PrefMFA, a solution taking the best of both internal and external preference mapping techniques

PrefMFA, a solution taking the best of both internal and external preference mapping techniques

Food Quality and Preference 30 (2013) 180–191 Contents lists available at SciVerse ScienceDirect Food Quality and Preference journal homepage: www.e...

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Food Quality and Preference 30 (2013) 180–191

Contents lists available at SciVerse ScienceDirect

Food Quality and Preference journal homepage: www.elsevier.com/locate/foodqual

PrefMFA, a solution taking the best of both internal and external preference mapping techniques Thierry Worch ⇑ QI Statistics Limited, Reading, Berkshire, United Kingdom

a r t i c l e

i n f o

Article history: Received 20 January 2013 Received in revised form 21 May 2013 Accepted 22 May 2013 Available online 1 June 2013 Keywords: Internal preference map External preference map Multiple Factor Analysis Hierarchical Multiple Factor Analysis Lg and RV coefficients

a b s t r a c t In product optimization, preference mapping techniques are widely used. Although their advantages and benefits are evident, these methods also show some limitations. For example, in external preference mapping (PrefMap), in general only two dimensions are used in the individual regression models, and there is no evidence that these dimensions are relevant for liking. Moreover, potentially important information which could explain the liking scores and which could be present on the third and higher dimensions of the sensory space is often not considered. For that reason, a new methodology, called PrefMFA, is proposed. This methodology based on Multiple Factor Analysis (MFA) is in between internal and external preference mapping since it takes the dimensions from the ‘‘common’’ product space between external (usually sensory) and the hedonic scores in the individual regressions. An extension to Hierarchical Multiple Factor Analysis (HMFA), when more than one external matrix is available, is also proposed. The PrefMFA methodology is applied to two datasets according to two different strategies of analyses, and the advantages of PrefMFA over PrefMap are shown. More precisely, the help for the interpretation as well as the various criteria proposed by MFA (such as the partial axes representation, the Lg, and the RV coefficients) help to better understand the strength of the underlying relationship between the external information and liking. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction In sensory science, common practice consists of asking assessors to rate the tested products on a list of predefined attributes. In order to decrease the error due to the variability (and subjectivity) between assessors, they are often trained (Meilgaard, Civille, & Carr, 2007). During these training sessions, the assessors are calibrated to use the scale in a similar way. This procedure is known as Quantitative Descriptive Analysis or QDAÒ (Stone, Sidel, Oliver, Woosley, & Singleton, 1974) and results in sensory product profiles. This information is useful for industries to understand how their products are perceived from a sensory point of view. Additionally, the same products are usually rated by consumers on overall liking using a Central Location Test (CLT). Finally, two types of information about the tested products are collected: the sensory profiles (from experts) and the liking ratings (from consumers). Although the information about the products gathered through these two independent tests is important, it takes its real power when the two blocks of data are combined. Many statistical methods have been developed to relate liking to sensory profiles such as Preference Mapping techniques ⇑ Tel.: +44 7557883091. E-mail address: [email protected] 0950-3293/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.foodqual.2013.05.009

(Danzart, 2009a; Greenhoff & MacFie, 1994; McEwan, 1996; Schlich & McEwan, 1992). There are two variants: internal preference mapping (MDPref; Carroll, 1972) and external preference mapping (PrefMap; Carroll, 1972). These two variants diverge in the point of view adopted (Jaeger, Wakeling, & MacFie, 2000; Van Kleef, Van Trijp, & Luning, 2006). In MDPref, the focus is on the liking scores: the product space obtained is based on the liking ratings of the consumers and the sensory attributes are regressed into this space in order to explain the differences in liking. In PrefMap, the focus is on the sensory profiles of the products: the product space obtained is based on the sensory differences between products. The first two sensory dimensions obtained by multivariate analysis (usually PCA) are extracted, and the liking scores of each consumer are regressed on them. Four different regression models can be used: linear, circular, elliptic or quadratic (Danzart, 2009b). This analysis results in the so-called surface plot (Mao & Danzart, 2008), which allows predicting the potentially most appreciated products within the sensory space for the group of consumers considered (Van Trijp, Punter, Mickartz, & Kruithof, 2007). This optimum product (often called ideal product) is then used as a reference to match in the product optimization. Although sensory profiles obtained from a trained panel are often used as a starting point for the product space, it is not restrictive.

