546 SECTIONAL TRANSACTIONS.—J. 
Dr. J. WisHarT.—Sampling error in the tetrad theory. 
Although much progress has been made with the establishment of the 
tetrad theory on a rigorous mathematical basis, the difficulties inherent in 
the study of the appropriate sampling errors have prevented a corresponding 
advance in this all-important part of the work. In deciding whether a given 
body of numerical data is in accordance with theory, we cannot expect the 
tetrads to be exactly zero, and must therefore make allowance for the random 
sampling error. This is usually done by forming the distribution of sample 
tetrads (necessarily symmetrical if each is to be counted twice, once positive 
and once negative), and comparing its standard deviation with an average 
theoretical value obtained by admittedly approximate methods. In this 
paper the inperfections of existing practice are noted, and some attempt 
is made to formulate more exactly the problems to be solved before the 
matter can be considered as settled. A full solution is not reached, but an 
extension of some earlier work of the author, in which a tetrad of product 
moments was used in place of that of the correlation coefficients, is sug- 
gested as a reasonable method of approach to a more exact solution. 
Illustrations are furnished from two series of numerical data supplied by 
Prof. Spearman. 
Dr. S. S. Witxs.—A criterion for testing the mutual independence 
of several sets of traits. 
Suppose that each of N persons has been measured on a set A of n 
traits, t;, to, . . . ¢,. Furthermore, let A be subdivided into k groups 
A,, Az, ... Ax, with the 7-th group A; having n; traits specified by 
ta,, tan, . . . tay;, The question with which we are concerned is the 
following : can this sample of Nn measurements be regarded as having 
come from a population in which there is no correlation between any trait 
of one group and any trait of another? For example, if several motor and 
several mental abilities are measured on a group of individuals, it might 
be important to ask if these two categories of abilities may be regarded as 
independent of each other. Again, in the problem of fitting factor patterns 
containing group factors to psychological data, which has been considered 
by T. Kelley and others, it would perhaps be useful in some cases to group 
the traits by a priori reasons and test for the significance of any dependence 
between the groups before attempting to find coefficients of the overlapping 
factors. Otherwise such coefficients may be insignificant. The same 
questions of independence will arise when linear transformations of the 
traits are considered. 
If rpq is the sample value of the correlation coefficient between tp and tg 
in A, D the determinant | rpq | of co:relation coefficients in A, and D; the 
determinant of correlation coefficients in A; (i= 1, 2... k), then the 
proposed criterion for testing the significance of the mutual independence 
of the groups A,, As, ... Az is OQ = D/(D, D, . . . Dy). When the 
hypothesis is true that these measurements have been made on a group of 
persons which has come from a normal population in which A, Ao, . . . Ap 
are mutually independent the sampling mioments are known, and in a 
number of cases exact expressions have been obtained for the probability 
integrals in terms of incomplete B-Functions. The QO criterion will be unity 
when, and only when, all of the 7’s vanish in D which do not occur in 
D,, Dz, . . . Dz—that is, when there is no correlation in the sample between 
any trait of one group and any trait of another. @Q becomes zero as the 
hypothesis of mutual independence becomes untenable, as far as the sample 
