Bayesian analysis of partial cladograms resulting from free-sorting tasks
The free-sorting task is increasingly being used to compare the sensory qualities (e.g., taste, smell) of food products. In this task, a participant initially sorts the products into groups based on their perceived similarities and then successively combines the two most similar groups until only two remain. These resulting cladograms are typically converted into an overall similarity matrix and analyzed using multidimensional scaling (MDS). While the relative efficiency of this task over pairwise evaluations increases with the number of food products, there is thought to be an upper limit on the number of products one can accurately sort. Thus, studies using this task have focused on 15 or fewer products.
In this paper, we propose methods to handle studies when the number of products is above this limit. We consider a design where each participant sorts partially overlapping subsets of products and propose a Bayesian modeling method to address the inferential challenge created by these partial cladograms. Our method facilitates the combination of information across product subsets for learning the underlying latent values for all products in a comprehensive manner. These latent values are then used to construct the similarity matrix for MDS. This model incorporates variability across participants and can be extended to include covariates to help explain this variability. We demonstrate the validity of this approach via simulation studies and apply it to a study involving 21 products that are combinations of different types and concentrations of astringent and bitter.