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Preference mapping using a latent class vector model
Food Quality and Preference 12 (2001) 369–372 www.elsevier.com/locate/foodqual
Preference mapping using a latent class vector model Ph. Courcoux *, P.C. Chavanne ENITIAA/INRA, Unite´ de Statistique Applique´e a` la Caracte´risation des Aliments BP 82225, 44 322 Nantes Cedex 03, France Received 7 September 2000; received in revised form 19 January 2001; accepted 22 January 2001
Abstract An application of a latent class vector model to preference data is presented. Analysing hedonic ratings, this technique realises simultaneously the clustering of consumers in homogeneous classes on the basis of their preferences and the joint representation of products and classes using a vector model. A probabilistic assumption allows performance of significance tests on the number of clusters and provides a useful tool for interpreting results of preference tests. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Latent class; Vector model; Preference data; EM algorithm
1. Introduction The internal preference mapping on hedonic rating data is generally realised using the classical MDPREF analysis (Caroll, 1972, 1980; Greenhoff & MacFie, 1994; Schiffman, Reynolds, & Young, 1981). This technique consists in the singular value decomposition of the data matrix (centred by consumer) and the resulting biplot allows a graphical interpretation of individual preferences. Unfortunately, such an analysis does not always reveal a clear clustering of individuals and the number of dimensions to observe is not easy to choose as the decrease of singular values is generally quite regular. As a response to these difficulties for the practitioner, the latent class vector model (De Soete & Winsberg, 1993) allows simultaneous performance of the clustering of consumers into a small number of segments and the joint representation of stimuli and classes using a vector model (as in the classical MDPREF model). This method provides tools for a direct and simple interpretation of the acceptance of products and of segmentation of the panel. Based on a probabilistic model, this technique can be followed by statistical significance tests on the number of classes and dimensions to
* Corresponding author. Tel.: +33-251-78-5438; fax: +33-251-785436. E-mail address: courcoux@enitiaa- nantes.fr (Ph. Courcoux).
explore. The resulting parsimonious biplot can help the practitioner to model the panel preferences using a small number of parameters.
2. The latent class vector model We are supposed to analyse the evaluation of the preferences of H consumers for N products on a rating scale using a complete design. The latent class vector model assumes the existence of T homogeneous classes whose weights are denoted lt (T is chosen a priori). In each class t, the preference vector yh of a subject h for the N products is the observation of a random variable normally distributed with mean ut and common variance 2I (where I denotes the identity matrix in RN). The mean vectors ut are related to a joint configuration of products and clusters based on the classical vector model: the preference for a product in a cluster is a linear function of the orthogonal projection of the point representing this product onto the vector representing this cluster. The problem appears as a finite mixture of multivariate normal distributions. Estimation procedure is based on maximum likelihood and a latent class approach is used for the determination of segments from the preference data, using an EM algorithm (Dempster, Laird, & Rubin, 1977; McLachlan & Basford, 1988). Details of the estimation procedure can be found in De Soete and Winsberg (1993).
0950-3293/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0950-3293(01)00027-1
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P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
model against the unconstrained model (i.e. with T dimensions). The efficiency of this significance test procedure has been studied on simulated data by De Soete and Winsberg (1993).
Parameters of this clustering procedure are the configurations of the N products and the T classes in an R-dimensional space (R is chosen a priori with R4T), the weights lt of these clusters and the common variance 2. The resulting biplot of stimuli and classes can be interpreted like a MDPREF configuration. One of the main advantages of the probabilistic assumption is to provide a likelihood ratio test which can be performed to determine the adequate number of latent classes and dimensions of the configuration. As in most mixture models, the sampling distribution of the likelihood ratio is unknown, but can be obtained by Monte Carlo simulations (Aitkin, Anderson, & Hinde, 1981). One can test first the number of classes by comparing the log likelihood obtained for T and T+1 classes models. When the appropriate number T of clusters is determined, it is possible to test the R-dimensional
3. Application on workshop data set We analysed the preferences of 320 consumers for 28 grape/raspberry beverages (only consumers with no missing data were selected from the original data). Consumers rated the overall liking for products on a ninbe-point hedonic scale. We applied a latent class vector model on this data set after centring each consumer preference vector. Table 1 gives the resulting log likelihood for models with 1–5 classes and 1–5 dimensions. As previously mentioned, when comparing models with different numbers of classes, these models are not nested and the asymptotic distribution of likelihood ratio statistic is not known. We performed 100 simulations based on a Monte Carlo procedure. Table 2 presents results of these significance tests. Significance tests show that the appropriate model has four classes and three dimensions. Configurations of products and classes are plotted in the Fig. 1. The weights of the four classes are, respectively, 0.07, 0.21, 0.04 and 0.68. The two main classes (C2 and C4) are discriminated on the second preference dimension.
