*Factor analysis* includes both *component
analysis* and *common factor analysis*. More than
other statistical techniques, factor analysis has suffered from
confusion concerning its very purpose. This affects my
presentation in two ways. First, I devote a long section to
describing what factor analysis does before examining in later
sections how it does it. Second, I have decided to reverse the
usual order of presentation. Component analysis is simpler, and
most discussions present it first. However, I believe common
factor analysis comes closer to solving the problems most
researchers actually want to solve. Thus learning component
analysis first may actually interfere with understanding what those
problems are. Therefore component analysis is introduced only
quite late in this chapter.

It was an interesting idea, but it turned out to be wrong. Today the College Board testing service operates a system based on the idea that there are at least three important factors of mental ability--verbal, mathematical, and logical abilities--and most psychologists agree that many other factors could be identified as well.

2. Consider various measures of the activity of the autonomic nervous system--heart rate, blood pressure, etc. Psychologists have wanted to know whether, except for random fluctuation, all those measures move up and down together--the "activation" hypothesis. Or do groups of autonomic measures move up and down together, but separate from other groups? Or are all the measures largely independent? An unpublished analysis of mine found that in one data set, at any rate, the data fitted the activation hypothesis quite well.

3. Suppose many species of animal (rats, mice, birds, frogs, etc.) are trained that food will appear at a certain spot whenever a noise--any kind of noise--comes from that spot. You could then tell whether they could detect a particular sound by seeing whether they turn in that direction when the sound appears. Then if you studied many sounds and many species, you might want to know on how many different dimensions of hearing acuity the species vary. One hypothesis would be that they vary on just three dimensions--the ability to detect high-frequency sounds, ability to detect low-frequency sounds, and ability to detect intermediate sounds. On the other hand, species might differ in their auditory capabilities on more than just these three dimensions. For instance, some species might be better at detecting sharp click-like sounds while others are better at detecting continuous hiss-like sounds.

4. Suppose each of 500 people, who are all familiar with different kinds of automobiles, rates each of 20 automobile models on the question, "How much would you like to own that kind of automobile?" We could usefully ask about the number of dimensions on which the ratings differ. A one-factor theory would posit that people simply give the highest ratings to the most expensive models. A two-factor theory would posit that some people are most attracted to sporty models while others are most attracted to luxurious models. Three-factor and four-factor theories might add safety and reliability. Or instead of automobiles you might choose to study attitudes concerning foods, political policies, political candidates, or many other kinds of objects.

5. Rubenstein (1986) studied the nature of curiosity by analyzing the agreements of junior-high-school students with a large battery of statements such as "I like to figure out how machinery works" or "I like to try new kinds of food." A factor analysis identified seven factors: three measuring enjoyment of problem-solving, learning, and reading; three measuring interests in natural sciences, art and music, and new experiences in general; and one indicating a relatively low interest in money.

- How many different factors are needed to explain the pattern of relationships among these variables?
- What is the nature of those factors?
- How well do the hypothesized factors explain the observed data?
- How much purely random or unique variance does each observed variable include?

The previous examples can be used to illustrate a useful
distinction--between *absolute* and *heuristic*
uses of factor analysis. Spearman's *g* theory of
intelligence, and the activation theory of autonomic functioning,
can be thought of as absolute theories which are or were
hypothesized to give complete descriptions of the pattern of
relationships among variables. On the other hand, Rubenstein
never claimed that her list of the seven major factors of curiosity
offered a complete description of curiosity. Rather those factors
merely appear to be the most important seven factors--the best way
of summarizing a body of data. Factor analysis can suggest either
absolute or heuristic models; the distinction is in how you interpret
the output.

A similar balancing problem arises in regression and analysis of variance, but it generally doesn't prevent different workers from reaching nearly or exactly the same conclusions. After all, if two workers apply an analysis of variance to the same data, and both workers drop out the terms not significant at the .05 level, then both will report exactly the same effects. However, the situation in factor analysis is very different. For reasons explained later, there is no significance test in component analysis that will test a hypothesis about the number of factors, as that hypothesis is ordinarily understood. In common factor analysis there is such a test, but its usefulness is limited by the fact that it frequently yields more factors than can be satisfactorily interpreted. Thus a worker who wants to report only interpretable factors is still left without an objective test.

A similar issue arises in identifying the nature of the factors. Two workers may each identify 6 factors, but the two sets of factors may differ--perhaps substantially. The travel-writer analogy is useful here too; two writers might each divide the US into 6 regions, but define the regions very differently.

Another geographical analogy may be more parallel to factor analysis, since it involves computer programs designed to maximize some quantifiable objective. Computer programs are sometimes used to divide a state into congressional districts which are geographically continguous, nearly equal in population, and perhaps homogeneous on dimensions of ethnicity or other factors. Two different district-creating programs might come up with very different answers, though both answers are reasonable. This analogy is in a sense too good; we believe that factor analysis programs usually don't yield answers as different from each other as district-creating programs do.

Another advantage of factor analysis over these other
methods is that factor analysis can recognize certain properties of
correlations. For instance, if variables A and B each correlate .7
with variable C, and correlate .49 with each other, factor analysis
can recognize that A and B correlate zero when C is held constant
because .7^{2} = .49. Multidimensional scaling and
cluster analysis have no ability to recognize such relationships,
since the correlations are treated merely as generic "similarity
measures" rather than as correlations.

