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Simple improvement of consumer ﬁt in external preference mapping Nicolaas (Klaas) M. Faber1, Jos Mojet*, Astrid A.M. Poelman ATO, Agrotechnological Research Institute, PO Box 17, 6700 AA Wageningen, The Netherlands Received 21 July 2002; received in revised form 13 November 2002; accepted 28 November 2002

Abstract In the common implementation of external preference mapping consumer preferences are ﬁtted as polynomial functions of the ﬁrst two principal components (PCs) of the sensory data. A major weakness of the method is the relatively small number of consumers that can be signiﬁcantly ﬁtted. Several researchers have proposed to improve the consumer ﬁt by including higher-numbered PCs in the analysis. We have explored the possibility of including higher-numbered PCs while restricting the model choice to the simplest polynomial function, i.e. the (linear) vector model. In addition, we have developed a heuristic decision rule for determining the number of PCs to keep in the ﬁt. A practical example is discussed where the consumer ﬁt improved from 51% (two PCs, polynomial) to 80% (ﬁve PCs, vector). # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Preference mapping; Sensory analysis; Consumers; Apples

1. Introduction Product optimisation is the goal of every food manufacturer. Several modelling procedures have been used for product optimisation and preference mapping is one of such methods. Depending on the data analysed, two modes of preference mapping exist. Internal preference mapping (IPM) models consumer preferences collected for a number of products, i.e. a single data set only, while external preference mapping (EPM) deals with two data sets simultaneously, i.e. consumer preference and sensory attribute ratings collected for the same products. This paper is concerned with EPM. The consumer preference for a product is at least to a certain extent determined by its sensory attributes. Thus, one has a typical regression set-up where the consumer data (Y-block) can be modelled as function of the sensory data (X-block). Before turning attention to the focus of this paper, i.e. the common implementation of EPM (McEwan, 1996; Schlich, 1995), we will mention two alternatives. The ﬁrst one, partial least squares regression (PLSR), has been used increasingly in recent years * Corresponding author. E-mail address: [email protected] (J. Mojet). 1 Present address. Chemometry Consultancy, Rubensstraat 7, 6717 VD Ede, The Netherlands.

(De Jong, 1991; Helgesen, Solheim, & Næs, 1997; Ka¨lvia¨inen, Salovaara, & Tuorila, 2002; Lawlor & Delahunty, 2000; Meullenet et al., 2001; Murray & Delahunty, 2000a, 2000b). The second one, which is highly popular in chemometrics, couples the data sets through a so-called inverse model (Jaeckle & MacGregor, 1998). We could not ﬁnd applications of this approach in the sensometrics literature. In the common implementation of EPM, the sensory data are decomposed using principal component analysis (PCA). Subsequently, the individual consumer preferences are modelled as polynomial functions of the ﬁrst two principal components (PCs). The relative ﬁtting ability of four functions, i.e. the so-called quadratic (also referred to as elliptical ideal point model with rotation, see Greenhoﬀ & MacFie, 1994; Schiﬀman, Reynolds, & Young, 1981), elliptical, circular and vector models, is examined in a stepwise manner, starting at the most complex one (i.e. quadratic). The choice for a particular function is based on a goodness-of-ﬁt test. This test will be referred to as the ﬁrst F-test in the remainder of this paper. To avoid presenting overoptimistic results, the ﬁnal ﬁt of the consumer data is validated using a second F-test. It is well documented that applications of EPM often suﬀer from the small number of consumers that is signiﬁcantly ﬁtted. Schlich (1995) comments that in his experience with several data

0950-3293/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0950-3293(03)00011-9

