368 THE FACTORS OF THE MIND

common use is entirely satisfactory. The method here
proposed turns on the fact that f, as a < measure of dis-
crepancy,' may be used to test the distribution, not only of
frequencies, but of the sums of squares of any sets of
variates presumed to be normally distributed with unit
standard deviation. Accordingly, the residuals (calculated
after making Fisher's z-transformation when necessary)
are squared, summed, and the sum expressed as a ratio of
their expected variance, i}(N — 3). The probability that
this total is the mere effect of sampling errors is then
ascertained in the usual way.1

8. A related problem is to prove or disprove the hier-
archical character of a given correlation matrix, For this
purpose, instead of calculating intercolumnar correlations
or tetrad differences in detail, it seems simpler and more
satisfactory to compute a single index to measure its hier-
archical tendency. One or other of the 'trace-ratios'
may be used for this purpose. These are virtually equiva-
lent to calculating the variances of the factor-saturations
and of the factor-variances respectively. The former
indicate the contributions of each factor to the total
variance ; the latter, or a simple function of the latter (the
condensation-ratio) may be used to indicate the amount of
flattening produced by rotating axes. Estimates of the
significance of the factors may also be based on these trace-
ratios.2

1 The use of ^2 is warranted only when the estimates of the parameters
employed are * efficient' ([no], p. 428), i.e. based on the 'method of
maximum likelihood' or an approximation thereto ([50], p. 15). The
method of least squares generally yields such estimates ([50], p. 23),
The centroid or simple summation methods do not.

2 I should like to express my indebtedness to Dr. Trubridge for his
Mndness in reading the manuscript of this paper and pointing out several
obscurities.