We are not saying these other methods should never be applied to correlation matrices; sometimes they yield insights not available through factor analysis. But they have definitely not made factor analysis obsolete. The next section touches on this point.

One possible meaning of the phrase about "differentiating"
is that a set of variables all correlate highly with each other but
differ in their means. A rather similar meaning can arise in a
different case. Consider several tests A, B, C, D which test the
same broadly-conceived mental ability, but which increase in
difficulty in the order listed. Then the highest correlations among
the tests may be between adjacent items in this list
(r_{AB}, r_{BC} and
r_{CD}) while the lowest correlation is between items
at the opposite ends of the list (r_{AD}). Someone
who observed this pattern in the correlations among the items
might well say the tests "can be put in a simple order" or "differ
in just one factor", but that conclusion has nothing to do with
factor analysis. This set of tests would *not* contain just
one common factor.

A third case of this sort may arise if variable A affects B,
which affects C, which affects D, and those are the only effects
linking these variables. Once again, the highest correlations would
be r_{AB}, r_{BC} and
r_{CD} while the lowest correlation would be
r_{AD}. Someone might use the same phrases just
quoted to describe this pattern of correlations; again it has nothing
to do with factor analysis.

A fourth case is in a way a special case of all the previous cases: a perfect Guttman scale. A set of dichotomous items fits a Guttman scale if the items can be arranged so that a negative response to any item implies a negative response to all subsequent items while a positive response to any item implies a positive response to all previous items. For a trivial example consider the items

- Are you above 5 feet 2 inches in height?
- Are you above 5 feet 4 inches in height?
- Are you above 5 feet 6 inches in height?
- Etc.

- Should our nation lower tariff barriers with nation B?
- Should our two central banks issue a single currency?
- Should our armies become one?
- Should we fuse with nation B, becoming one nation?

Applying multidimensional scaling to a correlation matrix
could discover all these simple patterns of differences among
variables. Thus multidimensional scaling seeks factors which
*differentiate* variables while factor analysis looks for
the factors which *underlie* the variables. Scaling may
sometimes find simplicity where factor analysis finds none, and
factor analysis may find simplicity where scaling finds none.

1.00 .72 .63 .54 .45 .72 1.00 .56 .48 .40 .63 .56 1.00 .42 .35 .54 .48 .42 1.00 .30 .45 .40 .35 .30 1.00Imagine that these are correlations among 5 variables measuring mental ability. Matrix R55 is exactly consistent with the hypothesis of a single common factor

r_{ab.g} = (r_{ab} - r_{ag}
r_{bg})/sqrt[(1-r_{ag}^{2})(1-r_{bg}^{2})]

This formula shows that r_{ab.g} = 0 if and only if
r_{ab} = r_{ag} r_{bg}.
The requisite property for a variable to function as a general factor
*g* is that any partial correlation between any two
observed variables, partialing out *g*, is zero. Therefore
if a correlation matrix can be explained by a general factor
*g*, it will be true that there is some set of correlations
of the observed variables with *g*, such that the product
of any two of those correlations equals the correlation between the
two observed variables. But matrix R55 has exactly that property.
That is, any off-diagonal entry r_{jk} is the product
of the *j*th and *k*th entries in the row .9 .8
.7 .6 .5. For instance, the entry in row 1 and column 3 is .9 x .7
or .63. Thus matrix R55 exactly fits the hypothesis of a single
common factor.

If we found that pattern in a real correlation matrix, what
exactly would we have shown? First, the existence of the factor
is *inferred* rather than *observed*. We
certainly wouldn't have *proven* that scores on these 5
variables are affected by just one common factor. However, that
is the simplest or most parsimonious hypothesis that fits the pattern
of observed correlations.

Second, we would have an estimate of the factor's correlation with each of the observed variables, so we can say something about the factor's nature, at least in the sense of what it correlates highly with or doesn't correlate with. In this example the values .9 .8 .7 .6 .5 are these estimated correlations.

Third, we couldn't measure the factor in the sense of deriving each person's exact score on the factor. But we can if we wish use methods of multiple regression to estimate each person's score on the factor from their scores on the observed variables.

Matrix R55 is virtually the simplest possible example of
common factor analysis, because the observed correlations are
perfectly consistent with the simplest possible factor-analytic
hypothesis--the hypothesis of a single common factor. Some other
correlation matrix might not fit the hypothesis of a single common
factor, but might fit the hypothesis of two or three or four
common factors. The fewer factors the simpler the hypothesis.
Since simple hypothesis generally have logical scientific priority
over more complex hypotheses, hypotheses involving fewer factors
are considered to be preferable to those involving more factors.
That is, you accept at least tentatively the simplest hypothesis (i.e.,
involving the fewest factors) that is not clearly contradicted by the
set of observed correlations. Like many writers, I'll let
*m* denote the hypothesized number of common
factors.

Without getting deeply into the mathematics, we can say
that factor analysis attempts to express each variable as the sum of
*common* and *unique* portions. The
common portions of all the variables are by definition fully
explained by the common factors, and the unique portions are
ideally perfectly uncorrelated with each other. The degree to
which a given data set fits this condition can be judged from an
analysis of what is usually called the "residual correlation
matrix".

