Relational structure

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169 The model We start by examining the structure which it is reasonable to assume for the measurement of effectiveness ...If R is the set of possible recall values and P is the set of possible precision values then we are interested in the set R x P with a relation on it ...Definition 1 ...1 Connectedness:either e 1 >e 2 or e 2 >e 1 2 Transitivity:if e 1 >e 2 and e 2 >e 3 then e 1 >e 3 We insist that if two pairs can be ordered both ways then R 1,P 1 R 2,P 2,i ...We now turn to a second condition which is commonly called independence ...Definition 2 ...All we are saying here is,given that at a constant recall precision we find a difference in effectiveness for two values of precision recall then this difference cannot be removed or reversed by changing the constant value ...We now come to a condition which is not quite as obvious as the preceding ones ...
171 In other words we are ensuring that the equation R,P R,P is soluble for R provided that there exist R,R such that R,P >R,P >R,P ...The fifth condition is not limiting in any way but needs to be stated ...Definition 5 ...Thus we require that variation in one while leaving the other constant gives a variation in effectiveness ...Finally we need a technical condition which will not be explained here,that is the Archimedean property for each component ...We now have six conditions on the relational structure <R x P,>>which in the theory of measurement are necessary and sufficient conditions for it to be an additive conjoint structure ...In our case we can therefore expect to find real valued functions [[Phi]]1 on R and [[Phi]]2 on P and a function F from Re x Re into Re,1:1 in each variable,such that,for all R,R [[propersubset]]R and P,P [[propersubset]]P we have:R,P >R,P <>F [[[Phi]]1 R,[[Phi]]2 P]>F [[[Phi]]1 R,[[Phi]]2 P]Note that although the same symbol >is used,the first is a binary relation on R x P,the second is the usual one on Re,the set of reals ...In other words there are numerical scales [[Phi]]i on the two components and a rule F for combining them such that the resultant measure preserves the qualitative ordering of effectiveness ...