Michael Meyners

On the Triangle Test with Replications

Abstract We consider the triangle test with replications. A commonly used test statistic for this situation is the sum of all correct assessments, summed over all assessors. Several authors argue that the binomial distribution cannot be used to analyse this kind of data. Brockhoff and Schlich [ Brockhoff, P.B., & Schlich, P. (1998). Handling replications in discriminations tests. Food Quality and Preference, 9, 303–312. ] propose an alternative model for the triangular test with replicates, where the assessors have different probabilities to correctly identify the odd sample even if the products are identical. Although we agree that assessors will have different probabilities of correct assessment if there are true differences, we do not think that Brockhoff and Schlichs model makes sense under the null hypothesis of equality of treatments. We show that all assessments are independent and have success probability 1 3 , if the null hypothesis is true and the experiment is properly randomized and properly carried out. This implies that the sum of all correct assessments is binomial with parameter p= 1 3 . Therefore the usual test based on this sum and the critical values of the binomial distribution is a level α test for the null hypothesis of equality of the products, even if there are replications.

Comparing generalized procrustes analysis and statis

We consider a model for sensory profiling data including translation, rotation and scaling. We compare two methods to calculate an overall consensus from several data matrices: GPA and STATIS. These methods are briefly illustrated and explained under our model. A series of simulations to compare their performance has been carried out. We found significant differences in performance depending on the variance of random errors and on the dimensionality of the true underlying consensus. Therefore we investigated on the dimensionality of the calculated group averages. We found both methods to give too many dimensions compared to the true consensus. This finding is supported by some theoretical considerations. Finally we propose a combined approach which takes advantage of both methods and which gave better results in the simulations.

Permutation tests: are there differences in product liking?

The target of our considerations is whether or not we can find significant differences between subgroups of consumers with respect to given hedonic variables. For this purpose a STATIS-consensus is computed for each group and the dissimilarities between groups are judged with the help of the RV-coefficient. Since the distribution of this coefficient is unknown and we do not make any assumptions on the distribution of the error terms, a permutation test is performed. This provides a simple possibility to test for significance of the dissimilarities in question. Some pre-treatment of the data is necessary to perform this statistical test. Afterwards subgroups according to the two sets of consumer tests, the different geographies and some of the classification variables within consumer test 1 are considered. When significant dissimilarities are found, a graphical representation of the respective consensuses is provided to interpret the differences.

On the number of perceivers in a triangle test with replications

In discrimination tests, two different questions usually arise: First of all, we are interested in deciding whether or not there are product differences at all that might be perceived by the assessors. However, often this is not our most important concern, since the main question is whether or not the consumers (in contradiction to e. g. a trained panel) might perceive the difference and, if so, how many of them are supposed to do so. While the first question has been addressed frequently in recent times, the known models for estimating the proportion of perceivers use strong conditions, e. g. that the assessors taste the difference always or never. We propose a more general model that allows the assessors to perceive differences once in a while and derive a method that takes this assumption into account. Several examples show that the estimates for the proportion of interest are quite reasonable.

Relating principal component analysis on merged data sets to a regression approach

A method for calculating a consensus of several data matrices on the same samples using a PCA is based on a mathematical background. We propose a model to describe the data which might be obtained e. g. by means of a free choice profiling or a fixed vocabulary in a sensory profiling framework. A regression approach for this model leads to a Principal Component Analysis on Merged Data sets (PCAMD), which provides a simple method to calculate a consensus from the data. Since we use less restrictions on the variables under investigation, the model is claimed to be more general than the model induced by GPA respectively STATIS, which are widely accepted methods to analyse this kind of data. Furthermore, the PCAMD provides also additional opportunities to compare and interpret assessor performances with respect to the variables of the calculated consensus. An example from a sensory profiling study of cider is provided to illustrate these possibilities.