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Indeed, since it has been shown that consumers can also profile products in a consistent way (Husson, Le Dien, & Pagès, 2001; Worch, Lê, & Punter, 2010), the sensory product space can also be obtained from consumers. This is for example the case in the Ideal Profile Method (Worch, Lê, Punter, & Pagès, 2013), in holistic approaches such as NappingÒ (Pagès, 2003, 2005), Free Sorting Task (Cadoret, Lê, & Pagès, 2009; Lawless, Sheng, & Knoops, 1995) or Sorted Napping (Pagès, Cadoret, & Lê, 2010). It can also be obtained from data from another nature, such as Check-All-That-Apply data (Piqueras-Fiszman, Ares, Alcaide-Marzal, & Diego-Mas, 2011), instrumental data (MacFie & Hedderley, 1993), words that consumers associated to the different products (Bécue-Bertaut, Alvarez-Esteban, & Pagès, 2008; Bécue-Bertaut & Lê, 2011). In general, any information that is related to products and which discriminates between them could be used. Although internal and external preference mapping techniques are widely used in practice, some limitations to these techniques have been pointed out. For instance, MDPref has been criticized for its lack in modeling saturation since only linear relationships between liking and the sensory attributes using correlation coefficients are considered (Rousseau, Ennis, & Rossi, 2012). PrefMap has been criticized that the regression models consider only two external dimensions (the first two dimensions of the space considered) and important information present in higher dimensions is omitted (Faber, Mojet, & Poelman, 2003). Such omission leads to ‘‘irrelevant models’’ for a non-negligible number of consumers. And increasing the number of sensory dimensions in the regression models seems not to fix the problem since it can over-fit the liking scores when circular, elliptic or quadratic models are used because only a low number of degrees of freedom are available for the estimation of the parameters. Finally, it should be mentioned that PrefMap is suitable for finding a local optimum within the product space, but cannot extrapolate outside the product space (Worch, Lê, Punter, & Pagès, 2012). For those reasons, several alternative techniques to these methodologies have been proposed. For the MDPref, we can mention Consumers’ Preference Analysis (Lê, Husson, & Pagès, 2006), Landscape Segmentation Analysis (Ennis, 2005) or BaSIC (Nestrud, 2012). For PrefMap, some methodologies based on PLS regression have been proposed (De Kermadec, Durand, & Sabatier, 1997; Tenenhaus, Pagès, Ambroisine, & Guinot, 2005; Verdun, Hanafi, Cariou, & Qannari, 2012). In this paper, we propose a new methodology which takes into consideration of the limitations mentioned above. This methodology is conceptually in between the internal and external preference mapping techniques. The advantages over the usual preference mapping techniques will be shown through two examples.

2. Material and method 2.1. Method 2.1.1. Presentation of the PrefMFA 2.1.1.1. Forewords. Internal and external preference mappings are useful (and widely used) for product optimization. However, as mentioned before, these techniques present some limitations: they are either not suitable to handle attribute saturation (MDPref) or the low number of external dimensions used in the individual regression models are not sufficient to fit the liking scores of a large number of consumers well (PrefMap). In PLS regression (Husson & Pagès, 2003) and derivative methods (e.g. L-PLS; Lengard & Kermit, 2006), the point of view adopted to avoid having a product space not related to liking consists in including liking information in the definition of the product space. In a recent paper (Verdun et al., 2012), quadratic effects were also

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added in the PLS models so that the final solution is more sensitive to the attribute saturation. It has been shown that this solution presents some advantages. However, the analysis is performed on the averaged liking scores and the individual variability is not used (the surface plot cannot be produced in this case). By following the same path, multi-block analysis in which external description of the products are considered on one hand and the individual hedonic scores are considered on the other hand can be used. The maximum part in common in the different matrices is then highlighted. In this case, any multiple block analysis such as Generalized Procrustes Analysis (Arnold, 1986; Arnold & Williams, 1986; Gower, 1975), STATIS (Lavit, Escoufier, Sabatier, & Traissac, 1994) or Multiple Factor Analysis (MFA; Escofier & Pagès, 1994, 1998) can be used. Couronne (1996) proposed a methodology using MFA in which the common space between the sensory and the hedonic configuration is studied as an alternative to preference mapping. The method presented here can be seen as an extension of Couronne’s work in which MFA and PrefMap are combined in the same analysis. In practice, the common space between hedonic and sensory data obtained from the MFA is used as starting point in the PrefMap regression procedure. 2.1.1.2. Description of MFA. MFA is a multivariate analysis for multiple block data proposed by Escofier and Pagès (1994). It is performed according to the following three steps:  A separate analysis is performed on each block of data;  in each group, the first eigenvalue kj1 of the separate analysis is extracted and its inverse (i.e. 1=kj1 ) is used as weight on the variables of that group;  an overall PCA is performed on all the weighted groups. Through this particular way of weighting the data, the shape of the configuration of each group is kept unchanged since the same weight 1=kj1 is applied to all the variables in each group. Moreover all the groups are equally balanced (the first eigenvalue of each separate analysis after weighting each group being 1). As a consequence, all the groups are potentially equally contributing to the construction of the first dimension of the MFA. In other words, the first dimension of the MFA is providing the maximum common information shared across the groups. One advantage of the MFA over other multiple block analyses relies in the nature of the data that can be considered in the analyses. Through this weighting procedure, MFA allows having groups of different nature simultaneously (quantitative, qualitative or frequency) (Bécue-Bertaut & Pagès, 2008; Pagès, 2004) in the same analysis. Another advantage of the MFA is the number of criteria and graphics which helps for the interpretation. For example, the first eigenvalue of the global analysis is a measure of similarity between the groups since it is comprised between 1 (all the groups are perfectly orthogonal) and the number of groups (all the groups are sharing the exact same information on their first dimension). Another example is the partial axes representation in which the main dimensions from the separate analyses are projected into the MFA space. Such graphical representation brings useful information since it shows how each group is related to the final MFA space. Other criteria such as the Lg and RV coefficients, or the overall quality of representation are also useful for the interpretation. 2.1.1.3. A solution in between internal and external preference mapping. As proposed by Couronne, an MFA is performed on both the external description of the products (let’s say the sensory profiles) on one hand and the liking scores provided by the consumers on the other hand (see Table 1).

