## Cluster Analysis

Cluster analysis comprises a set of statistical techniques that aim to group “objects” into homogenous subsets. The objects can be people or products. For example, cluster analysis can be used to segment people (consumers) into subsets based on their liking ratings for a set of products. Such consumer segmentation is an essential step in preference mapping

^{1}, where the goal is to understand drivers of consumer liking, and cluster analysis is used to summarize differences among consumers in their likes and dislikes. Cluster analysis can also be used to cluster products instead of people, in an effort to identify groups of similar products, for example on the basis of trained panel sensory evaluations.

There are two common types of clustering methods: hierarchical methods and partitioning methods^{2,3}. In the most common form of hierarchical clustering, the computation starts by assigning each object to its own cluster. Then, in a process of agglomeration, the most “similar” objects (in a statistical sense) are combined to form a new cluster, followed, in subsequent steps, by successively less similar objects. At each step in the hierarchy, the number of clusters decreases, until at the end all objects are combined into one large cluster. Hierarchical clustering programs typically produce a dendogram, a tree-like graphic representation of which objects are combined at each stage of agglomeration. The major limbs of the dendogram identify the major object groupings.

In the partitioning approach, the researcher must first specify the number of clusters that he/she is interested in. Objects are initially assigned to clusters on a random basis or on the basis of some prior knowledge or analysis. Using an iterative algorithm, the program reassigns each object to clusters until no further improvement in within-cluster homogeneity is achieved. The analysis is repeated for different numbers of clusters of interest to the researcher.

Among the several variants of agglomerative clustering, Ward’s method is one of the most popular and can be found in many statistical software packages. It is frequently used for consumer segmentation in sensory research. *K-means* is a widely used partitioning algorithm and is also available in many statistical software packages.

Each of the two methods of clustering (and each variation of the two methods) uses a different rule for forming clusters and therefore can lead to different answers when applied to the same set of data. Since all rules are essentially ad hoc, it is not possible to say in principle whether one approach is more correct than another. MacFie (2007)

^{1} discusses the strengths and weaknesses of different agglomerative hierarchical methods in the context of preference mapping. Wajrock et al. (2008)

^{4} compare hierarchical to partitioning methods for clustering.

None of the clustering approaches described above provide a definitive answer to the question of how many clusters exist. In practice, it is up to the researcher using these approaches to decide how many clusters make sense, considering the interpretability of the results and the number of objects per cluster that result from the analysis. For example, in segmenting a sample of 200 consumers based on their overall liking ratings for a set of products, a researcher might examine the results of analyses based on two to five clusters, and select the solution that is the most meaningful in light of the research objectives and that has sufficient objects (people) per cluster to be actionable.

Recently, latent class models

^{5,6} have been proposed as alternatives to traditional hierarchical and partitioning methods. Latent class models can be viewed as probabilistic extensions of k-means

^{7}, where objects are assigned to a cluster based on the probability of belonging to that cluster given the observed data. Latent class models offer several potential advantages over traditional methods: they are based on a statistical model (the mixture of underlying probability distributions); they provide various diagnostics that are useful in determining the number of clusters; and they allow for “fuzzy” cluster assignment, where objects can belong to more than one cluster (with different probabilities). Applications of latent class cluster models in sensory consumer research are described by Popper, Kroll & Magidson (2004)

^{8}, Séménou et al. 2007

^{9}, and Meullenet, Xiong & Findlay (2007)

^{10}.

Other approaches to cluster analysis include models that utilize external data in the process of clustering. For example, in a preference mapping application, such models can be used to cluster consumers’ liking ratings in a way that optimizes the predictability of liking ratings from trained panel evaluations (the external data). Cleaver & Wedel (2001)^{11} provide an example of a latent class regression model applied to sensory and consumer data. A different approach to utilizing external data was developed by Vigneau & Qannari (2002)^{12} and applied in a preference mapping context.

## References

^{1 }MacFie, H. (2007). Preference mapping and food product development. In H. MacFie (ed.), Consumer-Led Product Development. Cambridge, UK, Woodhead Publishing, pp. 551-592.

^{2 }Jacobsen, T. & Gunderson, R.W. (1986). Applied cluster analysis. In J.R. Piggott (ed.), Statistical Procedures in Food Research. New York, NY, Elsevier Science Publishing, pp. 361-408.

^{3} Myers, J.H. & Mullet, G. M. (2003). Managerial Applications of Multivariate Analysis in Marketing. Chicago, IL, American Marketing Association, pp.238-304.

^{4 }Wajrock, S., Antille, N., Rytz, A., Pineau, N. & Hager, C. (2008). Partitioning methods outperform hierarchical methods for clustering consumers in preference mapping. Food Quality and Preference, 19, 662-669.

^{5 }Courcoux,, P., & Chavanne, P.C. (2001). Preference mapping using a latent class vector model. Food Quality and Preference, 12, 369-372.

^{6 }Magidson, J. & Vermunt, J.K. (2004). Latent Class Models. In D. Kaplan (ed.), The Sage Handbook of Quantitative Methodology for the Social Sciences. Thousand Oaks, CA, Sage Publications, pp. 175-198.

^{7 }Magidson, J. & Vermunt, J. (2002). Data use: latent class modeling as a probabilistic extension of k-means clustering. Quirk’s Marketing Research Review, 20 (March), 77-80.

^{8} Popper, R., Kroll, J., & Magidson, J. (2004). Application of latent class models to food product development: A case study. Sawtooth Conference Proceedings, 155-170.

^{9 }Meullenet, J-F., Xiong, R., & Findlay, C.J. (2007). Multivariate and Probabilistic Analyses of Sensory Science Problems. Oxford, UK, Blackwell Publishing, pp. 111-127.

^{10 }Séménou, M., Courcoux, P., Cardinal, M., Nicod, H., & Ouisse, A. (2007). Preference study using a latent class approach. Analysis of European preferences for smoked salmon. Food Quality and Preference, 18, 720-728.

^{11} Cleaver, G. & Wedel, M. (2001). Identifying random-scoring respondents in sensory research using finite mixture regression models. Food Quality and Preference, 12, 373-384.

^{12} Vigneau, E. & Qannari, E.M. (2002). Segmentation of consumers taking account of external data. A clustering of variables approach. Food Quality and Preference, 13, 515-521.

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