46G 
Intra- Class and Inter-Class Correlations 
Problem V. To Determine Fractional Direct or Cross Inter-class Correlations 
from the First Two Moments of the Individual Classes. 
Under (IV) every x of the first class is compared with every y of the second 
class. In actual practice, however, cases are frequent where all the possible 
comparisons are not made. Such relationships may be designated as fractional 
inter-class correlations. They are more difficult to determine than the simple 
inter-class constants but they are of very great importance in many cases. Of 
course the method is quite general, but for clearness I take the determination of 
the avuncular correlations as an illustration. 
If x be the measure of a character in a parental array, family or sibship of p 
individuals and y the measure of the same or a different character in the offspring 
array (or family or families) of q individuals, then the inter-class correlation in- 
volving the comparison of every x with every y in the related classes of the 
population of m classes may be designated as an ascendant (or, if one chooses, 
descendant) correlation of the first order*. From such a table, actually formed 
by the method described under (I) or <m\y expressed algebraically as under 
(IV), one may easily derive the avuncular correlation — i.e., that between " uncles 
(aunts) " and " nephews (nieces) " — as follows : 
In such an ascendant correlation, the offspring individuals are associated with 
all the members of the parental array, both their own parents and their parent's 
siblings, or there are 8{pq) combinations. 
Hence it is given by formulae (xiii — -xv) under (IV). To obtain the avuncular 
correlation we have only to deduct from the five fundamental summations of the 
ascendant correlation those for the relationships between (weighted) individual 
parents and their S(q) individual offspring, and proceed to the correlation of r on 
the basis of S(pq - q) instead of S (pq) individuals. 
But the direct parental correlation is practically always wanted on its own 
account. It is most easily obtained from an ordinary or condensed correlation 
table, or by the method described under (II). Designating by the subscript i the 
measures entering into the correlations for individual parents (weighted with their 
offspring) and their offspring, we have for the avuncular relationship (the fractional 
inter-class correlation) by simple subtraction, 
S[$( X ')2(y>)]-S( X{ 'y ; ') 
S(pq)-S(q) ' V1) 
for the product moment coefficient, and 
SiqS&yi-SW S[qt(x'>)]-SW) 
S(pq)-S(q) ' S(pq)-S(q) ' 1 > 
S[p$(y')]-S(y-) 8[pX(y^]-S(y^) , .... 
S(pq)~S(q) ' S(pq)-S(q) ^ ; 
for the moment coefficients. 
* If correlation between parental and offspring arrays be designated as of the first order, that 
between grandparental and grandofispring arrays is of the second order, and so on. 
