Bootstrapping Multidimensional Scaling Output

Abstract of M.Sc. Project by Henry Potts
Department of Applied Statistics, University of Oxford 

This project uses data from my doctoral work and was supervised by Dr Francis Marriott.

This project presents a novel application of bootstrapping to the results from multidimensional scaling (MDS) analysis. To illustrate this, it uses a data set on the psychology of food rejection. Forty-one subjects answered a battery of thirteen questions - known as a Food Rejection Index (FRI) - about each of nineteen items that are generally rejected. Subjects made their answers on visual analogue scales, the data from which were transformed using an arcsine root transformation. A dissimilarity matrix for the items was constructed for each subject. The dissimilarity index used was based on the correlation between items over the transformed FRI data. MDS was applied to a matrix produced by averaging over all the subjects' dissimilarity matrices.

The new bootstrap procedure involves refitting one item at a time in the MDS output, conditional on the other items being held constant. For each item, bootstrap samples were drawn randomly from the forty-one subjects' dissimilarity matrices with replacement. A new, averaged dissimilarity matrix was constructed for each bootstrap sample and the item of concern refitted into the original solution, the fitting constrained on the other items remaining fixed. This produces a point scatter. An alpha percent confidence region was constructed by fitting a two-dimensional kernel density estimate to the point scatter and drawing an isocontour that excludes alpha percent of the points. Properties of the method were investigated.

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