12ü On the Theory of Association 



for we have at once 



ainl so Uli. TliL' case ot two attributes is tlius quite simple; the definition of 

 iiidependence is almost intuitive, and the criterion need only be applied to one 

 frequenc}' of tlie aggregate. The two attributes arc termed positively or uegativeiy 

 associated accordiiig ;us {AB) is greater or less than the value it would have 

 in the case of independeuce, or, to put the same thing aiiother way, accordiiig as 

 (AB)j{A) is greater or less than {B)IN or {AB)/(B) greater or less than {A)IN. 



If iiiorc than two attributes are noted in the record they must be dealt with, 

 in the lirst instance, pair by pair a-s above, but subsequentiy the assoeiation 

 between each pair should be observed in the sections of niateiial or sub-universes 

 defined by the other attributes. In the case of three attributes, for instance, we 

 have not only to deal with the assoeiation between A and B in the universe at 

 large but also in the universe of Cs and in the universe of 7's — associatiuns wliich 

 will be tested by comparing {ABG)I(AG) with (BC)l{C) or {ABC);iBC) with 

 (AC)l(C), and (ABy)j(Ay) with (ß7)/(7) and so on. Such "partial" associations 

 are of great practical importance, as a test for the correctncss or otherwise of 

 physical interpretations placed on any " total " associations observed. When 

 A and B are found to be associated it is a common form of argument to say that 

 the assoeiation is not " direct " but due to the assoeiation of A with G and B 

 with 0; the argument may be tested at once by finding whether A and B are 

 still associated in the separated universes C and 7. If they are the argument is 

 baseless. It has been said for instance that the assoeiation between vaccination 

 and protection from small-po-x is due to the assoeiation of both with sanitary 

 conditions (a larger proportion of the upper classes than of the lower classes being 

 vaccinated, accordiug to the argument). To test the argument the ''universe" or 

 material observed should be limited eitker to tliose liviug wholly in .sanitary 

 conditions or to those living wholly in insanitary conditions (it does not matter 

 which). 'Partial" associations again arc of importance to the biologist in the 

 theory of heredity. If an attribute be hcritable, its presence in the parent or the 

 grandparent is associated with its presence in the offspring; but the physical 

 Interpretation to be placed on such inheritauce depends very largely on whether 

 there is also a partial inheritance from the grandparent, the presence of the 

 attribute in the grandparent being associated with its presence in the offspring 

 even when the parents either all possess or all do not possess the attribute. 



It is important to notice that the test for assoeiation is necessarily based on a 

 comparison of perceiitagcs or proportions, e.g. {AB)'{A) and {B)IN. The mere 

 fact that a certain number of A'a are B gives no physical Information; besidcs 

 knowing how many A'^ are B you must kiiow also how niaiiy non-.4's are B 

 or what proportion of A's e.xists in the given universe at large. In an investigation 

 as to the causation of A it is therefore just iis important to observe non-.4's as A'a. 



