XIL—PARTIAL CORRELATION. l 
The student should notice that the set of three coefficients of 
order zero and value unity are only consistent if either one only, 
or all three, are positive, z.e. +1, +1, +1,0r = 1, —=1, +1; but 
not —1, —1, —1. On the other hand, the set of three coefficients 
of the first order and value unity are only consistent if one only, 
or all three, are negative: the only consistent sets are +1, +1, 
—land —1, —1, —1. The values of the two given 7’s need to 
be very high if even the sign of the third can be inferred ; if the 
two are equal, they must be at least equal to 4/05 or *707 . . . . 
Finally, it may be noted that no two values for the known 
coefficients ever permit an inference of the value zero for the 
third ; the fact that 1 and 2, 1 and 3 are uncorrelated, pair and 
pair, permits no inference of any kind as to the correlation 
between 2 and 3, which may lie anywhere between +1 and — 1, 
19. We do not think it necessary to add to this chapter a 
detailed discussion of the nature of fallacies on which the theory 
of multiple correlation throws much light. The general nature of 
such fallacies is the same as for the case of attributes, and was 
discussed fully in Chap. IV. §§ 1-8. It suffices to point out the 
principal sources of fallacy which are suggested at once by the 
form of the partial correlation 
7, =n eee ° 
a eT) & 
and from the form of the corresponding expression for r, in terms 
of the partial coefficients 
Premills b TTR, b 
li (1 = risa)(1 = 731) 2) 
From the form of the numerator of (a) it is evident (1) that even 
if 7, be zero, ry,, will not be zero unless either 7,5 or 7,, or 
both, are zero. If 7; and 7,, are of the same sign the partial 
correlation will be negative ; if of opposite sign, positive. = Thus 
the quantity of a crop might appear to be unaffected, say, by 
the amount of rainfall during some period preceding harvest : 
this might be due merely to a correlation between rain and low 
temperature, the partial correlation between crop and rainfall 
being positive and important. We may thus easily misinterpret 
a coefficient of correlation which is zero. (2) 7,55 may be, indeed 
often is, of opposite sign to 7, and this may lead to still more 
serious errors of interpretation, 
From the form of the numerator of (5), on the other hand, we 
see that, conversely, r,, will not be zero even though 7, , is zero, 
unless either r,, or ry, is zero. This corresponds to the theorem 
925°