Table 1 Log likelihood of the latent class vector model R T (number of classes) (number of dimensions) 1 2 3 1 2 3 4 5
18 350
18 328 -18 230
4
18 328 18 218 18 185
5 18 328 18 194 18 144 18 120
18 328 18 194 18 135 18 095 18 073
Table 2 Significance tests for (a) the number of classes and (b) the number of dimensions based on 100 Monte Carlo simulations T (number ofclasses)
1 2 3 4
Monte Carlo significance test T vs. T+1 classes
R (number of dimensions)
Likelihood ratio statistic
Probability
240 90 130 94
0.00 0.00 0.00 0.08
1 2 3 4
Fig. 1. Plot of the four-classes/three-dimensions solution.
Monte Carlo significance test R vs. 4 dimensions Likelihood ratio statistic
Probability
416 148 48
0.00 0.00 0.07
P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
371
Fig. 2. Centred preference average scores (vectors ut) for clusters 2 and 4.
Fig. 3. Projection of individual preferences for the four clusters of consumers on the first two dimensions of the latent class vector model solution.
Classes 1 and 3 seem to gather more marginal consumers. The products 414 and 813 seem to be the more discriminative among the four classes (product 414 is one of the more appreciated by consumers of class 2 and rejected in classes 3 and 4; the contrary occurs for product 813).
In each cluster t, the vector ut (means of the individual preferences weighted by the posterior membership probabilities) is helpful to describe the acceptance of products. Fig. 2 confirms that the main differences between the two major classes (clusters 2 and 4) concerns scores obtained by products 414 and 813.
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P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
Although the individual belonging to a class is based on posterior probability, the assignment of a consumer to a class is possible by taking the higher membership probability. Individual consumer preferences can be plotted separately for each cluster. Fig. 3 shows the representation of the four segments in the latent class solution. 4. Conclusion The latent class vector model provides an interesting tool in analysing preference ratings. The simultaneous clustering into a few number of classes and representation in a low dimension space provide a parsimonious model for preference data which can be interpreted in the same manner as the classical MDPREF technique. References Aitkin, M., Anderson, D., & Hinde, J. (1981). Statistical modelling
data on teaching styles. Journal of the Royal Statistical Society, Series A, 144, 419–461. Carroll, J. D. (1972) Individual differences and multidimensional scaling In R. N. Shepard, A. K. Romney, & S. B. Nerlove, Multidimensional scaling: theory and applications in the behavioral sciences (Vol. 1, pp. 105–155). New York: Seminar Press. Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance data). In X. Langermann, & H. Feger, Similarity and choice (pp. 234–289). Bern: Huber. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38. De Soete, G., & Winsberg, S. (1993). A latent class vector model for preference ratings. Journal of Classification, 10, 195–218. Greenhoff, K., & MacFie, H. J. H. (1994). Preference mapping in practice. In H. J. H. MacFie, & D. M.H Thomson, Measurement of food preferences (pp. 137–166). Glasgow: Blackie Academic and Professional. McLachlan, G. J., & Basford, K. E. (1988). Mixture models. New York: Marcel Dekker. Schiffman, S. S., Reynolds, M. L., & Young, F. W. (1981). Introduction to multidimensional scaling. Theory, methods and applications. Orlando: Academic Press.