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Table 1 Organisation of the dataset submitted to MFA.

pa of the products; the right part of the 665 table represents the hedonic scores provided by the consumers for the The left part of the table represents the sensory profiles y different products.

In this case, the separate analyses on the two groups are PCA on the correlation matrix for the sensory data (we give the same weight to each attribute) and PCA on the covariance matrix for the liking data (we give more weight to the consumers who discriminate more between products in terms of liking). The space obtained from the MFA is maximizing the inertia shared in both the sensory and hedonic block. In the product space obtained from the MFA, two products are close if (1) they have similar sensory profiles and (2) they are liked or disliked by the same consumers. The MFA space can be interpreted as a sensory space which can explain as much as possible the consumers’ similarities and differences in terms of liking. Basically this solution is similar to PLS where the liking scores are explained by the sensory description of the products. This is particularly true since PLS and MFA have some points in common (Pagès & Tenenhaus, 2001, 2002). To stay in line with the PrefMap philosophy which consists in using the external (sensory) space as starting point, the product configuration related to the sensory group within the MFA space is used. To do so, the coordinates of the products’ partial points related to the external group are extracted from the MFA. This sensory space is then used in the same way as in the PrefMap routine: the consumers’ hedonic scores are individually regressed on the coordinates of the products on the two first dimensions of the MFA space by using the vectorial, circular, elliptic or quadratic model. In order to evaluate the underlying relationship between the different groups, three criteria obtained from MFA are used: the Lg (or Ng) coefficient (Escofier & Pagès, 1998), the RV coefficient (Escoufier, 1973) and the overall quality of representation of group g by dimension a (noted VEðgÞ a ). For each group separately, the Ng coefficient (the Ng coefficient corresponds to the Lg coefficient when applied to a single group) informs about the dimensionality of the group itself. For a group in which the information is well balanced along the first S dimensions, the Ng coefficient is larger (between 1 and S) than for a group in which most of the inertia is explained along the first dimension only. When measured between pairs of groups, the Lg coefficient measures the richness of the common structure between the two groups: the larger the Lg coefficient, the larger the common structure. As an extension, the Lg coefficient is also measured between each group and the MFA dimensions obtained in order to measure the structure in common between each group with the MFA solution. The RV coefficient measures the link between groups of variables. It is comprised between 0 (the two groups are perfectly orthogonal) and 1 (the two groups are perfectly homothetic). As a third criterion, the overall quality of representation VEðgÞ a of the variables of each group g within the MFA solution is computed. It represents the quality of representation (in terms of percentage of variance explained) of each group g by each MFA dimension a. Such criterion is obtained through the squared correlation between the dimensions of the separate analysis and each MFA dimension, weighted by the percentage of variance explained by each dimension in the separate analysis (see Eq. (1)).

VEðgÞ a

¼

d¼D X d¼1

ðgÞ  2 k MFA ðgÞ r Dima ; Dimd  P d ðgÞ kd

! ð1Þ

In this equation, VEaðgÞ is the percentage of variance explained for group g by dimension a of the MFA, D the total number of dimenðgÞ

the dimension a of the MFA, VEd the sions in each group, VEðMFAÞ a ðgÞ P ðgÞ dimension d of the separate analysis on group g, and kd = kd the variance explained by dimension d in the separate analysis of ðgÞ

group g (kd being the eigenvalue associated to dimension d in the separate analysis of group g). The VEðgÞ coefficient should only be used for illustrative pura poses. Indeed, although it is interesting to evaluate the percentage of variance of each group that is explained by each MFA dimension, its interpretation is limited. In fact, a coefficient of 0.3 (=30% of the variance of the group g is explained by dimension i of the MFA) can either be obtained when all the attributes of g are slightly related to a or when few attributes of g are strongly related to a. 2.1.1.4. Regression of the individual liking scores on the MFA space. The coordinates of the product partial points associated with the external group on the first two dimensions are extracted from the MFA. For each individual consumer, the liking scores are regressed on these two dimensions. In this article, the quadratic model proposed by Danzart (1998) is considered (Eq. (2)):