The name of this matrix is somewhat misleading because the entries in the matrix are typically not correlations. If there is any doubt in your mind about some particiular printout, look for the diagonal entries in the matrix, such as the "correlation" of the first variable with itself, the second with itself, etc. If these diagonal entries are not all exactly 1, then the matrix printed is not a correlation matrix. However, it can typically be transformed into a correlation matrix by dividing each off-diagonal entry by the square roots of the two corresponding diagonal entries. For instance, if the first two diagonal entries are .36 and .64, and the off-diagonal entry in position [1,2] is .3, then the residual correlation is .3/(.6*.8) = 5/8 = .625.

Correlations found in this way are the correlations that
would have to be allowed among the "unique" portions of the
variables in order to make the common portions of the variables
fit the hypothesis of *m* common factors. If these
calculated correlations are so high that they are inconsistent with
the hypothesis that they are 0 in the population, then the hypothesis
of *m* common factors is rejected. Increasing
*m* always lowers these correlations, thus producing a
hypothesis more consistent with the data.

We want to find the simplest hypothesis (that is, the lowest
*m*) consistent with the data. In this respect, a factor
analysis can be compared to episodes in scientific history that took
decades or centuries to develop. Copernicus realized that the earth
and other planets moved around the sun, but he first hypothesized
that their orbits were circles. Kepler later realized that the orbits
were better described as ellipses. A circle is a simpler figure than
an ellipse, so this episode of scientific history illustrates the
general point that we start with a simple theory and gradually
make it more complex to better fit the observed data.

The same principle can be observed in the history of experimental psychology. In the 1940s, experimental psychologists widely believed that all the basic principles of learning, that might even revolutionize educational practice, could be discovered by studying rats in mazes. Today that view is considered ridiculously oversimplified, but it does illustrate the general scientific point that it is reasonable to start with a simple theory and gradually move to more complex theories only when it becomes clear that the simple theory fails to fit the data.

This general scientific principle can be applied within a single factor analysis. Start
with the simplest possible theory (usually *m* = 1), test the fit between that theory
and the data, and then increase *m* as needed. Each increase in *m*
produces a theory that is more complex but will fit the data better. Stop when you find a
theory that fits the data adequately.

Each observed variable's *communality* is its estimated squared correlation
with its own common portion--that is, the proportion of variance in that variable that is
explained by the common factors. If you perform factor analyses with several different
values of *m*, as suggested above, you will find that the communalities generally
increase with *m*. But the communalities are not used to choose the final value of
*m*. Low communalities are not interpreted as evidence that the data fail to fit the
hypothesis, but merely as evidence that the variables analyzed have little in common with one
another. Most factor analysis programs first estimate each variable's communality as
the squared multiple correlation between that variable and the other variables in the analysis,
then use an iterative procedure to gradually find a better estimate.

Factor analysis may use either correlations or *covariances*. The
covariance cov_{jk} between two variables numbered *j* and
*k* is their correlation times their two standard deviations: cov_{jk}
= r_{jk} s_{j} s_{k}, where r_{jk} is
their correlation and s_{j} and s_{k} are their standard
deviations.
A covariance has no very important substantive meaning, but it does have some very useful
mathematical properties described in the next section. Since any variable correlates 1 with
itself, any variable's covariance with itself is its variance--the square of its standard
deviation. A correlation matrix can be thought of as a matrix of variances and covariances
(more concisely, a covariance matrix) of a set of variables that have already been adjusted to
standard deviations of 1. Therefore I shall often talk about a covariance matrix when we
really mean either a correlation or covariance matrix. I will use R to denote either a
correlation or covariance matrix of observed variables. This is admittedly awkward, but the
matrix analyzed is nearly always a correlation matrix, and as explained later we need the
letter C for the common-factor portion of R.

The central theorem of factor analysis is that you can do something similar for an entire covariance matrix. A covariance matrix R can be partitioned into a common portion C which is explained by a set of factors, and a unique portion U unexplained by those factors. In matrix terminology, R = C + U, which means that each entry in matrix R is the sum of the corresponding entries in matrices C and U.

As in analysis of variance with equal cell frequencies, the explained component C can
be broken down further. C can be decomposed into component matrices c_{1},
c_{2}, etc., explained by individual factors. Each of these one-factor
components c_{j} equals the "outer product" of a column of "factor loadings".
The outer product of a column of numbers is the square matrix formed by letting entry
*jk* in the matrix equal the product of entries *j* and *k* in the
column. Thus if a column has entries .9, .8, .7, .6, .5, as in the earlier example, its outer
product is

.81 .72 .63 .54 .45 .72 .64 .56 .48 .40 cEarlier I mentioned the off-diagonal entries in this matrix but not the diagonal entries. Each diagonal entry in a c_{1}.63 .56 .49 .42 .35 .54 .48 .42 .36 .30 .45 .40 .35 .30 .25

In the example there is only one common factor, so matrix C for this example
(denoted C55) is C55 = c_{1}. Therefore the residual matrix U for this
example (denoted U55) is U55 = R55 - c_{1}. This gives the following
matrix
for U55:

.19 .00 .00 .00 .00 .00 .36 .00 .00 .00 U55 .00 .00 .51 .00 .00 .00 .00 .00 .64 .00 .00 .00 .00 .00 .75This is the covariance matrix of the portions of the variables unexplained by the factor. As mentioned earlier, all off-diagonal entries in U55 are 0, and the diagonal entries are the amounts of unexplained or unique variance in each variable.