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sets, about half of the consumers never ﬁt to the sensory space and are subsequently excluded from further analysis. Our experience is similar. Helgesen et al., (1997) report an example where only 36% of the consumers are signiﬁcantly ﬁtted. Pagliarini, Monteleone, and Wakeling (1997) do not even report EPM results because of the poor consumer ﬁt. By contrast, Pagliarini, Monteleone, and Ratti (2001) obtain a relatively good value of 61%, while Monteleone, Frewer, Wakeling, and Mela (1998) report a value of 68.5%. Guinard, Uotani, and Schlich (2001) even report an excellent value of 75%. However, they attribute this exceptional result to the large number of products (24 beers). Unfortunately, exposing untrained (‘naive’) consumers to such a large number of products is seldom practical. Greenhoﬀ and MacFie (1994) and Heyd and Danzart (1998), among others, have attributed the poor consumer ﬁt to the loss of information contained in higher-numbered PCs. Several investigators have reported an improved consumer ﬁt by including these otherwise discarded PCs (Arditti, 1997; Daillant-Spinnler, MacFie, Beyts, & Hedderley, 1996; Elmore, Heymann, Johnson, & Hewett, 1999). It is important to note that including higher-numbered PCs may easily lead to overﬁtting the consumer preferences, since applications of EPM are often concerned with modelling a small number of products (typically around 10). In other words, there are only a few degrees of freedom in the data and these are quickly depleted. The purpose of this paper is two-fold. First, the utility of including higher-numbered PCs is further explored. Unlike previous work, we restrict ourselves entirely to vector models, since the simplest model allows one to include more PCs before overﬁtting sets in. Clearly, for a ﬁxed number of PCs, the danger of overﬁtting increases with the complexity of the model. Besides, we have found the more complex models to be less useful for our applications (mostly fresh products such as apples and tomatoes). Our experience concurs with the observation of Schlich (1995) and McEwan (1996), that the elliptical and quadratic models are seldom used in sensory practice because they are often diﬃcult to interpret. The second purpose is to develop a decision rule for determining the number of PCs to include in the analysis. The need for this decision rule becomes apparent when realising that the ﬁrst F-test used in EPM (for choosing a particular function) is known to have problems (Draper, Guttman, & Kanemasu, 1971; Pope & Webster, 1972). Brieﬂy, the test statistic is not distributed as Fisher’s F under the null hypothesis. The ideas will be illustrated using a practical example.

2. Theory The sensory and consumer data are expressed in matrix notation as X and Y, respectively. The matrix X

is dimensioned Np Na where Np is the number of products and Na is the number of sensory attributes. Likewise, the matrix Y is dimensioned Np Nc where Nc is the number of consumers. In other words, the data collected for a speciﬁc product occupy corresponding rows of X and Y. The ﬁrst step of EPM consists of decomposing X using PCA. PCA is a multivariate technique that produces a least-squares approximation of a data matrix in a lower-dimensional space. Approximating X using the ﬁrst A PCs can be expressed as X ¼ TPT þ E where the A columns of T (Np A) and P (Na A) are known as scores and loadings respectively, the superscript ‘T’ symbolises matrix transposition and E (Np Na ) contains residuals. The standard convention is that the loadings are normalised, so that the scale of the data enters the scores. With normalised loadings, the Euclidean norm of a score vector is the percentage variance of X explained by a PC. Often a good approximation of X results with a small value for A, since the PCs are calculated in order of decreasing explained variance. Plotting the elements of the ﬁrst two columns of T yields a two-dimensional view (projection) of the position of the products in the multivariate space, likewise the ﬁrst two columns of P for the attributes. Relationships between products and attributes can be investigated by combining both scores and loadings in a single plot—a sensory map. This usually requires rescaling one set of vectors. Visualisation is an important aspect of preference mapping (internal as well as external mode). In the next step, the individual consumer preferences are ﬁtted as polynomial functions of the ﬁrst two PCs of X. Polynomial modelling allows one to calculate an optimum (global or local) for each individual consumer. The idea is that these optima provide useful information for product design. The following four functions are considered:

Vector: yn ¼ a þ b1 t1 þ b2 t2 þ en Circular: yn ¼ a þ b1 t1 þ b2 t2 þ cðt1 t1 þ t2 t2 Þ þ en Elliptical: yn ¼ a þ b1 t1 þ b2 t2 þ c1 t1 t1 þ c2 t2 t2 þ en Quadratic:yn ¼ a þ b1 t1 þ b2 t2 þ c1 t1 t1 þ c2 t2 t2 þ dt1 t2 þ en

where yn is the n-th column of Y (n ¼ 1; . . . ; Nc ), t1 and t2 are the ﬁrst two columns of T, en is a residual vector, ‘.‘ and ‘‘ symbolise the scalar and element-wise product, respectively. The parameters a, b1 , b2 , c, c1 , c2 , and d are estimated from the data at hand. For example, the intercept, a, is dealt with by mean-centring X and Y.

