# R-index Analysis

## Purpose

To analyse the results of an R-index test.

## Data Format

1. R_index_rank_withMD.xlsx

## Background

The R-index applies signal detection theory as an alternative approach to discrimination testing.  It is often used where there are a large number of products requiring a ‘yes/no’ response relative to a control or reference product.

It is also applicable where the ‘yes/no’ response is broken down into ‘degrees of certainty’ – e.g., ‘different, I am sure’; ‘different, I am not sure’; ‘different, I do not know and guess’; ‘the same, I do not know and guess’; ‘the same, I am not sure’; ‘the same, I am sure’.

In the above scenario, the R-index results from converting the rating data to a measure related to the area under the ROC (receiver operation characteristics) curve.

## Options

1. Reference Product: Select which product should be used as a reference.
2. Type of test: Similarity or Difference test
3. Difference: If the test is a similarity test, the threshold that will be considered non-negligible. That is, H0 is that the difference is greater than this threshold.
4. Significance level: Type-I error
5. Order of the Scale: Increase or Decrease. Should an increase or decrease in scores in the data set result in an increase in the certainty of difference. By default, a score of 1 means the assessor is sure the samples are the same, and a score at the top of the scale means the assessor is sure the samples are different.
6. Number of Decimals for Values: Required number of decimals for values given in the results.
7. Number of Decimals for P-Values: Required number of decimals for any p-values given in the results.

## Results and Interpretation

### Frequency table

This is a contingency table gives a summary of how many responses given for each of the scale categories. The products form the rows and there is a column for each level of the scale.
It is a useful visual look see.  It is also the basis for calculating the R-index using the traditional psychophysics principles from Thurstone.

### R-index table

The second tab is the R-Index table showing the estimated R-Index for each product. The following values are given:

1. N. The number of times each sample was evaluated.
2. An estimate of the R-index. R-index is expressed as a proportion. It is equivalent to the probability of correctly choosing one of two samples in a paired comparison, but is calculated from theoretical paired comparisons between each possible signal/noise pairing.
3. The critical value Z (at the significance threshold).
4. The p-value to conclude whether each product was significantly different from the reference product.

### Interpretation

At a basic interpretation level, the p-values are used to establish if each product is statistically different from (or similar to) the reference product.
The key point to note with the R-index is that it is based on Thurstonian scaling and the law of comparative judgement.
R index values can be analysed using parametric statistics e.g. ANOVA (as can d’).
Significance of R-index can be determined from critical value tables for ‘n’ responses.
‘Equivalent’ d’ can be determined for r-index values from published tables.

## Technical Information

1. The R-index is calculated manually.
1. Bi & O’Mahony (2007) Updated and extended table for testing the Significance of the r-index Journal of Sensory Studies 22 713–720.
2. Lawless & Heymann (1998) Sensory Evaluation of Foods Chapter 5 Discrimination Theories and  advanced topics 140-159.
3. Macmillan N & Creelman C D (2004) Detection Theory A User’s Guide.  Lawrence Arlbaum Associates Inc.
4. O’Mahony M (1992) Understanding Discrimination Tests: A user friendly treatment of response bias, rating and ranking R-index tests and their relationship to signal detection. Journal of Sensory Studies 7 1-47.
5. O’Mahony M & Rousseau B (2002) Discrimination testing: a few ideas, old and new. Food Quality and Preference 14 157-164.
6. Rousseau B (2011) Measuring product similarities: Are two indices, R-Index and d′, interchangeable? IFT Annual Meeting, New Orleans, USA
7. Brown J (1974) Recognition assessment by rating and ranking. Brit J Psycho 65 13-22.
8. Lee H-S, van Hout D Quantification of Sensory & Food quality: The R-Index Analysis Journal of Food Science 2009.
9. Bi, J. (2006).  Statistical Analyses for R-Index.  Journal of Sensory Studies, 21, 584-600.
10. Lawless, H.T. and Heymann, H. (2010).  Sensory Evaluation of Food – Principles and Practices.  Springer.
11. O’Mahony, M. (1986).  Sensory Evaluation of Food – Statistical Methods and Procedures.  CRC Press.