Preface to First Edition - lib

302 MULTIDIMENSIONAL SCALINGn in number, where n is the number of rows (and columns) of the proximitymatrix, and an associated measure of distance between pairs of points. Eachpoint is used to represent one of the stimuli in the resulting spatial model forthe proximities and the objective of a multidimensional approach is to determineboth the dimensionality of the model (i.e., the value of q) that providesan adequate ‘fit’, and the positions of the points in the resulting q-dimensionalspace. Fit is judged by some numerical index of the correspondence betweenthe observed proximities and the inter-point distances. In simple terms thismeans that the larger the perceived distance or dissimilarity between twostimuli (or the smaller their similarity), the further apart should be the pointsrepresenting them in the final geometrical model.A number of inter-point distance measures might be used, but by far themost common is Euclidean distance. For two points, i and j, with q-dimensionalcoordinate values, x i = (x i1 , x i2 , ...,x iq ) and x j = (x j1 , x j2 , ...,x jq ) the Euclideandistance is defined asq∑d ij = √ (x ik − x jk ) 2 .k=1Having decided on a suitable distance measure the problem now becomesone of estimating the coordinate values to represent the stimuli, and this isachieved by optimising the chosen goodness of fit index measuring how wellthe fitted distances match the observed proximities. A variety of optimisationschemes combined with a variety of goodness of fit indices leads to a variety ofMDS methods. For details see, for example, Everitt and Rabe-Hesketh (1997).Here we give a brief account of two methods, classical scaling and non-metricscaling, which will then be used to analyse the two data sets described earlier.17.2.1 Classical Multidimensional ScalingClassical scaling provides one answer to how we estimate q, and the n, q-dimensional, coordinate values x 1 , x 2 , ...,x n , from the observed proximitymatrix, based on the work of Young and Householder (1938). To begin wemust note that there is no unique set of coordinate values since the Euclideandistances involved are unchanged by shifting the whole configuration of pointsfrom one place to another, or by rotation or reflection of the configuration. Inother words, we cannot uniquely determine either the location or the orientationof the configuration. The location problem is usually overcome by placingthe mean vector of the configuration at the origin. The orientation problemmeans that any configuration derived can be subjected to an arbitrary orthogonaltransformation. Such transformations can often be used to facilitate theinterpretation of solutions as will be seen later.To begin our account of the method we shall assume that the proximitymatrix we are dealing with is a matrix of Euclidean distances D derived froma raw data matrix, X. Previously we saw how to calculate Euclidean distances© 2010 by Taylor and Francis Group, LLC