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The vector model is the multiple linear regression model that does not have a global optimum (Fig. 1a). However, when constraining the sensory space to a circle for which the radius is determined by the scores of the most extreme product, as proposed by Carroll (1972), it is possible to optimise the function locally on the circle, henceforth denoted as the preference circle. The circular model will have a global optimum, provided that the estimate for c is negative (Fig. 1b), otherwise it has a global minimum. The elliptical model is more ﬂexible than the circular model in the sense that the estimates for c1 and c2 can be diﬀerent. It will have a global optimum, provided that both estimates for c1 and c2 are negative. Conversely, a global minimum is obtained when both estimates are positive. A special situation arises when they are of opposite sign: this leads to a saddle point (Fig. 1c). Finally, the quadratic model is the elliptical model plus a cross-product term. The optimising conditions are the same as for the elliptical model, since the cross-product term merely amounts to a rotation of the surface (Fig. 1d). In cases where the circular, elliptical or quadratic models do not yield a global optimum within the preference circle, one may choose to replace the model by the vector model. McEwan (1996), who mentions that ‘‘In cases where the ideal point lies outside the sample space, the vector model may well be more appropriate’’, supports this solution. In our implementation the more complex model is kept and optimised locally on the preference circle. Both solutions are equivalent in the current context, where improvement of consumer ﬁt is the focus.

457

To determine the suitable model complexity, a sequential procedure is deployed. First, the quadratic and elliptical models are compared in an F-test. If deletion of the cross-term does not lead to a signiﬁcantly worse ﬁt, then one proceeds with comparing the elliptical and circular model, and so on. The size of the test is referred to as the level of choice. Schlich (1995) uses the value pchoice ¼ 0:10, because increasing it would complicate the models, whereas decreasing it would simplify them. Unfortunately, this procedure is rather problematic. It is known that stepwise procedures connected with F-tests can be highly misleading because the test statistic is not distributed as Fisher’s F under the null-hypothesis (Draper, Guttman, & Kanemasu, 1971; Pope & Webster, 1972). The reason for this problem is, that the deleted variable is not selected at random, but using a certain rule. In the current procedure, the deleted variable is ﬁxed beforehand, so the basic assumption behind an F-test is violated. The numerical example treated below will illustrate the problems with this F-test. One of the reviewers correctly noted that the various models could be tested from simple to complex (forwards stepwise regression). However, we will not further pursue this possibility, because the problem with the ﬁrst F-test remains. In the ﬁnal step, the adequacy of the ﬁt of the individual consumers is determined using a second F-test. This step is necessary because the ﬁnally selected model may not be adequate, although it is the best ﬁtting one of the four models considered. The size of this test is referred to as the level of selection. Schlich (1995) uses the value pselect ¼ 0:25 and comments: ‘‘The

Fig. 1. Illustration of the models considered in common EPM: (a) vector with local optimum (), (b) circular with global optimum (), (c) elliptical with saddle point (), and (d) quadratic with global minimum ().

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rationale of such a high P-value is to avoid false rejection of consumers, but we are aware that on the average one consumer out of four may just be selected by chance.’’ Determining the best values for pchoice and pselect is beyond the scope of this paper. Our goal is to improve the consumer ﬁt by including higher-numbered PCs, so the actual P-values employed are immaterial as long as they are kept ﬁxed in the comparison.

Table 1 Number of consumers (out of Nc ¼ 482) ﬁtted signiﬁcantly at the level pselect ¼ 0:25 for various models connected to diﬀerent levels of pchoice Model

pchoice ¼ 0:20

pchoice ¼ 0:10

pchoice ¼ 0:05

Quadratic Elliptical Circular Vector Total

49 42 0 163 254 (53%)

25 26 0 196 247 (51%)

14 12 0 212 238 (49%)

3. Materials and methods 3.1. Practical example A preference mapping study was conducted with apples. The sensory data was provided by 10 Dutch descriptive panellists, trained to evaluate apples on a total of twenty-eight odour, taste, mouthfeel and aftertaste attributes. From the 20 apples that were proﬁled, twelve apples were chosen for the consumer test. Eleven apples covered the sensory space well, and an additional apple from the centre of the PCA-plot was chosen to serve as a dummy in the test—to minimise ﬁrst-order eﬀects. Four hundred and eighty nine consumers (and buyers) of fresh apples (18–70 years of age), participated in the consumer test. They were recruited at ﬁve diﬀerent locations spread over Germany and they conducted the test on site. The consumers evaluated appearance, taste and overall liking on a nine-point hedonic scale in two blocks of six products. The order of presentation was randomised over the consumers, but the dummy product always came ﬁrst. After the taste evaluations the consumers ﬁlled in a questionnaire with questions on usage, attitude and demographics. The data on the liking of taste was used to relate to the sensory data. Seven consumers were excluded from the analysis because they expressed the same preference for all eleven products. In summary, the matrices X and Y are dimensioned 11 28 and 11 482, respectively.

