Theory. We are proponents of theory-driven
analysis. Researchers often
perform an EFA because someone designed an instrument to measure particular
constructs or factors. If the theoretical framework for the instrument
is sound, we should start with the expectation that we should see
that structure in the data. Instruments are rarely perfect (especially
the first time they are examined), and theoretical expectations are
not always supported. But unless one is on a fishing expedition in
a data set with no a priori ideas about how the analysis should turn
out, this is as good a place as any to start.
Regardless, the result of an EFA must be a sensible factor structure
that is easily understood, whether that final structure matches the
initial theoretical framework or not. The basic function of EFA, in
our mind, is to make meaning of data.
The Kaiser Criterion. The
Kaiser Criterion (Kaiser, 1960, 1970) proposes that an eigenvalue
greater than 1.0 is a good lower bound for expecting a factor to be
meaningful. This is because an eigenvalue represents the sum of the
squared factor loadings in a column, and to get a sum of 1.0 or more,
one must have rather large factor loadings to square and sum (e.g.,
four loadings of at least 0.50 each, three loadings of at least 0.60
each). But this criterion gets less impressive as more items are analyzed.
It is easy to get many unimportant factors exceeding this criterion
if you analyze 100 items in an analysis.
Despite the consensus
in the literature that this is probably the least accurate method
for selecting the number of factors to retain (Velicer, Eaton, &
Fava, 2000; see also Costello & Osborne, 2005), it is often implemented
as the default selection criterion in much statistical software (such
as SPSS). Prior to the wide availability of powerful computing, this
was a simple (and not unreasonable) decision rule. Toward the latter
part of the 20th century, researchers suggested that combining this
criterion with examination of the scree plot is better (Cattell, 1966).
Minimum
eigenvalue. The Kaiser Criterion was built with PCA in
mind. In PCA, each item contributes 100% of its variance, and thus
an eigenvalue of 1 would be equivalent to 1 variable loading on a
particular factor. When this is applied to factor analysis, the interpretation
is different. An eigenvalue of 1 can be more than the average contribution
of an item. For example, if eight items each contain 70% shared variance
and 30% unique variance, the average eigenvalue for the items would
be 0.70. The minimum eigenvalue criterion is a translation of the
Kaiser Criterion to the factor analysis context, where the default
minimum eigenvalue is the average amount of shared variance contributed
by an item. In the above example, the default minimum eigenvalue would
be 0.70. This means we would retain all factors with an eigenvalue
above this value. The estimation of the default minimum eigenvalue
differs slightly by extraction method, but they offer the same conceptual
solution (see SAS, 2015). SAS prints the average eigenvalue at the
top of any eigenvalue table for use in evaluating this criterion.
(See
Figure 3.1 Initial eigenvalue estimates below.) This
method will yield results that are identical to the Kaiser Criterion
when implemented in PCA (Kim, & Mueller, 1978).
Proportion of variance. The
proportion of variance criterion examines the proportion of common
or shared variance across all of the items. It proposes that when
preliminary factors are ordered by descending eigenvalue, we retain
all factors with a cumulative proportion of variance below or at a
predefined level. For example, if we set the proportion of variance
to 70%, we would extract factors until the sum of eigenvalues exceeds
70% of the common variance. The default proportion of variance used
in SAS is 100%. This can be interpreted as a criterion that will retain
the number of factors that explain all of the shared variance. The
default proportion of variance to be used can be changed using the PROPORTION
=
option.
Scree
plot. The scree test involves examining the graph of
the eigenvalues (available in all software) and looking for the natural
bend or “elbow” in the data where the slope of the curve
changes (flattens) markedly. The number of data points above the “elbow”
(i.e., not including the point at which the break occurs) is usually
considered a good estimate of the ideal number of factors to retain.
Although the scree plot itself is not considered sufficient to determine
how many factors should be extracted (Velicer et al., 2000), many
suggest that researchers examine solutions extracting the number of
factors ranging from one to two factors above the elbow to one or
two below. As this is an exploratory technique, one should be encouraged
to explore. Some scree plots do not have one clear bend. Some have
multiple possible points of inflection, and some have no clear inflection
point (for a good example of this, see the SDQ example below). Combining
theory, the Kaiser Criterion, and examination of the scree plot is
usually a good basis for deciding the number of factors to extract
in an exploratory factor analysis.
Parallel analysis was proposed by Horn
(1965). A procedure for conducting this analysis is not included in
most common statistical software, including SAS, and thus is not widely
used. However, it is considered advantageous over the more classic
approaches (although we will see in examples below that it is not
always better; see also Velicer et al., 2000). Parallel analysis involves
generating random uncorrelated data, and comparing eigenvalues from
the EFA to those eigenvalues from that random data. Using this process,
only factors with eigenvalues that are significantly above the mean
(or preferably, the 95th percentile) of those random eigenvalues should
be retained. Several prominent authors and journals have endorsed
this as the most robust and accurate process for determining the number
of factors to extract (Ledesma & Valero-Mora, 2007; Velicer et
al., 2000).
Minimum
average partial (MAP) analysis was proposed by Velicer
(1976) as another more modern methodology for determining the number
of factors to extract in the context of PCA. This procedure involves
partialing out common variance as each successive component is created;
this is a familiar concept to those steeped in the traditions of multiple
regression. As each successive component is partialed out, common
variance will decrease to a minimum. At that point, unique variance
is all that remains. Velicer argued that a minimum point should be
considered to be the criterion for the number of factors to extract
(Velicer et al., 2000). MAP has been considered superior to the “classic”
analysis, and probably is superior to parallel analysis, although
neither is perfect, and all must be used in the context of a search
for conceptually interpretable factors.
Which
method? Although we can understand the value of parallel
analysis or MAP analysis for deciding how many factors to extract,
we have to remember our mantra: EFA is an exploratory technique. No
criterion is perfect, and unless you are misusing EFA in a confirmatory
fashion, it seems to me that worrying over a slightly better extraction
criterion might be missing the point. The point is to get a reasonable
model within a representative sample (that is sufficiently large to
ensure a reasonable solution), and then to move into inferential statistics
available in confirmatory factor analysis. EFA is merely a first stopping
point on the journey, and researchers who forget this miss the point
of the process. Thus, use theory along
with parallel analysis or MAP analysis or any of the classic criteria
that suit you and that are defensible. The goal of creating theory-driven,
conceptually understandable solutions needs to prevail. And never
forget that your journey is not done until you confirm the results
of the EFA with different data in the context of CFA.