Often C is the sum of several matrices c_{j}, not just one as in this
example. The number of *c*-matrices which sum to C is the *rank* of
matrix C; in this example the rank of C is 1. The rank of C is the number of common
factors in that model. If you specify a certain number *m* of factors, a factor
analysis program then derives two matrices C and U which sum to the original correlation or
covariance matrix R, making the rank of C equal *m*. The larger you set
*m*, the closer C will approximate R. If you set *m* = *p*,
where *p* is the number of variables in the matrix, then every entry in C will
exactly equal the corresponding entry in R, leaving U as a matrix of zeros. The idea is to
see how low you can set *m* and still have C provide a reasonable approximation
to R.

The rules about number of variables are very different for factor analysis than for regression. In factor analysis it is perfectly okay to have many more variables than cases. In fact, generally speaking the more variables the better, so long as the variables remain relevant to the underlying factors.

Of the two rules that are discussed in this section, the first
uses a formal significance test to identify the number of common
factors. Let N denote the sample size, *p* the number
of variables, and *m* the number of factors. Also
R_{U} denotes the residual matrix U transformed into
a correlation matrix, |R_{U}| is its determinant, and
ln(1/|R_{U}|) is the natural logarithm of the
reciprocal of that determinant.

To apply this rule, first compute G = N-1-(2p+5)/6-(2/3)m. Then compute

Chi-square = G ln(1/|R_{U}|)

with

df = .5[(p-m)^{2}-p-m]

If it is difficult to compute ln(1/|R_{U}|), that
expression is often well approximated by
r_{U}^{2}, where the summation
denotes the sum of all squared correlations above the diagonal in
matrix R_{U}.

To use this formula to choose the number of factors, start
with *m* = 1 (or even with *m* = 0) and
compute this test for successively increasing values of
*m*, stopping when you find nonsignificance; that value
of *m* is the smallest value of *m* that is not
significantly contradicted by the data. The major difficulty with
this rule is that in my experience, with moderately large samples
it leads to more factors than can successfully be interpreted.

I recommend an alternative approach. This approach was
once impractical, but today is well within reach. Perform factor
analyses with various values of * m*,
complete with rotation, and choose the one that gives the most
appealing structure.

Now suppose a co-worker suggests summing each student's
verbal and math scores to obtain a composite "academic skill"
score I'll call AS, and taking the difference between each student's
verbal and math scores to obtain a second variable I'll call VMD
(verbal-math difference). The co-worker suggests running the
same set of regressions to predict grades in individual courses,
except using AS and VMD as predictors in each regression,
instead of the original verbal and math scores. In this example,
you would get exactly the same predictions of course grades from
these two families of regressions: one predicting grades in
individual courses from verbal and math scores, the other
predicting the same grades from AS and VMD scores. In fact,
you would get the same predictions if you formed composites of
3 math + 5 verbal and 5 verbal + 3 math, and ran a series of
two-variable multiple regressions predicting grades from these two
composites. These examples are all *linear functions* of
the original verbal and math scores.

The central point is that if you have *m* predictor
variables, and you replace the *m* original predictors by
*m* linear functions of those predictors, you generally
neither gain or lose any information--you could if you wish use the
scores on the linear functions to reconstruct the scores on the
original variables. But multiple regression uses whatever
information you have in the optimum way (as measured by the
sum of squared errors in the current sample) to predict a new
variable (e.g. grades in a particular course). Since the linear
functions contain the same information as the original variables,
you get the same predictions as before.

Given that there are many ways to get exactly the same
predictions, is there any advantage to using one set of linear
functions rather than another? Yes there is; one set may be
*simpler* than another. One particular pair of linear
functions may enable many of the course grades to be predicted
from just one variable (that is, one linear function) rather than
from two. If we regard regressions with fewer predictor variables
as simpler, then we can ask this question: Out of all the possible
pairs of predictor variables that would give the same predictions,
which is simplest to use, in the sense of minimizing the number of
predictor variables needed in the typical regression? The pair of
predictor variables maximining some measure of simplicity could
be said to have *simple structure*. In this example
involving grades, you might be able to predict grades in some
courses accurately from just a verbal test score, and predict grades
in other courses accurately from just a math score. If so, then you
would have achieved a "simpler structure" in your predictions than
if you had used both tests for all predictions.

In the extreme case of simple structure, each X-variable will have only one large entry, so that all the others can be ignored. But that would be a simpler structure than you would normally expect to achieve; after all, in the real world each variable isn't normally affected by only one other variable. You then name the factors subjectively, based on an inspection of their loadings.

In common factor analysis the process of rotation is actually somewhat more abstract that I have implied here, because you don't actually know the individual scores of cases on factors. However, the statistics for a multiple regression that are most relevant here--the multiple correlation and the standardized regression slopes--can all be calculated just from the correlations of the variables and factors involved. Therefore we can base the calculations for rotation to simple structure on just those correlations, without using any individual scores.