signiﬁcantly worse and is therefore preferable (the null hypothesis). Rather surprisingly, the circular model is not chosen at all. Here the violation of the underlying assumption of an F-test manifests itself, i.e. the deleted variable should be picked at random. The circular model can only alter the ﬁt of the consumer preferences by adding numbers to the vector model ﬁt that are all positive (c > 0) or negative (c < 0). By contrast, the quadratic and elliptical models have coeﬃcients c1 and c2 that can vary separately. It is the larger ﬂexibility of the quadratic and elliptical model that causes the test to seemingly behave as a genuine F-test. Closer examination shows that there is only a single quadratic model that attains its optimum inside the preference circle. In addition, there is one quadratic model that attains its minimum inside the preference circle. The remaining elliptical and quadratic models have a saddle point, either inside or outside the preference circle. Fig. 2 is the external preference map that is obtained by plotting the (global or local) optima of the individual consumers in the space spanned by the ﬁrst two PCs of the sensory data. Since these PCs explain 87% of the total variance in the sensory data, this plot should present a rather accurate overview.

3.2. Computations All calculations are performed in Matlab (The Mathworks, Natick, MA) using in-house written routines.

4. Results and discussion Table 1 summarises the consumer ﬁts obtained when using the common implementation of EPM. It is noted that the numbers of consumers ‘signiﬁcantly’ ﬁtted using the quadratic and elliptical model are close to what might be expected when the null hypothesis is true. For example, with pchoice ¼ 0:10 and 247 consumers one expects to stop at 0:10 247 25 quadratic models when, in reality, the simpler elliptical model does not ﬁt

Fig. 2. External preference map in the space spanned by PCs one and two of the sensory data: products (o), (scaled) key attributes (*) and global or local optima for consumers ﬁtted according to a vector (.), elliptical (e) or quadratic (q) model, respectively.

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459

Schlich (1995) and McEwan (1996) emphasise that the elliptical and quadratic models are seldom used in sensory practice because they are often diﬃcult to interpret. We mention another diﬃculty, which is of statistical nature. It is revealed when inspecting the variance-covariance matrix of the regression coeﬃcients. For the elliptical model, this matrix is proportional to 2 3 b2 c1 c2 b1 6 b1 50

8 0:3

0:1 7 6 7 6 120

0:5

0:3 7 Ve e6 b2 8 7 4 c1 0:3 0:5 0:02 0:01 5 0:05 c2 0:1 0:3 0:01 while for the quadratic model one 2 b1 b2 c1 6 b1 82

14 0:4 6 6 b2 14 121

0:6 V q e6 6 c1 0:4

0:6 0:02 6 4 c2

0:3

0:3 0:01 d

2 0:3 0:01

has c2

0:3

0:3

0:01 0:06 0:01

d

3

2 7 7 0:3 7 7

0:01 7 7 0:01 5 0:09

where ‘ ’ signiﬁes ‘proportional to’. The elements on the principal diagonal represent the variance of the estimates, while the oﬀ-diagonal elements measure the pair-wise correlation. One observes that the coeﬃcients c1 and c2 are anti-correlated. In other words, an increase of c1 accompanies a decrease of c2 and vice versa. (This eﬀect is well known for the slope and intercept in leastsquares straight-line ﬁtting.) Being anti-correlated, the coeﬃcients tend to have opposite sign, which favours the occurrence of saddle points. Saddle points indeed abound for the current data set because the coeﬃcients c1 and c2 are tiny compared to the coeﬃcients b1 and b2 . This is illustrated in Fig. 3 for the elliptical models (31 are chosen by the ﬁrst F-test, of which 26 yield a signiﬁcant ﬁt according to the second F-test). It is further observed that the coeﬃcients c1 and c2 scatter rather erratic around zero, while the coeﬃcients b1 and b2 follow smooth distributions. The coeﬃcients b1 and b2 are anti-correlated too. However, this does not inﬂuence the occurrence of saddle-points. It merely illustrates the fact that regression coeﬃcient estimates are diﬃcult to interpret for a multiple linear regression model. Finally, it is noted that the consumer ﬁt is hardly aﬀected by the value of pchoice . This can be explained from the relatively small contribution of the coeﬃcients c1 , c2 and d. The preceding results seem to imply that increasing the number of PCs, while restricting oneself to vector models, is a reasonable avenue for the data at hand. Starting at the third PC, the percentage consumers signiﬁcantly ﬁtted increases much faster than the explained variance of the sensory data (Fig. 4). Stated diﬀerently, this plot suggests that some of the higher-numbered PCs carry useful information. However, at the same time, it

Fig. 3. Frequency distribution of coeﬃcient estimates for 31 elliptical models (pchoice ¼ 0:10): (a) linear terms and (b) quadratic terms. The coeﬃcient estimates for the quadratic terms are multiplied by a factor 100 to bring them in the same range as the estimates for the linear terms.