Preference mapping using a latent class vector model Ph. Courcoux *, P.C. Chavanne ENITIAA/INRA, Unite´ de Statistique Applique´e a` la Caracte´risation des Aliments BP 82225, 44 322 Nantes Cedex 03, France Received 7 September 2000; received in revised form 19 January 2001; accepted 22 January 2001
Abstract An application of a latent class vector model to preference data is presented. Analysing hedonic ratings, this technique realises simultaneously the clustering of consumers in homogeneous classes on the basis of their preferences and the joint representation of products and classes using a vector model. A probabilistic assumption allows performance of significance tests on the number of clusters and provides a useful tool for interpreting results of preference tests. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Latent class; Vector model; Preference data; EM algorithm
1. Introduction The internal preference mapping on hedonic rating data is generally realised using the classical MDPREF analysis (Caroll, 1972, 1980; Greenhoff & MacFie, 1994; Schiffman, Reynolds, & Young, 1981). This technique consists in the singular value decomposition of the data matrix (centred by consumer) and the resulting biplot allows a graphical interpretation of individual preferences. Unfortunately, such an analysis does not always reveal a clear clustering of individuals and the number of dimensions to observe is not easy to choose as the decrease of singular values is generally quite regular. As a response to these difficulties for the practitioner, the latent class vector model (De Soete & Winsberg, 1993) allows simultaneous performance of the clustering of consumers into a small number of segments and the joint representation of stimuli and classes using a vector model (as in the classical MDPREF model). This method provides tools for a direct and simple interpretation of the acceptance of products and of segmentation of the panel. Based on a probabilistic model, this technique can be followed by statistical significance tests on the number of classes and dimensions to
* Corresponding author. Tel.: +33-251-78-5438; fax: +33-251-785436. E-mail address: courcoux@enitiaa- nantes.fr (Ph. Courcoux).
explore. The resulting parsimonious biplot can help the practitioner to model the panel preferences using a small number of parameters.
2. The latent class vector model We are supposed to analyse the evaluation of the preferences of H consumers for N products on a rating scale using a complete design. The latent class vector model assumes the existence of T homogeneous classes whose weights are denoted lt (T is chosen a priori). In each class t, the preference vector yh of a subject h for the N products is the observation of a random variable normally distributed with mean ut and common variance 2I (where I denotes the identity matrix in RN). The mean vectors ut are related to a joint configuration of products and clusters based on the classical vector model: the preference for a product in a cluster is a linear function of the orthogonal projection of the point representing this product onto the vector representing this cluster. The problem appears as a finite mixture of multivariate normal distributions. Estimation procedure is based on maximum likelihood and a latent class approach is used for the determination of segments from the preference data, using an EM algorithm (Dempster, Laird, & Rubin, 1977; McLachlan & Basford, 1988). Details of the estimation procedure can be found in De Soete and Winsberg (1993).
0950-3293/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0950-3293(01)00027-1
370
P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
model against the unconstrained model (i.e. with T dimensions). The efficiency of this significance test procedure has been studied on simulated data by De Soete and Winsberg (1993).
Parameters of this clustering procedure are the configurations of the N products and the T classes in an R-dimensional space (R is chosen a priori with R4T), the weights lt of these clusters and the common variance 2. The resulting biplot of stimuli and classes can be interpreted like a MDPREF configuration. One of the main advantages of the probabilistic assumption is to provide a likelihood ratio test which can be performed to determine the adequate number of latent classes and dimensions of the configuration. As in most mixture models, the sampling distribution of the likelihood ratio is unknown, but can be obtained by Monte Carlo simulations (Aitkin, Anderson, & Hinde, 1981). One can test first the number of classes by comparing the log likelihood obtained for T and T+1 classes models. When the appropriate number T of clusters is determined, it is possible to test the R-dimensional
3. Application on workshop data set We analysed the preferences of 320 consumers for 28 grape/raspberry beverages (only consumers with no missing data were selected from the original data). Consumers rated the overall liking for products on a ninbe-point hedonic scale. We applied a latent class vector model on this data set after centring each consumer preference vector. Table 1 gives the resulting log likelihood for models with 1–5 classes and 1–5 dimensions. As previously mentioned, when comparing models with different numbers of classes, these models are not nested and the asymptotic distribution of likelihood ratio statistic is not known. We performed 100 simulations based on a Monte Carlo procedure. Table 2 presents results of these significance tests. Significance tests show that the appropriate model has four classes and three dimensions. Configurations of products and classes are plotted in the Fig. 1. The weights of the four classes are, respectively, 0.07, 0.21, 0.04 and 0.68. The two main classes (C2 and C4) are discriminated on the second preference dimension.