A rotation which requires the factors to remain uncorrelated
is an *orthogonal* rotation, while others are
*oblique* rotations. Oblique rotations often achieve
greater simple structure, though at the cost that you must also
consider the matrix of factor intercorrelations when interpreting
results. Manuals are generally clear which is which, but if there
is ever any ambiguity, a simple rule is that if there is any ability
to print out a matrix of factor correlations, then the rotation is
oblique, since no such capacity is needed for orthogonal
rotations.

Verbal Numer- Visual Recog- ical nition General information .80 .10 -.01 -.06 Paragraph comprehension .81 -.10 .02 .09 Sentence completion .87 .04 .01 -.10 Word classification .55 .12 .23 -.08 Word meaning .87 -.11 -.01 .07 Add .08 .86 -.30 .05 Code .03 .52 -.09 .29 Counting groups of dots -.16 .79 .14 -.09 Straight & curved capitals -.01 .54 .41 -.16 Woody-McCall mixed .24 .43 .00 .18 Visual perception -.08 .03 .77 -.04 Cubes -.07 -.02 .59 -.08 Paper form board -.02 -.19 .68 -.02 Flags .07 -.06 .66 -.12 Deduction .25 -.11 .40 .20 Numerical puzzles -.03 .35 .37 .06 Problem reasoning .24 -.07 .36 .21 Series completion .21 .05 .49 .06 Word recognition .09 -.08 -.13 .66 Number recognition -.04 -.09 -.02 .64 Figure recognition -.16 -.13 .43 .47 Object-number .00 .09 -.13 .69 Number-figure -.22 .23 .25 .42 Figure-word .00 .05 .15 .37This table reveals quite a good simple structure. Within each of the four blocks of variables, the high values (above about .4 in absolute value) are generally all in a single column--a separate column for each of the four blocks. Further, the variables within each block all seem to measure the same general kind of mental ability. The major exception to both these generalizations comes in the third block. The variables in that block seem to include measures of both visual ability and reasoning, and the reasoning variables (the last four in the block) generally have loadings in column 3 not far above their loadings in one or more other columns. This suggests that a 5-factor solution might be worth trying, in the hope that it might yield separate "visual" and "reasoning" factors. The factor names in Table 1 were given by Gorsuch, but inspection of the variables in the second block suggests that "simple repetitive tasks" might be a better name for factor 2 than "numerical".

I don't mean to imply that you should always try to make every variable load highly on only one factor. For instance, a test of ability to deal with arithmetic word problems might well load highly on both verbal and mathematical factors. This is actually one of the advantages of factor analysis over cluster analysis, since you cannot put the same variable in two different clusters.

The central concept in PCA is representation or summarization. Suppose we want to replace a large set of variables by a smaller set which best summarizes the larger set. For instance, suppose we have recorded the scores of hundreds of pupils on 30 mental tests, and we don't have the space to store all those scores. (This is a very artificial example in the computer age, but was more appealing before then, when PCA was invented.) For economy of storage we would like to reduce the set to 5 scores per pupil, from which we would like to be able to reconstruct the original 30 scores as accurately as possible.

Let *p* and *m* denote respectively the original and reduced number
of variables--30 and 5 in the current example. The original variables are denoted X, the
summarizing variables F for factor. In the simplest case our measure of accuracy of
reconstruction is the sum of *p* squared multiple correlations between X-variables
and
the predictions of X made from the factors. In the more general case we can weight each
squared multiple correlation by the variance of the corresponding X-variable. Since we can
set
those variances ourselves by multiplying scores on each variable by any constant we choose,
this
amounts to the ability to assign any weights we choose to the different variables.

We now have a problem which is well-defined in the mathematical sense: reduce
*p* variables to a set of *m* linear functions of those variables which best
summarize the original *p* in the sense just described. It turns out, however, that
infinitely many linear functions provide equally good summaries. To narrow the problem to
one
unique solution, we introduce three conditions. First, the *m* derived linear
functions
must be mutually uncorrelated. Second, any set of *m* linear functions must
include
the functions for a smaller set. For instance, the best 4 linear functions must include the best
3, which include the best 2, which include the best one. Third, the squared weights defining
each linear function must sum to 1. These three conditions provide, for most data sets, one
unique solution. Typically there are *p* linear functions (called *principal
components*) declining in importance; by using all *p* you get perfect
reconstruction of the original X-scores, and by using the first *m* (where
*m* ranges from 1 to *p*) you get the best reconstruction possible for
that
value of *m*.

Define each component's *eigenvector* or *characteristic vector* or
*latent vector* as the column of weights used to form it from the X-variables. If
the
original matrix R is a correlation matrix, define each component's *eigenvalue* or
*characteristic value* or *latent value* as its sum of squared correlations
with
the X-variables. If R is a covariance matrix, define the eigenvalue as a weighted sum of
squared
correlations, with each correlation weighted by the variance of the corresponding X-variable.
The sum of the eigenvalues always equals the sum of the diagonal entries in R.

