THE MISMEASURE OF MANThurstone therefore calculated the Spearman-Burt principal com-
ponents and then rotated them to different positions until they lay
as close as they could (while still remaining perpendicular) to actual
clusters of vectors. In this rotated position, each factor axis would
receive high positive projections for the few vectors clustered near
it, and zero or near zero projections for all other vectors. When
each vector has a high projection on one factor axis and zero or
near zero projections on all others, Thurstone referred to the result
as a simple structure. He redefined the factor problem as a search for
simple structure by rotating factor axes from their principal com-
ponents orientation to positions maximally close to clusters of vec-
tors.
Figs. 6.6 and 6.7 show this process geometrically. The vectors
are arranged in two clusters representing verbal and mathematical
tests. In Fig. 6.6 the first principal component (g) is an average of
all vectors, while the second is a bipolar, with verbal tests projecting
negatively and arithmetic tests positively. But the verbal and arith-
metic clusters are not well defined on this bipolar factor because
most of their information has already been projected upon g, and
little remains for distinction on the second axis. But if the axes are
rotated to Thurstone's simple structure (Fig. 6.7), then both clus-
ters are well defined because each is near a factor axis. The
arithmetic tests project high on the first simple structure axis and
low on the second; the verbal tests project high on the second and
low on the first.
The factor problem is not solved pictorially, but by calculation.
Thurstone used several mathematical criteria for discovering sim-
ple structure. One, still in common use, is called "varimax," or the
search for maximum variance upon each rotated factor axis. The
"variance" of an axis is measured by the spread of test projections
upon it. Variance is low on the first principal component because
all tests have about the same positive projection, and the spread is
limited. But variance is high on rotated axes placed near clusters,
because such axes have a few very high projections and other zero
or near zero projections, thus maximizing the spread.*
The principal component and simple structure solutions are
* Readers who have done factor analysis for a course on statistics or methodology in
the biological or social sciences will remember something about rotating axes to
varimax positions. Like me, they are probably taught this procedure as if it were a
mathematical deduction based on the inadequacy of principal components in find-