 ¼ a1 Dim1 þ a2 Dim2 þ a11 Dim2 þ a22 Dim2 Hjp  h j 1 2 þ a12 Dim1 Dim2

ð2Þ

 is the centered liking scores for consumer j In this equation, Hjp  h j (the liking scores are centered so that the mean is set to 0) and ai is the regression weight associated to the corresponding (linear, quadratic or interaction) dimension i. From this regression, an individual response matrix is estimated for each consumer. This individual response matrix mimics the sensory space set as a grid. For each point of the grid, the liking score is estimated using the model defined previously. If for a consumer, the estimation of the (centered) liking score of a point in the space is larger than a predefined level of acceptance (usually, the level is set to 0 corresponding to the averaged liking score for that consumer), the cell in the response matrix which corresponds to that point in the space takes a 1, otherwise a 0. Once all individual response matrices are estimated, they are summed together and the proportion of consumers accepting the product present in each point of the space is computed. This results in the surface plot representation. 2.1.2. Extension to the PrefHMFA In some situations, more information about the products is available. For instance, in addition to the sensory profiles of the products, one might have instrumental descriptions of the products. The sensory profiles might also be obtained with different panels (expert and consumers), or from different locations (crosscultural studies). In this case, a larger hierarchy between groups

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needs to be considered. In the first level of the partition the different external blocks are balanced together (profiles from different panels or sensory and instrumental data). In the second level of the partition, the common ‘‘external’’ space thus obtained is balanced with the hedonic scores. Hierarchical Multiple Factor Analysis (HMFA) can be used for such analysis (Le Dien & Pagès, 2003a, 2003b). It should be noted that HMFA must also be considered if the sensory space is obtained from holistic approaches such as NappingÒ or Sorted Napping in which variability between consumers is used in the determination of the product space. A theoretical representation of the possible hierarchy for HMFA is given in Fig. 1. 2.1.3. PrefMFA and clustering For the products’ optimization, it is important to consider the variability between consumers and more precisely their behavior upon liking. For that reason, it is recommended to see if the consumers can be segmented into different homogeneous clusters. With preference mapping techniques, segmenting consumers beforehand is not necessary since consumers are regressed individually in the sensory space. If the panel of consumers is segmented in let’s say two clear and equilibrated segments, the resulting preference map will highlight these two segments by providing two optimums. In practice, two different strategies are often adopted: – The panel of consumer is segmented upon liking beforehand and a preference mapping is performed for each cluster separately. In this case, the MFA solution used as starting point in the PrefMFA procedure is different for each cluster, since the block of consumers differs for each cluster, and the drivers of liking might be different from one cluster to another; – The entire panel of consumers is used in the preference mapping simultaneously. In this case, the variability within the consumer block provides clues about the existence of different segments. In this article, two different datasets are used for illustration. The two strategies will be adopted. 2.2. Comparison with PrefMap Since the MFA space is based on both external information about the products and the liking ratings, this space will be more suitable for fitting the consumers’ liking scores on the dimensions. Let’s consider that for the large majority of the consumers, parts of dimensions 1, 2 and 3 of the sensory space are relevant. In the classical approach, since only dimensions 1 and 2 are being used, an important part of the information from dimension 3 is not considered. With MFA, since the space summarizes the maximum common inertia between the external information and the liking scores, the first plan of the MFA considers the main dimension of the liking information and parts of dimensions 1, 2 and 3 of the external space, which would lead to better fit of the individual models.

Fig. 1. Possible hierarchy in the data to perform HMFA. E1 to En represent the external matrix (sensory obtained from different panel, sensory and instrumental, or the Nappes coordinates of each panelist); H represents the matrix of hedonic scores. At the first level of the partition P1, the external matrices E1 to En are balanced together and the common configuration E is obtained. At the second level P2 of the partition, the common space between E and H is defined. This product space (HMFA) is used in PrefHMFA.

The comparison of the results obtained with PrefMap and PrefMFA are based on the interpretation and adequacy of the maps and on the criteria obtained from these two techniques. Additionally, in order to verify the assumption that MFA provides a space that better fits the hedonic scores, and to quantify this improvement, the quality of the individual models obtained with PrefMap and PrefMFA are compared graphically. The adjusted R2 of the individual models are used.

2.3. Illustration 2.3.1. The sauce dataset In a first step, 13 sauces were tested by 10 experts in triplicates. The experts described the sauces on 35 predefined sensory attributes (noted Attr. 1 to Attr. 35). The same 13 sauces were also rated by 123 Dutch consumers on overall liking. For confidentiality reasons, more information concerning this project cannot be given and both the names of the products and attributes are made anonymous. This is the classical approach of preference mapping in which the sensory profiles of the products are obtained from experts and the liking scores from consumers. In this analysis, the block of sensory descriptions (matrix with 13 rows and 35 columns) and the block of hedonic ratings (matrix with 13 rows and 123 columns) are balanced in PrefMFA. For this analysis, we adopt the strategy consisting in performing cluster analysis first and create one preference map for each cluster separately.

2.3.2. The soup dataset In the second example, 9 soups were rated by 109 Dutch consumers on 24 sensory attributes following the Ideal Profile Method procedure (Worch et al., 2013). The same consumers also rated the products on overall liking. A list of the attributes is given in Table 2.