Nonunique solutions arise only when two or more eigenvalues are exactly equal; it then turns out that the corresponding eigenvectors are not uniquely defined. This case rarely arises in practice, and I shall ignore it henceforth.

Each component's eigenvalue is called the "amount of variance" the component explains. The major reason for this is the eigenvalue's definition as a weighted sum of squared correlations. However, it also turns out that the actual variance of the component scores equals the eigenvalue. Thus in PCA the "factor variance" and "amount of variance the factor explains" are always equal. Therefore the two phrases are often used interchangeably, even though conceptually they stand for very different quantities.

If the last paragraph's line of reasoning seems to contain a gap, it is in the failure to distinguish between sampling error and measurement error. Significance tests concern only sampling error, but it is reasonable to hypothesize that an observed correlation of, say, .8 differs from 1.0 only because of measurement error. However, the possibility of measurement error implies that you should be thinking in terms of a common factor model rather a component model, since measurement error implies that there is some variance in each X-variable not explained by the factors.

An alternative method called the *scree test* was suggested by Raymond B.
Cattell. In this method you plot the successive eigenvalues, and look for a spot in the plot
where
the plot abruptly levels out. Cattell named this test after the tapering "scree" or rockpile at
the
bottom of a landslide. One difficulty with the scree test is that it can lead to very different
conclusions if you plot the square roots or the logarithms of the eigenvalues instead of the
eigenvalues themselves, and it is not clear why the eigenvalues themselves are a better
measure
than these other values.

Another approach is very similar to the scree test, but relies more on calculation and less on graphs. For each eigenvalue L, define S as the sum of all later eigenvalues plus L itself. Then L/S is the proportion of previously-unexplained variance explained by L. For instance, suppose that in a problem with 7 variables the last 4 eigenvalues were .8, .2, .15, and .1. These sum to 1.25, so 1.25 is the amount of variance unexplained by a 3-factor model. But .8/1.25 = .64, so adding one more factor to the 3-factor model would explain 64% of previously-unexplained variance. A similar calculation for the fifth eigenvalue yields .2/(.2+.15+.1) = .44, so the fifth principal component explains only 44% of previously unexplained variance.

1. Sum of eigenvalues = p

if the input matrix was a correlation matrix

Sum of eigenvalues = sum of input variances

if the input matrix was a covariance matrix

2. Proportion of variance explained = eigenvalue / sum of eigenvalues

3. Sum of squared factor loadings for *j*th principal component

= eigenvalue_{j}

4. Sum of squared factor loadings for variable *i*

= variance explained in variable *i*

= C_{ii} (diagonal entry *i* in matrix C)

= communality_{i} in common factor analysis

= variance of variable *i* if *m* = *p*

5. Sum of crossproducts between columns *i* and
*j* of factor loading matrix

= C_{ij} (entry *ij* in matrix C)

6. The relations in #3, #4 and #5 are still true after rotation.

7. R - C = U. If necessary, rule 4 can be used to find the diagonal entries in C, then rule 7 can be used to find the diagonal entries in U.

Actually, several different questions might be phrased as questions about the similarity of two factor analyses. First we must distinguish between two different data formats:

1. *Same variables, two groups*. The same set of measures might be taken
on men and women, or on treatment and control groups. The question then arises whether
the two factor structures are the same.

2. *One group, two conditions or two sets of variables*. Two test batteries
might be given to a single group of subjects, and questions asked about how the two sets of
scores differ. Or the same battery might be given under two different conditions.

The next two sections consider these questions separately.

The question, "Do these two groups have the same factor structure?" is actually quite different from the question, "Do they have the same factors?" The latter question is closer to the question, "Do we need two different factor analyses for the two groups?" To see the point, imagine a problem with 5 "verbal" tests and 5 "math" tests. For simplicity imagine all correlations between the two sets of tests are exactly zero. Also for simplicity consider a component analysis, though the same point can be made concerning a common factor analysis. Now imagine that the correlations among the 5 verbal tests are all exactly .4 among women and .8 among men, while the correlations among the 5 math tests are all exactly .8 among women and .4 among men. Factor analyses in the two groups separately would yield different factor structures but identical factors; in each gender the analysis would identify a "verbal" factor which is an equally-weighted average of all verbal items with 0 weights for all math items, and a "math" factor with the opposite pattern. In this example nothing would be gained from using separate factor analyses for the two genders, even though the two factor structures are quite different.

Another important point about the two-group problem is
that an analysis which derives 4 factors for group A and 4 for
group B has as many factors total as an analysis which derives 8
in the combined group. Thus the practical question may be not
whether analyses deriving *m* factors in each of two
groups fit the data better than an analysis deriving *m*
factors in the combined group. Rather the two separate analyses
should be compared to an analysis deriving 2*m* factors
in the combined group. To make this comparison for component
analysis, sum the first *m* eigenvalues in each separate
group, and compare the mean of those two sums to the sum of the
first 2*m* eigenvalues in the combined group. It would
be very rare that this analysis suggests that it would be better to do
separate factor analyses for the two groups. This same analysis
should give at least an approximate answer to the question for
common factor analysis as well.