Fig. 4. Percentage of cumulative variance of sensory data explained by adding PCs (^) and corresponding percentage of consumers signiﬁcantly ﬁtted using pselect ¼ 0:25 ().

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illustrates the danger of over-ﬁtting the consumer data. With 11 products, 10 PCs can exactly represent the sensory as well as the consumer data owing to the mean centring. As a result, the consumer ﬁt must always reach the value 100% for 10 PCs. The observed percentage increases rapidly until the seventh PC is included (95%), after which it levels oﬀ. However, with 11 products, seven PCs and mean centring, there are only 11 7

1 ¼ 3 degrees of freedom. Consequently, this excellent ﬁt (95%) should be regarded with extreme caution. Thus, a decision rule for acceptability of an additional PC is required. We have indicated that the F-test for model choice is problematic, so we prefer not to use it: selecting PCs using this F-test would lead to the same problems. As an alternative for this test, we have developed the following heuristic decision rule. The percentage consumer ﬁt is calculated for all combinations of two out of ﬁve PCs. (None of the individual PCs ﬁtted the consumers in the present study signiﬁcantly.) With only two PCs included in the model, the risk of overﬁtting should be minimal. The results are displayed graphically (Fig. 5). One expects that PCs that carry negligible information will behave similarly. By contrast, information rich PCs should stand out or form a trend. It is seen that the second PC contributes most to consumer ﬁt. There is a clear trend (decreasing consumer ﬁt) starting at the second PC and ending at PC six, followed by a plateau. This result suggests that one should stop at ﬁve PCs, otherwise additional consumers are predominantly ﬁtted by chance. Including ﬁve PCs leads to 80% of the consumers signiﬁcantly ﬁtted. Optimising the consumers (locally) on the preference circle leads to a plot similar to Fig. 2 (not shown). Consequently, the interpretation of e.g. consumer segmentation does not alter much. However, being based on more consumers, it may lead to more conﬁdence in the modelling procedure. Some ﬁnal remarks seem to be in order. First, including higher-numbered PCs need not work in all cases. In

other words, it is not a panacea. For example, including the third PC did not substantially increase the number of consumers ﬁtted by the model in the study of Monteleone et al. (1998). Second, the exclusive choice for vector models is too restrictive if one expects that a global optimum is contained in the product range, see Hough and Sa´nchez (1998) for an illustrative example. Third, Jaeger, Wakeling, and MacFie (2000) improve the consumer ﬁt by weighing the sensory data and representing the consumer data in a reduced space using PCA. Their work is important because it enables bringing in expert knowledge into an empirical modelling procedure.

5. Conclusions Common EPM considers polynomial models of the ﬁrst two PCs of the sensory data. The current results suggest that the consumer ﬁt in EPM can be substantially improved by using vector models based on more PCs (similar results have been observed for other data sets). The beneﬁt of improved consumer ﬁt lies in a higher stability of post-processing results such as consumer segmentation. Conversely, when this higher stability is not required, one may decide to recruit fewer consumers. This may translate into substantial cost reduction. We have argued against the ﬁrst F-test in common EPM, because the test statistic is not distributed as Fisher’s F under the null-hypothesis. At this point, we have only been able to develop a heuristic decision rule for the number of PCs to keep in the ﬁt. We believe that developing a formal decision rule is an interesting subject for future research. One of the reviewers noted that ANOVA on the sensory PCs (assuming at least duplicate measurements have been made) ensures that only discriminatory PCs are included in the modelling. This would compliment the current PC selection procedure. We recommend that practitioners of EPM consider several variations of EPM to arrive at their ﬁnal results. Including the third PC has been proposed in combination with the following models: vector and elliptical (Arditti, 1997) or vector, circular and elliptical (Elmore et al., 1999). The current variation is simplest in the sense that it is restricted to (linear) vector models. This will often allow one to go higher than three PCs without exhausting the degrees of freedom and subsequently overﬁtting the consumer data.

Acknowledgements Fig. 5. Percentage of consumers signiﬁcantly ﬁtted (pselect ¼ 0:25) using two PCs versus the number of one of the PCs involved in the ﬁt. The numbers identify the companion PC that has led to the best ﬁt.

This research was carried out with ﬁnancial support of Inova Fruit bv. The constructive criticism of the reviewers is appreciated by the authors.

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