some items highly correlated, some not).
The Bartlett's test evaluates whether or not our correlation matrix is an
identity matrix (1 on the diagonal & 0 on the off-diagonal). Here, it indicates
that our correlation matrix (of items) is not an identity matrix--we can verify
this by looking at the correlation matrix. The off-diagonal values of our
correlation matrix are NOT zeros, therefore the matrix is NOT an identity
matrix.
A
communality (*h*²) is the sum of the
squared factor loadings and represents the amount of variance in that variable
accounted for by all the factors. For example, all five *extracted* factors
account for 33.4% of the variance in variable y1 (*h*²
= .334).
The next table displays the amount of variance accounted for in the
variables' or items' variance-covariance matrix by each of the factors and
cumulatively by all the factors. Here we see that all 5 extracted factors (those
with an eigenvalue greater than 1) account for 32.209% of the variance in the
items' variance-covariance matrix.
The scree plot graphically displays the information in the previous table;
the factors' eigenvalues.
The next table displays each variable's loading on each factor. We notice
from the output, we have two items (y14 & y15) which do not load on the first
factor (always the strongest without rotation) but create their own retained
factor (also with eigenvalue greater than 1). We know a factor should have, as a
minimum, 3 items/variables; but let's reserve deletion of items until we can
discover whether or not our factors are related.
Finally, we have the goodness-of-fit table; which gives an indication of how
well our 5 factors reproduce the variables' or items' variance-covariance
matrix. Here, the test shows that the reproduced matrix is NOT significantly
different from the observed matrix -- which is what we would hope to find;
indicating *good fit*.
Next, we will use the saved factor scores to determine if our factors are
correlated, which would suggest an oblique rotation strategy would be
appropriate. If the factors are not correlated, then an orthogonal rotation
strategy is appropriate. Keep in mind, if literature and theory suggest the
factors are related then you may use an oblique rotation, regardless of what you
find in your data.
In the Data Window, click on Analyze, Correlated, Bivariate...
Next, highlight and use the arrow button to move all the REGR factor scores
to the Variables: box. Then click the OK button.
The output should display the correlation matrix for the 5 factor scores as
below.
Here, we see that none of the factor scores are related, which suggests the
factors themselves are not related -- which indicates we should use an
orthogonal rotation in subsequent factor analysis.
**FA 2**.
VARIMAX rotation imposed. Next, we re-run the FA specifying 5 factors to
be retained. We will also specify the VARIMAX rotation strategy, which is a form
of orthogonal rotation.
Begin by clicking on Analyze, Dimension Reduction,
Factor...
Next, you should see that the previous run is still specified; variables y1
through y15. Next click on Descriptives...and select the following; we no longer
need the univariate descriptives, the correlation matrix, or the KMO and
Bartlett's tests. Then click the Continue button. Next, click on the
Extraction... button. We no longer need the scree plot; but we do need to change
the number of factors to extract. We know from the first run, there were 5
factors with eigenvalues greater than one, so we select 5 factors to extract.
Then click the Continue button.
Next, click on Rotation... and select Varimax. Then click the Continue
button. Then click on the Scores... button and remove the selection for Save as
Variables. Then click the Continue button. Then click the OK button.
The first 3 tables in the output should be identical to what is displayed
above from FA 1; accept, now we have two new tables at the bottom of the
output.
The rotated factor matrix table shows which items/variables load on which
factors after rotation. We see that the rotation cleaned up the interpretation
by eliminating the global first factor. This provides a clear depiction of our
factor structure (marked with red ellipses).
Again, the Factor Transformation Matrix simply displays the factor
correlation matrix prior to and after rotation.
**FA 3.** Finally, we can eliminate the two
items (y14 & y15) which (a) by themselves create a factor (factors should have
more than 2 items or variables) and (b) do not load on the un-rotated or initial
factor 1. Again, click on Analyze, Dimension Reduction, then Factor...
Again, you'll notice the previous run is still specified, however we need to
remove the y14 and y15 variables. Next, click on Extraction... and change the
number of factors to extract from 5 to 4. Then click the Continue button and
then click the OK button.
The output should be similar to what is displayed below.
The communalities
are lower than we would prefer (generally would like to see at least 0.450).
The four extracted (and rotated) factors account for 35.483 % of the variance
in the items' variance-covariance matrix.
The Factor Matrix table displays factor loadings for each item (prior to
rotation).
The goodness-of-fit table; which gives an indication of how well our 5
factors reproduce the variables' or items' variance-covariance matrix. Here, the
test shows that the reproduced matrix is NOT significantly different from the
observed matrix -- which is what we would hope to find; indicating *good fit*.
The Rotated Factor Matrix displays the loadings for each item on each rotated
factor, again clearly showing the factor structure.
And again, the Factor Transformation Matrix displays the correlations among
the factors prior to and after rotation.
As a general conclusion, we can say we have four factors accounting for
35.483 % of the variance in our 13 items. In the Rotated Factor Matrix table we
see clear factor structure displayed; meaning, each item loads predominantly on
one factor. For instance, the first four items load virtually exclusively on
Factor 1. Furthermore, if we look at the communalities we see that all the items
displayed a communality of 0.30 or greater, with one exception. The exception is
y4, which is a little lower than we would like and given that Factor 1 has three
other items which load substantially on it, we may choose to remove item y4 from
further analysis or measurement in the future.
*o = T + e*
*obs = True + err*
It is important to note that traditional factor analysis assumes the
Classical Test Theory of measurement, which states that observed scores (*obs*)
are a result of true scores (*True*) and error (*err*). Therefore, in
most factor model diagrams, the arrows point to the observed variables (note, in
the diagram below the coefficients are not present) reflecting the assumption
that the true value of the factor and the error combines to result in the
observed score(s).
REFERENCES / RESOURCES
Horn, J. (1965). A rationale and test for the number of
factors in factor analysis. *Psychometrika, 30, *179 – 185.
O'Rourke, N., Hatcher, L., & Stepanski, E.J. (2005). A step-by-step approach
to using SAS for univariate and multivariate statistics, Second Edition. Cary,
NC: SAS Institute Inc.
Patil,
V. H., Singh, S. N., Mishra,
S., & Donavan, D. T. (2007). Parallel Analysis Engine to Aid Determining
Number of Factors to Retain [Computer software]. Retrieved 08/23/2009 from
http://ires.ku.edu/~smishra/parallelengine.htm
Zwick, W. R., & Velicer, W. F. (1986). Factors influencing
five rules for determing the number of components to retain. *Psychological
Bulletin, 99, *432 – 442. |