134 Oll the Theorii of Association 



Write 



{AB\=^{A){B), (4C), = -i(4)(C), (ßC')„=^,(ß)(0), 



subtract {AB)a froiii lioth sides of the above eqiuition, simplify, and we have 



Mm (An\ N[{AC)-{ AGU[{ BG)-{BC),-\ 

 (AB) - {AB\ : ü [F -TQ] • 



'I'hat is to say, tkere will he apparent assodation hetween A and B in the universe 

 at large unless either Ä or B is independeiit of C. Tlius, in the iniaginary case 

 of inhcritance given above, if A and B stand for the presence of the «attribute 

 in the parents and the offspring respectively, and C for tlie male sex, we find a 

 positive association between A and B in the universe at large (the poolcd results) 

 becausc A and B are both po.sitively associated with C, i.e. the niales of both 

 generations possess the attributo niore fre(iuently thaii the females. The " parents 

 with attribute " are mostly males ; as we have only noted offspring of the sanie 

 sex as the parents, their offspring invist hc mostly niales in the sanio proportion, 

 and therefore more liable to the attribute than the niostly-fomale offspring of 

 " parents without attribute." It follows obviously that if we had found no 

 inheritance to exist in any one of the four possible liues of descent (niale-male, 

 male-femaie, female-male, and female-female), no fictitious inheritance could have 

 been introduced by the pooling of the four records. The pooling of the two 

 records for the crossed-sex lines would give rise to a fictitious negative inherit- 

 ance — disinhoritanco — cancelling the positive inheritance created by the pooling 

 of the records for the same-sex lines. I leave it to the reader to verify these 

 Statements by following out the arithmetical example just given should he so 

 desire. 



The fallacy might lead to seriously niisleading results in several cases where 

 mixtures of the two sexes occur. Suppose for instance experiments were being 

 made with sonie new antitoxin on patients of both sexes. Tliere would nearly 

 always be a difference between the case-rates of mortality for the two. If the 

 female cases terminated fatally with the greater frequency and the antitoxin were 

 administered most often to the niales, & fictitious association between " antitoxin" 

 and "eure" would be created at oncc. The general expression ioT{AB) — {AB\ 

 shews how it may be avoided ; it is only neces.sary to administer the antitoxin 

 to the same proportion of patients of both sexes. This should be kept constantly in 

 mind ;us an essential rule in such experiments if it is desired to niake the most use 

 of the results. 



The fictitious association caused by mixing records finds its counterpart in the 

 spuriiius correlation to which the .saine proeess may give rise in the ca-se of 

 continuous variables, a case to which attention was drawn and which was fully 

 discussed by Professor Pearsoii in a recent momoir*. If two separate records, for 

 each of which the correlation is zero, be pooled together, a spnrious correlation 

 will necessiirily be created unless the niean of one of the variables, at leiist, be the 

 same in tlie two cases. 



• Phil. Tram. A, Vol. 192, p. 277. 



