A variety of joint space multidimensional scaling (MDS) methods have been utilized for the spatial analysis of two- or three-way dominance data involving subjects’ preferences, choices, considerations, intentions, etc. so as to provide a parsimonious spatial depiction of the underlying relevant dimensions, attributes, stimuli, and/or subjects’ utility structures in the same joint space representation. We demonstrate that care must be taken with respect to a key assumption in existent joint space MDS models such that all estimated dimensions are utilized by each and every subject in the sample, as this assumption can lead to serious distortions with respect to the derived joint spaces. We develop a new Bayesian dimension selection methodology for the multidimensional unfolding model which accommodates heterogeneity with respect to such dimensional utilization at the individual subject level for the analysis of two or three-way dominance data. A consumer psychology application regarding the preference for Over-the-Counter (OTC) analgesics is provided. We conclude by discussing the practical implications of the results, as well as directions for future research.
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Note that PREFSCAL assumes Euclidean distance while all other models, including the proposed model, assume squared Euclidean distance. As such, we inputted Euclidean distances and added the same amount of error for PREFSCAL analysis.
We thank an anonymous reviewer for this point.
These derived joint spaces are usually referred as degenerate solutions meaning that the derived joint space is extremely uninformative despite good fit to the data (Heiser 1989). In MDU models, this usually takes the form of a wide separation between row and column points. Note that we use 0.00001 for default convergence criteria for ALSCAL as the default convergence option would keep the final solution close to the initial solution.
One reviewer suggested a different specification of PREFSCAL to mimic the proposed model with a three-way dimension weighting option combined with almost missing values. Here, one can replicate the two-way data I=100 times where row i of the i-th replication has a weight one and very small numbers (i.e., 1e–06) elsewhere. We tried this approach but didn’t find any significant improvement over the more traditional PREFSCAL approach.
Theoretically, the weighted unfolding model should be able to account for such dimensional heterogeneity where dimensions not utilized for a particular subject would have an associated weight equal to zero. However, we observed from the small synthetic example presented earlier that this does not always occur in practice.
As noted by one reviewer, many nonmetric unfolding models can yield degenerate solutions when the transformation can become a constant combined with a solution that has all between set distances equal to the same constant (see, e.g., Borg & Groenen 2005). The proposed model avoids this type of degeneracy problem as there is no slope parameter on the utility U ir .
Also note that BIC (Bayesian information criterion) could be a rough approximation to the logarithm of the Bayes factor (see Kass & Raftery 1995, for complete review on Bayes factor).
Note that the proposed HDMDU is uniquely determined as discussed in Section 3.1 As such, no transformation was applied to the results of HDMDU model.
Bayes factor between HDMDU and BSMDU would not be significant (see Kass & Raftery 1995, for review of the Bayes factor).
The computing time for this data is approximately 3 hours run on a 3.3 GHZ computer with a Windows operating system.
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Park, J., Rajagopal, P. & DeSarbo, W.S. A New Heterogeneous Multidimensional Unfolding Procedure. Psychometrika 77, 263–287 (2012). https://doi.org/10.1007/s11336-012-9256-6
- multidimensional unfolding
- dimension selection
- Bayesian multidimensional scaling
- consumer psychology