Table 2 List of the 24 attributes used in the Soup study. Odor

Appearance

Taste

Mouth feel

After taste

Intensity Odor 2 Odor 3

Color Fatty Thickness Shine Ingredient variation Ingredient coarseness

Intensity Saltiness Sourness Sweetness Bitterness Taste2 Taste3 Fullness Natural

Thickness Fatty Floury Ingredient firmness

Intensity Length

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The ideal intensity scores measured for each attribute are not used here. Here, the sensory profiles and the hedonic scores of the products are provided by the same consumers. If a sensory description of the products was also provided by experts, HMFA would be considered. Such analysis would first balance the consumer and experts descriptions of the products before balancing the common external space thus obtained with the hedonic differences between products. However, since the expert information is not available, PrefMFA is considered. This PrefMFA is obtained by associating the averaged sensory profiles (9 rows and 25 columns) obtained

from consumers with the consumer liking ratings (9 rows and 109 columns). In this case, we adopt the strategy consisting in using the entire panel of consumers simultaneously in the analysis. If segmentation is observed within the PrefMFA solution, the clusters highlighted will be compared to the cluster solutions. 2.4. Statistical analyses The statistical analyses were performed using R 2.15.2 (R Core Team, 2012) and the SensoMineR (Lê & Husson, 2008) and FactoMineR (Lê, Josse, & Husson, 2008) packages. A PrefMFA and an

Fig. 2. Dendrogram defining three clusters of consumers in the sauce study.

Fig. 3. Differences in averaged liking scores between clusters in the sauce study.

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Fig. 4. First two dimensions of the sensory space obtained from the expert panel in the sauce study. In this figure are represented (a) the product space; (b) the variables representation; (c) the individual liking scores projected as illustrative and colored according to the cluster the consumers belong to; (d) the cluster means projected as supplementary variables.

adapted PrefMap function have been created by the author. These functions are at the disposition of the reader upon request. 3. Results 3.1. Sauce dataset Before generating the preference maps, a cluster analysis is performed in order to find out if there are different segments of consumers based on liking. A PCA is performed on the hedonic matrix crossing the consumers in rows and the products in columns. On the consumer space thus obtained, hierarchical clustering on principal component (HCPC; Husson, Josse, & Pagès, 2010) is carried out and three clusters were found (Fig. 2). Consumers in the cluster 1 (51 consumers) like products 10, 12, 1, and 5 and dislike products 11 and 8. Moreover, the cluster 1 likes (vs. dislikes) products 2, 10, and 9 (vs. 4, 8, and 11) significantly more than the rest of the panel. Consumers in cluster 2 (39 consumers) like products 11, 8, 9, and 3 significantly more and dislike products 2, 7, 5, and 13 significantly more than the rest of the pa-

nel. Finally, consumers in cluster 3 (33 consumers) like products 5, 7, and 1 significantly more and dislike products 9, 10, and 3 significantly more than the rest of the panel (Fig. 3). In order to evaluate the relationship between the sensory profiles and the hedonic scores, a PCA is carried out on the sensory profiles (Fig. 4a and b) of the products obtained from the expert panel and the hedonic scores are projected as supplementary variables (Fig. 4c). As a summary, the mean scores for each cluster are also projected as supplementary variables (Fig. 4d). From this projection, it appears that the cluster 1 is strongly positively correlated with the second dimension of the sensory space (r = 0.91). The second and third clusters are negatively correlated with the first dimension (r = 0.70 and r = 0.53 respectively). The separation between cluster 2 and cluster 3 is done along the third dimension: the cluster 3 is positively correlated with dimension 3 (0.57) while cluster 2 is not correlated with it (results not shown here). For each cluster, PrefMap and PrefMFA are carried out. For clusters 1 and 2, dimensions 1 and 2 are considered in the PrefMap. For cluster 3, dimensions 1 and 3 are used.

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Fig. 5. PrefMap (a) and PrefMFA (b) solutions obtained for cluster 1 in the sauce study.

Fig. 6. Partial axes representation obtained in the PrefMFA for cluster 1 in the sauce study.

For cluster 1, the PrefMap and PrefMFA are giving very similar results (Fig. 5) after rotation: the second dimension of the sensory space (PrefMap, Fig. 5a) corresponds to the first dimension of the MFA (PrefMFA, Fig. 5b). In both cases, the product that would be accepted by a maximum of consumers is located close to products 5, 1, and 10. This result is consistent with the description of the cluster 1 as described in the previous paragraph. This result is highlighted by the partial axes representation of the MFA (Fig. 6): the consumers liking pattern is mainly explained by the second dimension of the sensory space. The Ng coefficient shows that the sensory space is much more multidimensional (Ng (Sensory) = 2.08) than the hedonic space (Ng (Hedonic) = 1.08) which is clearly one-dimensional. This result is expected since the panel of consumers has been segmented into homogeneous clusters. For this cluster, most of the hedonic structure is also present in the sensory one (Lg(Sensory,Hedonic) = 0.85)

