Extending the Bradley–Terry model for paired comparisons to accommodate weights

Abstract

In the method of paired comparisons (PCs), treatments are compared on the basis of qualitative characteristics they possess, in the light of their sensory evaluations made by judges. However, there may emerge the situations where in addition to qualitative merits/worths, judges may assign quantitative weights to reflect/specify the relative importance of the treatments. In this study, an attempt is made to reconcile the qualitative and the quantitative PCs through assigning quantitative weights to treatments having qualitative merits using/extending the Bradley-Terry (BT) model. Behaviors of the existing BT model and the proposed weighted BT model are studied through the test of goodness-of-fit. Experimental and simulated data sets are used for illustration.

Bayesian Analysis of the Weibull Paired Comparison Model Using Numerical Approximation

“…More than 500 consumers' preference datasets were taken for analysis of the model. Abbas and Aslam [4] analyzed various factors which generally influence the judges' opinions about the objects using the mixture model for the preference data. Probabilistic features of the mixture distribution besides inferential and computational problems arising out of maximum likelihood estimation are discussed.…”

Eliciting hyperparameters of prior distributions for the parameters of paired comparison models

“…/* 'C' codes to elicit hyperparameters of Dirichlet prior for Chi-square model */ # include <stdio.h> # include <math.h> # include <conio.h> # define pi 3.141592653589793 void main() { inti,j; double p_value,t1,t2,t3,t4,t5, h1,h2,h3,h4,h5,p12,p13,p14, p15,p23,p24,p25,p34,p35,p45,p21,p31,p41,p51,p32,p42,p52,p43,p53,p54,ord, pp12,pp13,pp14,pp15,pp23,pp24,pp25,pp34,pp35,pp45,pp21,pp31,pp41,pp51,pp32, pp42,pp52,pp43,pp53,pp54,lf,prior,post,integ,integc,e [5][5],chiold=50.50,chi=0.0, chi1,pij,est [5],obs [5][5]={0,15,12,10,15,4,0,3,9,6,6,7,0,6,6,4,8,11,0,3,9,7, 10,6,0},integt1,integt2,integt3,integt4,integt5,dl=0.05; clrscr(); printf("Start of Program..."); for (h1=0.01;h1<=1.0-dl;h1+=dl) for (h2=0.01;h2<=1.0-h1-dl;h2+=dl) for (h3=0.01;h3<=1.0-h1-h2-dl;h3+=dl) for (h4=0.01;h4<=1.0-h1-h2-h3-dl;h4+=dl) { h5=1.0-h1-h2-h3-h4; // Finding the Normalizing Constant integ=0.0; integt1=0.0; integt2=0.0; integt3=0.0; integt4=0.0; for (t1=0.01;t1<=1.0-dl;t1+=dl) for (t2=0.01;t2<=1.0-t1-dl;t2+=dl) for (t3=0.01;t3<=1.0-t1-t2-dl;t3+=dl) for (t4=0.01;t4<=1.0-t1-t2-t3-dl;t4+=dl) { t5=1.0-t1-t2-t3-t4; p12=t1/(t1+t2); p13=t1/(t1+t3); p14=t1/(t1+t4); p15=t1/(t1+t5); p23=t2/(t2+t3); p24=t2/(t2+t4); p25=t2/(t2+t5); p34=t3/(t3+t4); p35=t3/(t3+t5); p45=t4/(t4+t5); lf=pow(p12,15)*pow(1.0-p12,4)*pow(p13,12)*pow(1.0-p13,6)*pow(p14,10)*pow(1.0-p14, 4)* pow(p15,15)*pow(1.0-p15,9)*pow(p23, 3)*pow(1.0-p23,7)*pow(p24,9)*pow(1.0-p24, 8)*pow(p25, 6)*pow(1.0-p25,7)*pow(p34, 6)*pow(1.0-p34,11)*pow(p35, 6)*pow(1.0-p35,10)* pow(p45, 3)*pow(1.0-p45,6); prior=pow(t1,h1 -1.0)*pow(t2,h2-1.0)*pow(t3,h3-1.0)*pow(t4, h4-1.0)* pow(t5,h5-1.0); post=lf*prior; integ+=post*pow(dl,4); integt1+=post*t1*pow(dl,4); integt2+=post*t2*pow(dl,4); integt3+=post*t3*pow(dl,4);integt4+=post*t4*pow(dl,4); } integt1=integt1/integ; integt2=integt2/integ; integt3=integt3/integ; integt4=integt4/integ; integt5=1.0-integt1-integt2-integt3-integt4; est[0]=integt1; est [1] (chi1>chiold) continue; // To proceed with the better chi-square value // Calculating p-value p_value = 0.0; for (t1=0.0;t1<=chi1;t1+=0.005) p_value += 0.005*t1*t1*exp(-t1/2.0)/(2.0*8.0); // The Chi-square pdf clrscr(); printf("\nThe pref. probs\t\tObs.…”