Table 1 Log likelihood of the latent class vector model R T (number of classes) (number of dimensions) 1 2 3 1 2 3 4 5
18 350
18 328 -18 230
4
18 328 18 218 18 185
5 18 328 18 194 18 144 18 120
18 328 18 194 18 135 18 095 18 073
Table 2 Significance tests for (a) the number of classes and (b) the number of dimensions based on 100 Monte Carlo simulations T (number ofclasses)
1 2 3 4
Monte Carlo significance test T vs. T+1 classes
R (number of dimensions)
Likelihood ratio statistic
Probability
240 90 130 94
0.00 0.00 0.00 0.08
1 2 3 4
Fig. 1. Plot of the four-classes/three-dimensions solution.
Monte Carlo significance test R vs. 4 dimensions Likelihood ratio statistic
Probability
416 148 48
0.00 0.00 0.07
P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
371
Fig. 2. Centred preference average scores (vectors ut) for clusters 2 and 4.
Fig. 3. Projection of individual preferences for the four clusters of consumers on the first two dimensions of the latent class vector model solution.
Classes 1 and 3 seem to gather more marginal consumers. The products 414 and 813 seem to be the more discriminative among the four classes (product 414 is one of the more appreciated by consumers of class 2 and rejected in classes 3 and 4; the contrary occurs for product 813).
In each cluster t, the vector ut (means of the individual preferences weighted by the posterior membership probabilities) is helpful to describe the acceptance of products. Fig. 2 confirms that the main differences between the two major classes (clusters 2 and 4) concerns scores obtained by products 414 and 813.
372
P. Courcoux, P.C. Chavanne / Food Quality and Preference 12 (2001) 369–372
Although the individual belonging to a class is based on posterior probability, the assignment of a consumer to a class is possible by taking the higher membership probability. Individual consumer preferences can be plotted separately for each cluster. Fig. 3 shows the representation of the four segments in the latent class solution. 4. Conclusion The latent class vector model provides an interesting tool in analysing preference ratings. The simultaneous clustering into a few number of classes and representation in a low dimension space provide a parsimonious model for preference data which can be interpreted in the same manner as the classical MDPREF technique. References Aitkin, M., Anderson, D., & Hinde, J. (1981). Statistical modelling
data on teaching styles. Journal of the Royal Statistical Society, Series A, 144, 419–461. Carroll, J. D. (1972) Individual differences and multidimensional scaling In R. N. Shepard, A. K. Romney, & S. B. Nerlove, Multidimensional scaling: theory and applications in the behavioral sciences (Vol. 1, pp. 105–155). New York: Seminar Press. Carroll, J. D. (1980). Models and methods for multidimensional analysis of preferential choice (or other dominance data). In X. Langermann, & H. Feger, Similarity and choice (pp. 234–289). Bern: Huber. Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38. De Soete, G., & Winsberg, S. (1993). A latent class vector model for preference ratings. Journal of Classification, 10, 195–218. Greenhoff, K., & MacFie, H. J. H. (1994). Preference mapping in practice. In H. J. H. MacFie, & D. M.H Thomson, Measurement of food preferences (pp. 137–166). Glasgow: Blackie Academic and Professional. McLachlan, G. J., & Basford, K. E. (1988). Mixture models. New York: Marcel Dekker. Schiffman, S. S., Reynolds, M. L., & Young, F. W. (1981). Introduction to multidimensional scaling. Theory, methods and applications. Orlando: Academic Press.