although the two global structures are not so much linked (RV = 0.57). This is probably due to the fact that the main variation in the sensory data is not related to hedonic data. Finally, it appears that the first dimension of the MFA captures 55% of the hedonic inertia (VEhedonic ¼ 0:55) and 18% of the sensory inertia 1 (VEsensory ¼ 0:18) while the second dimension captures 6% of the he1 donic inertia (VEhedonic ¼ 0:06) and 28% of the sensory inertia 2 (VEsensory ¼ 0:28). 2 For cluster 2, the results are similar to the one obtained with cluster 1 (not shown here). For cluster 3, the PrefMap highlights a zone of maximum liking in the area close to products 1, 5, and 12 (Fig. 7a). Similar results are obtained with the PrefMFA (Fig. 7b). However, a disagreement between the two methods is observed for product 8 which seems quite accepted in the PrefMap (between 70% and 80% of the consumers would accept it) and clearly rejected in the PrefMFA (less than 20% of the consumers would accept it). Oppositely, product 4 is quite rejected in the PrefMap (only 40% of the consumers accept it) and still well accepted in the PrefMFA (around 55% of the consumers would accept it). Fig. 3 supports the results of the PrefMFA since product 4 scores higher than product 8. From the attribute points of view, the PrefMap shows no relationship between the attributes 10 and 27 with the cluster preferences while PrefMFA does show a relationship (consumers belonging to cluster 3 like more the products that are intense for those two attributes). These differences can be explained through the partial axes representation from PrefMFA (Fig. 8). The first dimension of the MFA is related to the first dimension of the hedonic space and almost equally to dimensions 1, 2 and 3 of the sensory space. Hence, by discarding dimension 2 in PrefMap for cluster 3, relevant information has been discarded. In PrefMFA, the first dimension of the MFA is associated with these three dimensions. Finally, PrefMFA shows that for cluster 3, the hedonic structure (Ng (Hedonic) = 1.14) is also present in the sensory data (Lg (Sensory, Hedonic) = 0.99), and both structures are well linked (RV = 0.64). The first dimension PrefMFA explains 46% of the

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Fig. 7. PrefMap (a) and PrefMFA (b) solutions obtained for cluster 3 in the sauce study.

Fig. 8. Partial axes representation obtained from the MFA for cluster 3 in the sauce study.

Hedonic inertia (VEhedonic ¼ 0:46) and 22% of the sensory inertia 1 (VEsensory ¼ 0:22) while dimension 2 explains 8% of the hedonic 1 inertia (VEhedonic ¼ 0:08) and 22% of the sensory inertia 2 (VEsensory ¼ 0:22). 2 Finally, the two preference maps are compared through the quality of the individual models using the adjusted R2 coefficients. As expected, the adjusted R2 are higher for PrefMFA than for PrefMap for the large majority of the consumers (Fig. 9) although the improvement is small. 3.2. Soup dataset For this dataset, the strategy adopted consists in realizing the analyses on the entire panel of consumers without segmenting the panel beforehand. From the average liking ratings, it seems that the most liked products (also with the best consensus, i.e. the lowest standard deviation) are products 7, 5, and 9 while the least liked products are products 4, 8, and 1.

Fig. 9. Scatter plot of the adjusted R2 coefficients associate with the individual models for both the PrefMap and PrefMFA techniques for cluster 1 in the sauce study.

The PrefMap performed for the entire panel of consumers highlights two potential maxima, suggesting the presence of two clusters (Fig. 10a). The first area integrates products 5, 7, and 1 and is positioned in the centre of the space while the second area integrates products 2 and 9 and is located in the top right part of the space. This result is somewhat surprising since product 1 is one of the least appreciated products but it is highlighted in one of the most liked area. The corresponding PrefMFA performed on the same dataset shows one dominant area of maximum appreciation (Fig. 10b). This area is close to products 5 and 7 (positive side of the second dimension) which are the most appreciated products. Another small area is observed close to product 9 (bottom right part of

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Fig. 10. PrefMap (a) and PrefMFA (b) solutions obtained in the soup study.

Table 3 LSD results between products obtained for each cluster separately for the Soup study. Cluster 1

Cluster 2

Product

A

7 5 9 8 6 2 4 3 1

7.58 7.11

B 7.11 6.82

C

6.82 6.34

D

E

Product

A 7.62 7.21

5.52 5.52 5.47 5.40

2 7 9 6 5 3 1 4 8

6.34 6.24

B 7.21 7.09 7.02 6.83 6.74

C

D

E

6.83 6.74 6.43 5.89 5.04

Two product means belonging to the same group (i.e. in the same column) are not significantly different at 5% w.r.t. LSD post’hoc test.

Fig. 11. Variables representation for both the PrefMap (a) and PrefMFA (b) in the soup study. In PrefMap, the individual liking scores as well as the cluster means are projected as supplementary variables and are color-coded according to the different clusters. In PrefMFA, the individual liking scores are active in MFA. These consumers are color-coded according to the different clusters. The cluster means are projected as supplementary in this analysis.

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Fig. 12. Differences in averaged liking scores between clusters in the soup study.

Fig. 14. Scatter plot of the adjusted R2 coefficients associate with the individual models for both the PrefMap and PrefMFA techniques in the soup study.

