Department of Statistics, University of California, Riverside, CA 92521

1. 2500 2000 1500 Frequency 1000 500 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Q Bradley-Terry Model Analysis of Cat Food Recipes Hongjie Deng1, Daniel R. Jeske2 and Ted Younglove3 • Department of Statistics, University of California, Riverside, CA 92521 • 1Graduate Student, 2Faculty and Director of Collaboratory, 3Manager of Collaboratory Lana Cinnamon Wheaties Wheaties Introduction Model Goodness Of Fit Multiple Comparison Procedure To identify which food recipes are different with respect to cat preference, the procedure uses an algorithm to generate hypothetical tables of data under the null hypothesis . Test The Del Monte Pet Products Division of Del Monte Foods conducted palatability studies of dry cat food, wet cat food, and cat treats using paired comparison consumption tests. The Statistical Consulting Collaboratory at the University of California, Riverside was consulted to improve the analysis of the paired comparison experiments, focusing initially on the experiments that used dry cat food. Our goal was to apply Bradley-Terry modeling and analysis techniques to the experimental data. In particular, we wanted to estimate a quality score for each food recipe, test whether the scores were significantly different, and explore the power of alternative paired comparison designs. Ho: Bradley-Terry model fits the data Ha: Bradley-Terry model does not fit the data Test statistic: = -2() where is the saturated likelihood function, reduced by the Bradley-Terry link function. Test Result: p-value = 0.1093. Do not reject Ho at = 0.05. Algorithm Randomly generate a table of data 104 times (see below) Calculate ( i =1,…,t ) for each table How Well The Model Fit Obtain for each table Experimental Design For each (i, j) pair, calculate Monte-Carlo p-value = ( # of Q > ) /104 The experiment was conducted using a colony of 300 cats, male and female, of various breeds and ages. Each cat in a panels of 30 randomly selected cats was given two different bowls of food on each of two days. On the first day food A was placed to the left and food B was placed to the right. On the second day, the left-right orientation was reversed. The relative amount of each food, A and B, that the cats consumed over the two days was used to indicate which food they preferred. In this poster, we show how to analyze the data and select the food recipe that is most attractive to the cats. Compare the Monte-Carlo p-value with Observed Frequencies and ( Expected Frequencies ) Power Analysis Conclusion How To Randomly Generate A Table Of Data Two Alternative Designs D1: 10 panels comparing all pairs of recipes with 30 cats each D2: 4 panels comparing (A,B) , (B,C) , (C,D) , (D,E), each with 75 cats D2 has the minimum number of comparisons needed to estimate all the ratings and is motivated by being a simpler experiment to manage. Power Comparison Ho: v1 = v2 = v3 = v4 = v5 (i.e., no difference in recipes) Ha: not Ho Power levels for each design of a 5% test of Ho using 1000 simulated data sets are presented below. Data Histogram Of Q Note: Cellij=number of cats who prefer food i over food j Bradley-Terry Model Suppose there are t treatments in an experiment involving paired comparisons. Each pair of treatments is compared by k different judges. Define Pr ( treatment i is preferred over treatment j ) = . Define rijk = rank of the i-th treatment when compared with j-th treatment by judge k. The saturated likelihood function is The Bradley-Terry link function is , where vi= true rating of treatment i. Rank treatments based on . Pr (Q>0.6512)=0.05 ComparisonOf Standard Deviation Of Contrasts Test Results Food E Food A Food D Food C Food B Note: Foodsconnected by a line are not significantly different at =0.05 Estimation Of True Ratings vi Estimates Of Maximum Likelihood Estimates of the true ratings vi were obtained using the R software package and are presented below. Based on 1000 simulations under Ho. Results are relatively invariant to what is assumed for the true vi values . Conclusion Although D2 is simpler to manage, it has less power than D1. The loss of information by using less panels is not compensated for by using more cats in each panel. Contrasts under D1 have equal precision while for D2 they do not. Special Thanks To: Javier Suarez and Hua Yu of the UCR Statistical Consulting Collaboratory. Graduate Students of the Fall 2005 offering of STAT 293: Yingtao Bi, Mike Huang, Steward Huang, Sungsu Kim, Scott Lesch, Rupam Pal, Jose Sanchez, Jason Wilson, Rui Xiao, Karen Huaying Xu, and Qi Zhang.