574 



Journal of Agricultural Research 



Vol. XX, No. 7 



The general formula for partial correlation can easily be expressed in 

 the present terminology. 



L^ X — DCB ^ Xl 1 



J 



r 2 = 



DCB' XA 



A. = i — 



cf>(XABCD)ct>(BCD) 

 4>{ABCD)<t>{XBCD) 



In some cases it may be of interest to find the degree of determination 

 when a number of factors not in the direct path between cause and effect 

 are assumed constant. 



3°^X-A — 



(o.-.rTS-.-CB " xXuTsfr a) 

 (o".uts... C b°' 2 a)(uts ' x) 



J>(XBC...STU...O)4>(ASTU) 

 <j>(ABC...STU)<fi(XSTU) 



RELATION TO MULTIPLE CORRELATION 



The expressions denned as <f>(XABC...), etc., suggest the expansion of 

 determinants. It is in fact easy to show that <f>(XABC. ..N) =A. 



Where 



A = 



The formula for Pearson's coefficient of multiple correlation has already 

 been given, ^ x(ABC0 ) = -J i - ~ L 



where A xx is the minor made by 



deleting row X, column X. 



Evidently in this class of cases the coefficient of determination degen- 

 erates into a function of the coefficient of multiple correlation. For the 

 degree of determination by residual factors we have 



_ <f>(XABC.) 

 ix '°~ <t>(ABC.) 



= i-R 2 



X(ABO") 



in agreement with Pearson's results. 



For the degree of determination by a known factor we have 



0(XgC...O) _ 0(XgC...)-dx.o0(gC...) A AA A XX -AA AA „ 

 flx ' A 4>{ABC...O) 4>(ABC.) A 2 IX 



Px-A 



