278	THE FACTORS OF THE MIND
quality which all the tests are presumed more or less to measure.
But frequently it is possible to cross-classify or sub-classify one and
the same set of measurements by two or more criteria simulta-
neously. In such cases the analysis of variance goes on to split the
residual variance (or rather residual square-sums) into two or more
portions, much as it began by splitting the total. The ' analysis of
variance with two criteria ' corresponds to those factorial inquiries
in which we are interested both in the general factor for tests and in
the general factor for persons : with two criteria, therefore, the
problem is to demonstrate the significance of the differences between
the means both for the columns (persons) and for the rows (tests) of
the measurement matrix. When these two sources of variation
have been removed, we are left with a doubly centred bipolar table
of * discrepance/ the means of both rows and columns all equal to
zero, analogous to tlie doubly centred bipolar table of residual
measurements and residual correlations obtained in factor-analysis
after the general factor or factors have been removed. In the
analysis of variance these residual figures are utilized for the study of
what is called ' interaction,' that is to say, of the differential rela-
tions between the items in the rows and in the columns. This
corresponds to the study of secondary factors in multiple-factor
analysis. In the analysis of variance, as the term ^< interaction '
implies, its reciprocity is taken for granted. When these further
developments are borne in mind it becomes obvious that there must
be a large variety of psychological problems which could be attacked
with, profit by these newer statistical methods.^1
^1 In planning experiments for tke subsequent analysis of variance, there is
one important device which has proved of great use in agricultural research
and which, might also be applied with advantage for numerous psychological
investigations, namely, the * randomization * of the arrangement of the
selected items. We have found this principle of special value in overcoming
more effectively many of the familiar difficulties in experiments on the
efficiency of different methods of training, testing, and the like (e.g. more
particularly in ruling out the irrelevant effect of the lapse of time, difference
in difficulty of test-materials, etc.) ; it also enables us to obtain valid results
from smaller groups and often to solve several questions simultaneously :
(an illustrative research by W. D, Seymour — on the influence of visual adapta-
tion on speed of reading with different persons — will shortly be published).
It is suggestive to note that Fisher, in his recent book on the Design of
Experiments, actually chooses a psycho-physical experiment to introduce the
principles he has elaborated, and to illustrate the need for randomization
Interaction, it should be added, may be of various kinds.   That noted in
the text corresponds to the interaction of the items in the rows with the