Fig. 13. Partial axes representation obtained from the MFA for the soup study.

the space). In this case, product 1, previously highlighted in the PrefMap analysis, is separated from the two other products. The results obtained both with PrefMap and PrefMFA highlight the presence of (at least) two clusters of consumers. Cluster analysis is performed (HCPC) and the two cluster solutions is kept. The two clusters differ significantly for products 8, 7, and 5 (liked more by cluster 1) and for products 1, 3, and 2 (liked more by cluster 2). The significant differences between products within each cluster are given in Table 3. The cluster means are projected as supplementary variables in the two spaces (Fig. 11). In PrefMap, both the individual scores and the cluster means are projected as supplementary variables while in the PrefMFA, only the cluster means are projected as a supplementary group since the consumer liking scores are already used as active in the analysis. For PrefMap, the clustering results are not matching well the cluster suggestion from PrefMap (Fig. 11a). Indeed, cluster 1 seems to be associated with another dimension of the space, while cluster 2 is related to dimension 1 (the consumers belonging to this cluster are still segmented along

dimension 2). For PrefMFA (Fig. 11b), the cluster solution is closer to the clusters suggested by HCPC. Cluster 1 points in the direction of products 5 and 7 while cluster 2 points in the direction of product 2. This corresponds to the products the cluster respectively liked the most (Fig. 12, Table 3). The difference between the results obtained from the two preference mapping techniques can be highlighted by the partial axes representation of PrefMFA (Fig. 13). The partial axes representation shows that dimension 1 of the hedonic space (correlation of 0.92 with cluster 1) is correlated to both dimensions 2 and 3 (r = 0.50 and r = 0.67 respectively) of the sensory space while dimension 3 of the hedonic space (correlation of 0.94 with cluster 2) is related to dimension 1 of the sensory space (r = 0.87). Dimension 2 of the hedonic space seems not to be related to the sensory data. In PrefMap, since the third sensory dimension is not taken into account in the individual models, relevant information is omitted. The Ng coefficient associated to the hedonic group is larger than for the sensory group (Ng (Hedonic) = 2.21; Ng (Sensory) = 1.80) meaning that the hedonic group is more multidimensional than the sensory one. A consequence of that is that the entire hedonic differences cannot be explained by liking. However, both groups have a large structure in common (Lg (Sensory,Hedonic) = 1.53 and RV = 0.77). Still, MFA summarizes a big part of the information present in both groups. The first dimension of the MFA explains 29% of the inertia related to the sensory group (VEsensory ¼ 0:29) 1 and 21% of the inertia related to the hedonic group (VEhedonic ¼ 0:21). The second dimension of the MFA explains 24% 1 (VEsensory ¼ 0:24) and 20% (VEhedonic ¼ 0:20) of the inertia for the sen2 2 sory and hedonic group respectively. Finally, the quality of fit of the individual models is compared across the two methodologies. As expected, the PrefMFA is associated to larger adjusted R2 coefficient than PrefMap for the majority of consumers. For a non negligible amount of consumers (about 20%), the opposite is observed (Fig. 14). 4. Conclusions PrefMFA is a novel technique considered to be as a combination of internal and external preference mapping since the product

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space considered for the individual regression models is related to the external (usually sensory) description of the products that is the most linked with the hedonic differences. The philosophy behind this technique is similar to techniques based on PLS regression since the external space considered is maximizing the liking information gathered from consumers. As shown through the examples, the major advantage of PrefMFA over PrefMap relies on a better selection of the dimensions for the individual models. In the hypothetical situation where the most important external dimension is the third one (the first dimension being of lower interest for explaining hedonic differences), PrefMap searches models on less relevant information (since dimensions 1 and 2 are often used) while PrefMFA adapts to the situation by increasing the importance of the third and subsequent dimensions and by decreasing the importance of dimension 1 of the external (i.e. sensory) space. Such situation will also be shown through the value of the Ng/Lg coefficient. In such case, the common structure between the first hedonic dimension and the sensory configuration will show that a big part of the hedonic data can be explained by the sensory data. However, it will also show that a big part of the sensory data cannot be explained by the hedonic data (e.g. analysis of each cluster in the sauce example). When the analyses (PrefMap and PrefMFA) are performed on each homogeneous cluster of consumers separately, the results between PrefMap and PrefMFA are similar in the following cases: (1) when the sensory space considered in PrefMap is rigourously well selected beforehand (by looking at the link between the sensory space and each cluster result), and (2) when the relationship between sensory and hedonic description is explained by one or two sensory dimensions only. In the case where the hedonic scores for a cluster are related to more than 2 dimensions (e.g. cluster 3 in the sauce example), PrefMFA is performing better than PrefMap since some important information might be discarded in PrefMap and kept in PrefMFA. When the analyses are performed on the entire panel of consumers, PrefMFA returns more relevant results since the maximum of common information present in the two blocks (sensory and hedonic) is used. In the case of PrefMap, the first two dimensions of the sensory space are often used since we give ourselves the maximum chances (in terms of inertia) to find relationships with hedonic data although some other dimensions of lower inertia might be also (or more) relelvant. In these two situations, with PrefMFA there is no need of searching and selecting the best dimensions for the individual models since the first two dimensions of the MFA will always be used (by construction, the first two dimensions of MFA correspond to the largest common structure in the two matrices). Another advantage of PrefMFA stands in the large number of criteria provided that help understanding the underlying relationship between sensory and hedonic data. These criteria can be very useful for the interpretation of the results. Among these criteria, the partial axes representation highlights the relationship existing between each dimension of the external space and each dimension of the hedonic space. Additionally, the Ng and Lg coefficients help to understand the structure of each group separately (Ng) and quantifies the structure in common between groups (Lg). These coefficients are often associated with the RV coefficient which measures the similarity between the product configurations obtained from each group. Finally, the VE coefficient summarizes the part of inertia within each group that is explained by each dimension of the MFA. It should also be mentioned that when using the MFA solution in the regression, the individual models are associated to a better fit than the one obtained from the PCA space (the adjusted R2 has been used here). This result is expected since in PrefMFA, the

