R-index is developed to measure the area under an empirical receiver operation characteristics (ROC) curve in signal detection theory

^{3}^{,6}. R-index defines the degree of difference between two samples in term of the probability of discriminating paired samples. An R-index value of 1.0 indicates paired samples are easily distinguishable while an R-index value of 0.5 shows that paired stimuli are extremely difficult to discriminate. For those samples with R-index between 0.5 and 1.0, a larger R-index reflects better discrimination

^{4}.

R-index can be collected and calculated by rating or ranking tasks, where a panelist is presented with a sample and determines if the sample is the signal or noise; in the rating tasks, the panelist responds by selecting one category on a scale (e.g. “signal sure”, “signal unsure”, “noise unsure” or “noise sure”). In the ranking task, the stimuli are scored against each other simultaneously and the samples should be ranked by panelist based on the strength of signal

^{3}. In practice, the sample with lower attribute intensity is normally designed as the “noise” sample, while the sample with higher intensity is designated as “signal” sample

^{4}.

The validation and advantage of R-index are greatly facilitated by the fact that the area under the ROC curve, when the trapezoidal rule is applied, is closely related to the

Mann-Whitney U statistics, a version of the Wilcoxon statistic

^{1}. The relationship of R-index and Mann-Whitney-Wilcoxon (MWW) make various statistical tools useful in R-index. According to Bi

^{1}, the application of R-index in the field of consumer and sensory research should be encouraged for the following reasons:

First, the R-index is one of most powerful non-parametric statistic. R-index was found comparable to the t-test^{2}. When the distribution of data is exactly normal, t-test is slightly more powerful than R-index while for those data close to normal distribution, there is no measurable difference in power between these two tests. In practice, researchers normally don’t have a priori knowledge about the distribution of data and sometime the assumption of normal distribution doesn’t hold; moreover, The hedonic ratings data from consumer tests are frequently skewed and bimodal than normal^{7}.Under these conditions, analysis by R-index is more reasonable than by t-test^{1}.

Second, R-index statistic is distribution free and more robust. It can work on ordinal measurements rather than interval scales. Strictly speaking, ratings in consumer science are typically considered as ordinal rather than interval measurements

^{5}. Therefore, t-test/ANOVA is to some extent inappropriate for some ratings obtained in consumer and sensory research.

Third, R-index is a measurement which is more direct, intuitive and interpretable since it shows the estimate of a probability of two products being distinguishable; moreover, R-index is not be affected by the decision criteria or number of categories in rating data.

However, some disadvantage of R-index should be mentioned. The test practice and data collection by R-index are time-consuming since more samples will be served; In addition, results analyzed by R-index do not show the extent or direction of the difference between two stimuli.

## References

^{1} Bi J. 2006. Statistical analyses for R-index. Journal of Sensory Studies 21, 584–600.

^{2 }Bickel PJ., Doksum KA. 1977. Mathematical Statistics: Basic Ideas and Selected Topics. Pp.350-353. Holden-Day, Inc., San Francisco, CA.

^{3} Brown J. 1974. Recognition assessed by rating and ranking. Br. J. Psychol.65, 13–22.

^{4 }Cliff MA., O’mahony M., Fukumoto L., King MC. 2000. Development of a ‘bipolar’ R-index. Journal of Sensory Studies. 15, 219-229.

^{5 }Cliff N., Keats, J.A. 2003. Ordinal Measurement in the Behavioral Sciences. Mahwah, NJ. Lawrence Erlbaum Associates, Publishers.

^{6 }Lawless HT., Heymann H. 1998. Sensory Evaluation of Food: principles and practices. New York, NY. Chapman&Hall; Press.

^{7} O’Mahony M. 1982. Some assumptions and difficulties with common statistics for sensory analysis. Food Technol. 32, 75–82.