Regression Methods for Pairwise Comparisons Data with Applications to Forestry

We present a regression formulation of the analytic hierarchy process (AHP) introduced by Saaty (1977) as a method of deriving a ratio scale of preferences (or priorities) concerning a set of m entities or attributes. This method involves a quantification of all m(m-1)/2 pairwise comparisons between the entities. The ratio scale is derived using an eigenvalue calculation on a matrix formed from the quantified comparisons. Saaty’s approach causes computational difficulties when m is large. This can be avoided in regression approach, which allows a smaller amount of comparisons. The main difficulty in the regression approach is with the creation of a nonstandard design matrix. We have developed a Mathematica package AHP.m for this purpose. For ease of language we will speak of judges evaluating photographs at different locations. The same setting appears in many other contexts. The “photographs” could be replaced by cars and the “judges” by buyers, for example. Suppose we have judges k K = 1,..., whose task is to evaluate photographs taken at locations i I = 1,..., . A treatment j J = 1,..., is applied to each photograph to reflect possible future use of the landscape depicted. In practice this is done by a digital manipulation of a computerized image. We will speak of photograph ( , ) i j for short. Suppose the value of ( , ) i j for k is of the loglinear form v i j k L i j L j k ( , , ) exp( ( , ) ( , )) = + + μ 1 2 , where μ is an intercept term, L i j 1( , ) measures a baseline value of ( , ) i j and L j k 2 ( , ) shows how the background characteristics of k influence his/her evaluation of j relative to the baseline. The relative value of treatment j, in location i, is given by