CHAPTER VI
THE DIFFERENCES BETWEEN P-, Q-, AND R-TECHNIQUES

Inverted-factor Theory. — One further theory calls
for mention — the c theory of inverted factors,3 as it has
been termed. It stands on a different footing, being not
so much a theory as a method or technique. Until
recently most psychologists have for the most part confined
themselves to factors obtained by correlating tests or
traits ; and nearly all the theories reviewed above were
originally elaborated on this basis. Almost from the outset,
however, correlations have also been calculated between
persons ; and from time to time such correlations have
been expressly studied and analysed from a factorial
standpoint.

With this modification of the usual procedure, the roles of
persons and traits become interchanged or transposed?-
Thus, when correlating traits by the ordinary product-sum
formula, the expression S#!^2 means — multiply the

1 To talk of inverting the factors or the theorems is misleading alike to the
mathematician and to the logician ; and the use of the term has prompted
a good deal of criticism that is really irrelevant to the principle essentially
involved. As I have often pointed out, the theorems required for analysing
correlations between persons are not i inversions ' of those required for the
older procedure ; they are formally identical with them, and materially their
analogues. Similarly, the matrix of measurements with which we start is
not the inverse of, but a transpose of, that which is correlated in the usual
way (the rows are merely rewritten as columns, and if necessary re-
standardized). And to describe the resulting factorial matrices as inver-
sions of each other is incorrect, except in certain special cases. With my
own method of calculation, the matrix of factor-measurements, obtained by
.covariating traits, was described as being (under certain conditions) the
inverse of the matrix of regression-cosines obtained by covariating persons ;
but that was merely because the transpose of an orthogonal matrix happens
* to be identical with its inverse. This statement seems to have led later
writers who adopted the same or a similar procedure to suppose that it
rested essentially on an * inversion * (cf. [130], p. 406).

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