A logistic regression-based pairwise comparison method to aggregate preferences

Abstract

In a group decision making process, several individuals or a committee have the responsibility to choose the best alternative from a set. The problem addressed in this paper is how to aggregate personal preferences to arrive at an optimal group decision. New technologies allow individuals that may seldom or never meet to make group decisions. This paper proposes a methodology to obtain the group preference ordering in two steps. Firstly, each individual studies the problem isolated, and then, in a possibly virtual meeting, the group must agree on the preferences on some pairs of alternatives. Then, the group criterion is achieved by using a logistic regression model within the pairwise comparison framework proposed here. Properties of the procedure are studied and two illustrative examples are presented.

Appendix. Logistic regression model

This appendix introduces the notation and uses of the logistic regression model. Logistic regression is part of a category of statistical models called Generalized Linear Models (GLM). Excellent treatments of GLMs are presented, for example, in Agresti (1996) and Ryan (1997).

Binary logistic regression is a variation of ordinary regression, useful when the dependent variable Y is restricted to two values, which usually represent the occurrence (coded as 1) or non-occurrence (coded as 0) of some outcome event. It produces a formula that predicts the probability of the event occurrence π as a function of independent variables X ₁,X ₂,...,X _n . The independent or predictor variables can take any form (continuous, discrete, dichotomous,...) and no assumption is necessary about their distribution. The expression for π is given by

$$ \pi=\frac{\hbox{e}^{\beta_1x_1+\beta_2x_2+\cdots+\beta_nx_n}}{1+\hbox{e}^{\beta_1x_1+\beta_2x_2+\cdots+\beta_nx_n}}. $$