Consensus partition

The inputs of both the hierarchical and the partitioning algorithms are the overall dissimilarities obtained at the panel level. We will also outline strategies of analysis where the input data are the individual (i.e. subjects) dissimilarity matrices. In this strategy of analysis, the aim is to obtain what is usually referred to as a consensus partition. That is, a partition that agrees as much as possible with the partitions given by the subjects. 

The problem of finding a consensus partition that reflects the general point of view of a set of partitions has been studied by several authors. One of the first works is due to R gnier (1983). A bibliographical review of this problem was undertaken by Leclerc and Cucumel (1987). 

Let us denote by U, (/j,...,(/, the partitions of the n products respectively associated with the J subjects. We denote by U the consensus partition of the products that we seek to obtain. Choosing the ARI as a measure of agreement between partitions, the problem of determining the consensus partition leads us to the following criterion to be maximized ... 

Since the number, k, of classes of the consensus partition is not known a priori, Courcoux et al. (2014) suggest performing the previons algorithm with increasing values of k and retaining the value k that corresponds to the maximnm of criterion C. An illustration of this strategy of analysis is outlined when discnssing the case study. 

As pointed out in the section devoted to the statistical treatment of free sorting data, the determination of a consensus partition is based on the maximization of a criterion that reflects the extent to which the subjects agree with this consensus partition. Figure 7.8 shows the evolution of the optimization criterion according to the number of groups of the consensus partition. The maximum value is obtained with seven groups of car body styles. 

Figure 7.8 Evolution of the optimization criterion according to the number of groups of the consensus partition.

Figure 7.9 shows the additive tree resulting from the analysis of dissimilarities between car body styles. The seven groups of the consensus partition are superimposed on the tree. 

Figure 7.9 Additive tree and representation of the consensus partition groups.

Courcoux, R, Faye, P. and Qannari, E. M. (2014). Determination of the consensus partition and cluster analysis of subjects in a free sorting task experiment. Food Quality and Preference, 32, 107-112. 

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