External Preference Mapping

External Preference Mapping

Purpose

External preference mapping (EPM) is a way of combining sensory, or analytical data on a set of products with consumer liking measured on the same products.  The aim is to form a product map based solely on the sensory/analytical data, and then to predict the consumer liking of products at all points on the map.

Data Format

The data should be supplied in a format similar to the example data file on breads <Link to UK_Breads2.xlsx>.  The main “dataset” tab of the data file needs to have a variable giving the raw liking scores for each judge and each product.  If your data set contains other consumer variables, then make sure that these are excluded at the “Visualization and Selection” screen, otherwise an error will occur.  The external data should be supplied as supplementary columns of variables on the “products” tab which should appear in the 3rd column onwards following the usual product codes and product labels in the first two columns.  Typically, these supplementary variables would be mean scores from a sensory panel, or analytical measurements made in the laboratory from which the external axes will be derived by PCA.  Alternatively, you may provide just 2 supplementary columns with the product coordinates of the map that you want to fit the consumers to. In this latter case, it only makes sense to provide (X, Y) product coordinates that are mean centered and uncorrelated with each other.

Background

EPM is a hybrid method with two distinct steps. The first is to obtain the ‘external axes’, often these are the 1st two principal component dimensions of a PCA computed on sensory panel mean scores.  This constitutes the product map to which consumer liking data is projected using response surface regression analysis.  A separate regression model is computed for each consumer, where their liking scores (Z), are regressed onto the product coordinates (X & Y).  The software offers a choice of the regression model used which may be one of the following options:
      
      Vector :  Z = aX + bY + c
      Elliptic :  Z = aX + bY + cX2 + dY2 + e
      Quadratic :  Z = aX + bY + cX2 + dY2 + eXY + f

In the early days of the EPM method users would try to interpret the estimated parameters in one of the models above, however this is a complex task as there is a separate set of estimates for each consumer and for the 2 models with curvature there is the possibility of either maxima or minima.  These days it is more usual to summarise all these models with a contour plot, Danzart (1998) and Danzart et al. (2004).  The product map is divided into a n x n grid, and for each cell in the grid the number of consumers whose models exceed a threshold liking value is counted. A coloured contour plot is then used to highlight areas where these counts are highest, as these are the same areas where the greatest number of consumers are predicted to be accepting of a product.

The main graphical outputs are shown below.  The first two plots are the PCA product scores and variables loadings plots for the first two dimensions; these are only generated if the user chooses the ‘pre-transformation’=PCA option.  The third plot shows the external preference map itself, which clearly shows that ‘Daily Delight’ bread is well placed in the darkest red region, while the ‘Sliced Right’ bread is poorly liked as it is in the darkest blue region.  All three plots were generated using the example breads data set, using a correlation PCA on all attributes, the elliptical model option, and a grid size of 100. 






Options

  1. Sensory Attributes – using the drop down list of variables names, either (a) select all of the variables from which you want to derive the product map by PCA, or (b) select just two variables that define the (X,Y) product coordinates on the external axes that you have pre-computed.
  2. Pre-transformation – If you wish the software to compute a PCA, that is option (a) above, then select ‘PCA’, or if you want no pre-transformation, that is option (b) above then select ‘None’.
  3. Type of PCA  – The PCA may be performed on either the correlation or covariance matrix.  Only choose the covariance matrix if all your supplementary variables have been measured on the same scale, and you wish to give more importance to variables that use the full range of the scale.
  4. Threshold – the threshold determines how many consumers are counted at each point on the product map, it defaults to zero which is defined as the average consumer score, since the liking scores for a consumer are mean centred prior to the response surface regression.  Useful values are probably in the range 0 to 3.  So, a threshold of 1 would only count consumers who are predicted to give a score that is 1 point higher than their own average score.  See the background section for more details on thresholding.
  5. Regression model – The type of response surface regression model that you would like to fit. See the background section for more details. 
  6. Resolution – The size of the n x n grid for the contour plot.  The contour surface plot that is created will be made up of n2 cells which are all independently coloured.
  7. ​Number of decimals for values – The number of decimals to show in all numeric output.

Results and Interpretation

  1. Products tab show results relating to the observations or products in the PCA.
    1. Coord shows the principal component scores on each dimension.
    2. Cos2 shows a table of squared cosines, the sum of the squared cosines in each row is 1, so they can be interpreted as the amount that each product is attributable to each dimension.
    3. Contrib shows a table of contributions, these sum to 100 for each dimension and therefore summarise the relative importance of products on each dimension.
    4. Graph shows a plot of the principal component scores on the first two dimensions.
  2. Attributes tab show results relating to the variables in the PCA.
    1. Coord shows the principal component loadings on each dimension.
    2. Cor shows a table of correlation coefficients between each variable and each PC dimension.
    3. Cos2 shows a table of squared cosines, the sum of the squared cosines in each row is 1, so they can be interpreted as the amount that each variable is attributable to each dimension.
    4. Contrib shows a table of contributions, these sum to 100 for each dimension and therefore summarise the relative importance of the consumers on each dimension.
    5. Graph shows a plot of the principal component loadings on the first two dimensions. The plot is allows a visual summary of the correlations between variables.
  3. Surface Plot tab shows a coloured contour plot summarising the external preference map. This superimposes contours of equal consumer liking on top of the external product map (this is the same product map that can be seen under Products/Graph).  The darkest red area corresponds to the region on the map where the largest number of consumers are predicted to scores above the threshold.  The darkest blue region on the map corresponds to the region on the map where the least number of consumers are predicted to score above the threshold.  
  4. Individual Models tab shows a summary of the individual regression analyses (see Background section above).  The table gives the r-squared and adjusted r-squared values for each consumer which both summarise the degree of model fit.

Technical Information

  1. The R package Factominer is used.

References

  1. Danzart, M. (1998) Quadratic model in preference mapping. 4th Sensometrics meeting, Copenhagen.
  2. Danzart, M., Sieffermann, J.-M. & Delarue, J. (2004) New developments in preference mapping techniques: Finding out a consumer optimal product, its sensory profile and the key sensory attributes. 7th Sensometrics Meeting, Davis, CA, USA.
  3. Lawless, H.T. and Heymann, H. (2010).  Sensory Evaluation of Food – Principles and Practices.  Springer.
  4. McEwan, J.A. (1996).  Preference Mapping for Product Optimization.  In Naes, T. and Risvik, E. (Eds).  Multivariate Analysis of Data in Sensory Science.

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