Preference mapping

Extended preference arrow for a subject. The popularity (blue numbers) of the sixth product results from an orthogonal projection onto the extension of the arrow.

Preference mapping is a statistical method from the field of sensometry , i. H. the application and further development of statistical methods for measuring sensory impressions.

Experts describe products objectively using a large number of sensory variables. These variables are reduced to two essential variables with the help of dimension-reducing statistical methods, which serve as the basis for creating the preference map. The products are then subjectively assessed by untrained test persons. Linear models are created for the test persons, which model their respective preferences. The vector model, circular, elliptical and rotated models are possible as linear models:

${\displaystyle y=\alpha +\beta _{1}x_{1}+\beta _{2}x_{2}}$ (Vector model)

${\displaystyle y=\alpha +\beta _{1}x_{1}+\beta _{2}x_{2}+\delta x_{1}^{2}+\delta x_{2}^{2}}$ (Circular model)

${\displaystyle y=\alpha +\beta _{1}x_{1}+\beta _{2}x_{2}+\delta _{1}x_{1}^{2}+\delta _{2}x_{2}^{2}}$ (Elliptical model)

${\displaystyle y=\alpha +\beta _{1}x_{1}+\beta _{2}x_{2}+\delta _{1}x_{1}^{2}+\delta _{2}x_{2}^{2}+\gamma _{1,2}x_{1}x_{2}}$ (Rotated model)

Here, the target variable is a measure of the popularity of the product, and the two dimensional systems and new variables , , , , and the regression coefficients. The listed linear models represent a generalization from top to bottom. In the preference map, the vector model is represented by a preference arrow and the three other models by an ideal point or an anti-ideal point, if one exists. In addition to the (anti) ideal points or preference arrows of the consumers, the products are also drawn on the preference map. By combining this information, it is possible to see which product features are preferred by the majority of consumers, which ideally enables products that are popular for the target group to be designed. ${\displaystyle y}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle \alpha }$ ${\displaystyle \beta _{1}}$ ${\displaystyle \beta _{2}}$ ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$ ${\displaystyle \gamma _{1,2}}$

Choice of linear models

Contour diagram of a rotated model with an ideal point at (−2.1). The gradient points in the direction of increasing popularity.

Following the dimensional reduction, one of the above-mentioned linear models is adapted for each subject. There are various options for choosing the linear models.

A simple variant is to select the same model type for all test subjects. In this way, the preferences of all test subjects can be displayed and compared on the same preference map. For example, if you choose the circular model, you get the preference map with the ideal and anti-ideal points. If the vector model is selected, the preferences can be indicated by arrows. In this case, it is possible to draw in the length of the arrows proportionally to the corrected coefficient of determination in order to visualize the quality measure at the same time.

Another possibility is to make the choice of the model type depending on statistical tests. For example, one could opt for the circular model if the Wald test rejects the null hypothesis and otherwise for the vector model. Another possibility is to choose the model with the larger corrected coefficient of determination. ${\displaystyle \delta =0}$

It does not always make sense to show the preferences of each subject in the preference map. If the corrected coefficient of determination of a participant's model falls below a predetermined limit, his or her ideal point or preference arrow is removed from the preference map. Ideal points that lie far outside the sample area are also removed.

example

In the correlation circle, the correlations of 13 variables with the two main components are shown by arrows.

External preference map in the circular model with ideal points (green) and anti-ideal points (red).

In the SensoMineR R package , 16 cocktails with different proportions of the fruits orange, banana, mango and lemon were examined. These were described by 12 experts using 13 characteristics. These variables are color intensity, odor intensity, orange odor, banana odor, mango odor, lemon odor, taste intensity, sweetness, acidity, bitterness, aftertaste, pulp content and thickness. These were then reduced to two variables with the help of the principal component analysis, which explain more than 80% of the variance. The first main component differentiates thick, sweet cocktails with a distinct banana aroma on the left side from sour and bitter cocktails on the right side (see correlation circle). In the circular model, most of the ideal points are in the left area, so sweet cocktails with a high proportion of bananas are preferred. In contrast, most of the anti-ideal points are in the right area, since most of the test subjects did not like sour and bitter cocktails.

Individual evidence

1. ^ McEwan, Jean A .: Preference Mapping for Product Optimization. In: Multivariate Analysis of Data in Sensory Science 16 (1996), 71-102. https://books.google.de/books?id=xlpzm5lsUMMC . ISBN 9780080537160
2. ↑ Husson, Francois; Le, Sebastien; Cadoret, Marine: SensoMineR: Sensory data analysis with R. http://CRAN.R-project.org/package=SensoMineR . Version: 2014. - R package version 1.20