hedonic scores are regressed on a space that also includes the hedonic data, while in PrefMap, this is not the case. However, it can be noted that the improvement in models fit is observed for a large majority of consumers. Although PrefMFA shows some good properties, it also has some limitations: it can be noticed that PrefMFA handles slightly better multiple relevant dimensions than PrefMap. However, it is still limited to a small number of dimensions in the individual models (usually two when quadratic models are used). For dataset in which many sensory dimensions should be considered to explain liking, this low number of dimensions can be a limitation. Moreover, since PrefMFA is also based on regressions, good extrapolation outside the product space is not possible. Recently, Yenket, Chambers, and Adhikari (2011) have also shown that in preference mapping, consumers are not always closer to their most preferred product. A similar result has been obtained with the PrefMap for the soup dataset (one of the least liked products being in a highly accepted area). In that case, PrefMFA seems to give more relevant results. A more systematic verification needs to be done before giving more general conclusions. Acknowledgements The author would like to thank Pieter Punter, OP&P Product Research, Utrecht, the Netherlands for providing the dataset and allowing its use in this paper. References Arnold, G. M. (1986). A Generalized Procrustes macro for sensory analysis. Genstat Newsletter, 18, 61–80. Arnold, G. M., & Williams, A. A. (1986). The use of Generalized Procrustes Techniques in sensory analysis. In J. R. Piggot (Ed.), Statistical procedures in food research (pp. 233–253). London: Elsevier Applied Science. Bécue-Bertaut, M., Alvarez-Esteban, R., & Pagès, J. (2008). Rating of products through scores and free-text assertions: Comparing and combining both. Food Quality and Preference, 19, 122–134. Bécue-Bertaut, M., & Lê, S. (2011). Analysis of multilingual labeled sorting tasks: Application to a cross-cultural study in wine industry. Journal of Sensory Studies, 26, 299–310. Bécue-Bertaut, M., & Pagès, J. (2008). Multiple factor analysis and clustering of a mixture of quantitative, categorical and frequency data. Computational Statistics & Data Analysis, 52, 3255–3268. Carroll, J. D. (1972). Individual differences and multidimensional scaling. In R. N. Shepard, A. K. Romney, & Si Nerloves (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences. New York: Academic Press. Cadoret, M., Lê, S., & Pagès, J. (2009). A Factorial Approach for Sorting Task data (FAST). Food Quality and Preference, 20, 410–417. Couronne, T. (1996). Application de l’analyse factorielle multiple à la mise en relation de données sensorielles et de données de consommateurs. Sciences des Aliments, 16, 23–35. Danzart, M. (1998). Quadratic model in preference mapping. In 4th sensometric meeting, Copenhagen, Denmark, August 1998. Danzart, M. (2009a). Cartographie des préférences. In SSHA 3eme (Ed.), Evaluation sensorielle. Manuel méthodologique (pp. 443–450). Paris: Lavoisier. Danzart, M. (2009b). Recherche d’un point optimal. In SSHA 3eme (Ed.), Evaluation sensorielle. Manuel méthodologique (pp. 431–439). Paris: Lavoisier. De Kermadec, F. H., Durand, J. F., & Sabatier, R. (1997). Comparison between linear and nonlinear PLS methods to explain overall liking from sensory characteristics. Food Quality and Preference, 8, 395–402. Ennis, D. M. (2005). Analytic approaches to accounting for individual ideal points. IFPress, 8(2). Escofier, B., & Pagès, J. (1994). Multiple factor analysis (AFMULT package). Computational Statistics & Data Analysis, 18, 121–140. Escofier, B., & Pagès, J. (1998). Analyses factorielles simples et multiples. Paris: Dunod. Escoufier, Y. (1973). Le traitement des variables vectorielles. Biometrics, 29, 751–760. Faber, N. M., Mojet, J., & Poelman, A. A. M. (2003). Simple improvement of consumer fit in external preference mapping. Food Quality and Preference, 14, 455–461. Gower, J. C. (1975). Generalized procustes analysis. Psychometrika, 20, 33–51. Greenhoff, K., & MacFie, H. J. H (1994). Preference mapping in practice. In H. J. H. MacFie & D. M. H. Thompson (Eds.), Measurement of food preferences (pp. 137–166). London: Blackie Academic & Professionals. Husson, F, Josse, J., & Pagès, J. (2010). Principal component methods – hierarchical clustering – partitional clustering: Why would we need to choose for visualizing data? Technical report – Agrocampus. Retrieved from: .

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