Response Surface Fitting Using a Generalization of the Bradley-Terry Paired Comparison Model

Abstract

SUMMARY A paired comparison technique is presented for fitting response surfaces. This is especially useful when subjective responses are involved, where it is often difficult to justify the basic assumptions of the classical procedure. This report discusses aspects of the estimation of parameters and their properties, tests of relevant hypotheses and the selection of experimental designs. The method is applied to an example in food testing. PAIRED comparison techniques have been used for many years for the subjective comparison of several treatments (David, 1963), and extensions are available for the comparison of factorial treatment combinations. This report presents a further extension of the paired comparison method, the analogue of multiple regression analysis. Although it is possible to formulate a general paired comparison regression model this report will concentrate on the analogue of multiple linear regression mainly because of its application to response surface fitting. In the field of sensory assessment, it is frequently required to determine the effect of varying product composition. The classical response surface approach is useful here, despite many of its basic assumptions being invalid in the sensory situation, but occasionally it becomes virtually impossible to implement the classical method, particularly when carry-over and sensory organ fatigue effects are present. The degree to which these affect the results depends on the sense used, taste and smell being usually the more prone. Classical response surface designs, even with blocking, frequently require more units than may be reliably assessed in one sensory assessment session. Further, the assessors may not, due to the above effects-and others-be able reliably to score their impressions on a continuous scale. If any of these factors is present, the method of paired comparisons becomes extremely useful. A number of models have been proposed for the analysis of paired comparison data in the analogue of one-way analysis of variance. The specific analogue one-way ANOVA paired comparison model which this paper adapts to analogue multiple regression is that due to Rao and Kupper (1967), itself a generalization of the Bradley-Terry model (1952). A parallel generalization of the Bradley-Terry model, due to Davidson (1970), could also be extended for the present purpose in an exactly similar manner.