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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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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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one unit of precision for an increase of one unit of recall,but will not sacrifice another unit of precision for a further unit increase in recall,i
...R 1,P 1 >R,P but R 1,P >R 2,P 1 We conclude that the interval between R 1 and R exceeds the interval between P and P 1 whereas the interval between R 1 and R 2 is smaller
...Finally,we incorporate into our measurement procedure the fact that users may attach different relative importance to precision and recall
...Definition 6
...Can we find a function satisfying all these conditions?If so,can we also interpret it in an intuitively simple way?The answer to both these questions is yes
...The scale functions are therefore,[[Phi]]1 P [[alpha]]1 P,and [[Phi]]2 R 1 [[alpha]]1 R
...We now have the effectiveness measure
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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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the different contributions made to the measure by the different cells
...Discrimination gain hypothesis In the derivation above I have made the assumption of independence or dependence in a straightforward way
...P xi,xj P xi,xj w 1 P w 1 P xi,xi w 2 P w 2 P xi P xj [P xi w 1 P w 1 P xi,w 2 P w 2][P xj w 1 P w 1 P xj,w 2 P w 2]If we assume conditional independence on both w 1 and w 2 then P xi,xj P xi,w 1 P xj,w 1 P w 1 P xi w 2 P xj w 2 P w 2 For unconditional independence as well,we must have P xi,xj P xi P xj This will only happen when P w 1 0 or P w 2 0,or P xi w 1 P xi w 2,or P xj w 1 P xj w 2,or in words,when at least one of the index terms is useless at discriminating relevant from non relevant documents
...Kendall and Stuart [26]define a partial correlation coefficient for any two distributions by |
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