THE REAL ERROR OF CYRIL BURT
Oblique axes and second-order g
Since Thurstone pioneered the geometrical representation of
tests as vectors, it is surprising that he didn't immediately grasp a
technical deficiency in his analysis. If tests are positively correlated,
then all vectors must form a set in which no two are separated by
an angle of more than 900 (for a right angle implies a correlation
coefficient of zero). Thurstone wished to put his simple structure
axes as near as possible to clusters within the total set of vectors.
Yet he insisted that axes be perpendicular to each other. This cri-
terion guarantees that axes cannot lie really close to clusters of vec-
tors—as Fig. 6 .11 indicates. For the maximal separation of vectors
is less than 90°, and any two axes, forced to be perpendicular, must
therefore lie outside the clusters themselves. Why not abandon this
criterion, let the axes themselves be correlated (separated by an
angle of less than 900 ), and permit them to lie right within the clus-
ters of vectors?
Perpendicular axes have a great conceptual advantage. They
are mathematically independent (uncorrelated). If one wishes to
identify factor axes as "primary mental abilities," perhaps they had
best be uncorrelated—for if factor axes are themselves correlated,
then doesn't the cause of that correlation become more "primary"
than the factors themselves? But correlated axes also have a differ-
ent kind of conceptual advantage: they can be placed nearer to
clusters of vectors that may represent "mental abilities." You can't
have it both ways for sets of vectors drawn from a matrix of positive
correlation coefficients: factors may be independent and only close
to clusters, or correlated and within clusters. (Neither system is
"better"; each has its advantages in certain circumstances. Corre-
lated and uncorrelated axes are both still used, and the argument
continues, even in these days of computerized sophistication in fac-
tor analysis.)
Thurstone invented rotated axes and simple structure in the
early 1930s. In the late 1930s he began to experiment with so-
tory of 'basic abilities' is already waning. The continuous difficulties with factor anal-
ysis over the last half century suggest that there may be something fundamentally w
rong with models which conceptualize intelligence in terms of a finite number of
"near dimensions. To the statistician's dictum that whatever exists can be measured,
*e factorist has added the assumption that whatever can be 'measured' must exist,
ut the relation may not be reversible, and the assumption may be false."
