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structures are decomposable
...A further simplification of the measurement function may be achieved by requiring a special kind of non interaction of the components which has become known as additive independence
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P where F is simply the addition function
...R,P >R,P <>[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P [[Phi]]1 R [[Phi]]2 P It can be shown that starting at the other end given an additively independent representation the properties defined in 1 and 3,and the Archimedean property are necessary
...Here the term [[Phi]]1 [[Phi]]2 is referred to as the interaction term,its absence accounts for the non interaction in the previous condition
...We are now in a position to state the main representation theorem
...Theorem Suppose <R x P,>>is an additive conjoint structure,then there exist functions,[[Phi]]1 from R,and [[Phi]]2 from P into the real numbers such that,for all R,R [[propersubset]]R and P,P [[propersubset]]P:R,P >R,P <>[[Phi]]1 R [[Phi]]2 P >[[Phi]]1 R [[Phi]]2 P If [[Phi]]i []are two other functions with the same property,then there exist constants [[Theta]]>0,[[gamma]]1,and [[gamma]]2 such that [[Phi]]1 [][[Theta]][[Phi]]1 [[gamma]]1 [[Phi]]2 [][[Theta]][[Phi]]2 [[gamma]]2 The proof of this theorem may be found in Krantz et al
...Let us stop and take stock of this situation
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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
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conjoint structure
...The analysis is not limited to the two factors precision and recall,it could equally well be carried out for say the pair fallout and recall
...Presentation of experimental results In my discussion of micro,macro evaluation,and expected search length,various ways of averaging the effectiveness measure of the set of queries arose in a natural way
...In this section the discussion will be restricted to single number measures such as a normalised symmetric difference,normalised recall,etc
...The measurements we have therefore are Za Q 1,Za Q 2,... |
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