Suppose the question really is whether the two factor structures are identical. This question is very similar to the question as to whether the two correlation or covariance matrices are identical--a question which is precisely defined with no reference to factor analysis at all. Tests of these hypotheses are beyond the scope of this work, but a test on the equality of two covariance matrices appears in Morrison (1990) and other works on multivariate analysis.

As in the case of two separate samples of cases, there is a question which often gets phrased in terms of factors but which is better phrased as a question about the equality of two correlation or covariance matrices--a question which can be answered with no reference to factor analysis. In the present instance we have two parallel sets of variables; that is, each variable in set A parallels one in set B. In fact, sets A and B may be the very same measures administered under two different conditions. The question then is whether the two correlation matrices or covariance matrices are identical. This question has nothing to do with factor analysis, but it also has little to do with the question of whether the AB correlations are high. The two correlation or covariance matrices within sets A and B might be equal regardless of whether the AB correlations are high or low.

Darlington, Weinberg, and Walberg (1973) described a test of the null hypothesis that the covariance matrices for variable sets A and B are equal when sets A and B are measured in the same sample of cases. It requires the assuption that the AB covariance matrix is symmetric. Thus for instance if sets A and B are the same set of tests administered in years 1 and 2, the assumption requires that the covariance between test X in year 1 and test Y in year 2 equal the covariance between test X in year 2 and test Y in year 1. Given this assumption, You can simply form two sets of scores I'll call A+B and A-B, consisting of the sums and differences of parallel variables in the two sets. It then turns out that the original null hypothesis is equivalent to the hypothesis that all the variables in set A+B are uncorrelated with all variables in set A-B. This hypothesis can be tested with MANOVA.

FACTOR will accept data in standard rectangular format. It will automatically compute a correlation matrix and use it for further analysis. If you want to analyze a covariance matrix instead, enter

TYPE = COVARIANCE

If you later want to analyze a correlation matrix, enter

TYPE = CORRELATION

The "correlation" type is the default type, so you need not enter that if you want to analyze only correlation matrices.

A second way to prepare data for a factor analysis is to compute and save a correlation or covariance matrix in the CORR menu. SYSTAT will automatically note whether the matrix is a correlation or covariance matrix at the time it is saved, and will save that information. Then FACTOR will automatically use the correct type.

A third way is useful if you have a correlation or covariance matrix from a printed source, and want to enter that matrix by hand. To do this, combine the INPUT and TYPE commands. For instance, suppose the matrix

.94 .62 .47 .36 .62 .89 .58 .29 .47 .58 .97 .38 .36 .29 .38 .87

is the covariance matrix for the four variables ALGEBRA, GEOMETRY, COMPUTER, TRIGONOM. (Normally enter correlations or covariances to more significant digits than this.) In the DATA module you could type

SAVE MATH

INPUT ALGEBRA, GEOMETRY, COMPUTER, TRIGONOM

TYPE COVARIANCE

RUN

.94

.62 .89

.47 .58 .97

.36 .29 .38 .87

QUIT

Notice that you input only the lower triangular portion of the matrix. In this example you input the diagonal, but if you are inputting a correlation matrix so that all diagonal entries are 1.0, then enter the command DIAGONAL ABSENT just before RUN, then omit the diagonal entries.

The fourth way, which *won't* work, is to enter
or scan the correlation or covariance matrix into a word processor,
then use SYSTAT's GET command to move the matrix into
SYSTAT. In this method SYSTAT will not properly record the
matrix TYPE, and will treat the matrix as a matrix of scores rather
than correlations or covariances. Unfortunately, SYSTAT willgive you output in the format
you expect, and there will be no
obvious sign that the whole analysis has been done
incorrectly.

FACTOR ALGEBRA, GEOMETRY, COMPUTER, TRIGONOM

To choose common factor analysis instead of principal components, add the option IPA for "iterated principal axis". All options are listed after a slash; IPA is an option but the variable list is not. Thus a command might read

FACTOR ALGEBRA, GEOMETRY, COMPUTER, TRIGONOM / IPA

The ITER (iteration) option determines the maximum
number of iterations to estimate communalities in common factor
analysis. Increase ITER if SYSTAT warns you that communality
estimates are suspect; the default is ITER = 25. The TOL option
specifies a change in communality estimates below which
FACTOR will stop trying to improve communality estimates;
default is TOL = .001. The PLOT option yields plots of factor
loadings for pairs of factors or components. The number of such
plots is m(m-1)/2, which may be large if *m* is large.
A command using all these options might read

FACTOR / IPA, TOL = .0001, ITER = 60, PLOT

These are the only options to the FACTOR command; all other instructions to the FACTOR program are issued as separate commands.

There are two commands you can use to control the number of factors: NUMBER and EIGEN. The command

NUMBER = 4

instructs FACTOR to derive 4 factors. The command

EIGEN = .5

instructs FACTOR to choose a number of factors equal to the number of eigenvalues above .5. Thus when you factor a correlation matrix, the command

EIGEN = 1

implements the Kaiser rule for choosing the number of factors. The default is EIGEN = 0, which causes FACTOR to derive all possible factors. If you use both NUMBER and EIGEN commands, FACTOR will follow whichever rule produces the smaller number of factors.

The one-word command SORT causes FACTOR to sort the variables by their factor loadings when printing the factor loading matrix. Specifically, it will make FACTOR print first all the variables loading above .5 on factor 1, then all the variables loading above .5 on factor 2, etc. Within each block of variables, variables are sorted by the size of the loading on the corresponding factor, with highest loadings first. This sorting makes it easier to examine a factor structure matrix for simple structure.

The ROTATE command allows you to choose a method of rotation. The choices are

ROTATE = VARIMAX

ROTATE = EQUAMAX

ROTATE = QUARTIMAX

The differences among these methods are beyond the scope of this chapter. In any event, rotation does not affect a factor structure's fit to the data, so you may if you wish use them all and choose the one whose results you like best. In fact, that is commonly done. The default method for rotation is varimax, so typing just ROTATE implements varimax.

There are three options for saving the output of factor analysis into files. To do this, use the SAVE command before the FACTOR command. The command

SAVE MYFILE/SCORES

saves scores on principal components into a file named MYFILE. This cannot be used with common factor analysis (the IPA option) since common factor scores are undefined. The command

SAVE MYFILE/COEF

saves the coefficients used to define components. These coefficients are in a sense the opposite of factor loadings. Loadings predict variables from factors, while coefficients define factors in terms of the original variables. If you specify a rotation, the coefficients are the ones defining the rotated components. The command

SAVE MYFILE/LOADING

saves the matrix of factor loadings; it may be used with either common factor analysis or component analysis. Again, if you specify a rotation, the loadings saved are for rotated factors.

- eigenvalues
- factor loading matrix (called factor pattern for IPA)
- variance explained by factors (usually equal to eigenvalues)
- proportion of variance explained by factors
- rotated factor loadings
- variance explained by rotated factors
- proportion of variance explained by rotated factors

- initial communality estimates
- an index of changes in communality estimates
- final communality estimates

- Input correlation or covariance matrix R
- Matrix of residual covariances--the off-diagonal part of U

- a scree plot
- plots of factor loadings, two factors at a time

use usdata

rotate = varimax

sort

print long

number = 2

factor cardio, cancer, pulmonar, pneu_flu, diabetes, liver / ipa, plot

Except for a scree plot and a plot of factor loadings, which have been omitted, and a few minor edits I have made for clarity, these commands will produce the following output:

MATRIX TO BE FACTORED CARDIO CANCER PULMON PNEU_FLU DIAB LIVER CARDIO 1.000 CANCER 0.908 1.000 PULMONAR 0.441 0.438 1.000 PNEU_FLU 0.538 0.358 0.400 1.000 DIABETES 0.619 0.709 0.227 0.022 1.000 LIVER 0.136 0.363 0.263 -0.097 0.148 1.000 INITIAL COMMUNALITY ESTIMATES 1 2 3 4 5 6 0.901 0.912 0.297 0.511 0.600 0.416 ITERATIVE PRINCIPAL AXIS FACTOR ANALYSIS ITERATION MAXIMUM CHANGE IN COMMUNALITIES 1 .7032 2 .1849 3 .0877 4 .0489 5 .0421 6 .0372 7 .0334 8 .0304 9 .0279 10 .0259 11 .0241 12 .0226 13 .0212 14 .0201 15 .0190 16 .0181 17 .0054 18 .0009 FINAL COMMUNALITY ESTIMATES 1 2 3 4 5 6 0.867 1.000 0.256 1.000 0.525 0.110 LATENT ROOTS (EIGENVALUES) 1 2 3 4 5 6 2.831 0.968 0.245 -0.010 -0.052 -0.223 FACTOR PATTERN 1 2 CANCER 0.967 0.255 CARDIO 0.931 -0.011 DIABETES 0.620 0.374 PNEU_FLU 0.563 -0.826 LIVER 0.238 0.231 PULMONAR 0.493 -0.113 VARIANCE EXPLAINED BY FACTORS 1 2 2.831 0.968 PERCENT OF TOTAL VARIANCE EXPLAINED 1 2 47.177 16.125 ROTATED FACTOR PATTERN 1 2 CANCER 0.913 0.409 DIABETES 0.718 0.098 CARDIO 0.718 0.593 PNEU_FLU -0.080 0.997 PULMONAR 0.313 0.397 LIVER 0.330 -0.030 VARIANCE EXPLAINED BY ROTATED FACTORS 1 2 2.078 1.680 PERCENT OF TOTAL VARIANCE EXPLAINED 1 2 34.627 28.002 MATRIX OF RESIDUALS CANCER DIABETES CARDIO PNEU_FLU PULMONAR LIVER CANCER 0.000 DIABETES 0.011 0.000 CARDIO -0.019 -0.010 0.000 PNEU_FLU 0.005 0.024 0.030 0.000 PULMONAR 0.046 0.014 -0.037 -0.017 0.000 LIVER -0.083 0.074 0.172 -0.040 -0.087 0.000

Gorsuch, Richard L. (1983) *Factor Analysis*. Hillsdale, NJ: Erlbaum

Morrison, Donald F. (1990) *Multivariate Statistical
Methods*. New York: McGraw-Hill.

Rubenstein, Amy S. (1986). An item-level analysis of questionnaire-type measures of intellectual curiosity. Cornell University Ph. D. thesis.

click here to read a Belorussian translation of this article by Galina Miklosic.