Preface to First Edition - lib

ANALYSIS USING R 305that they differed. For such situations, what is needed is a method of multidimensionalscaling, the solutions from which depend only on the rank order ofthe proximities, rather than their actual numerical values. In other words thesolution should be invariant under monotonic transformations of the proximities.Such a method was originally suggested by Shepard (1962a,b) andKruskal (1964a). The quintessential component of the method is the use ofmonotonic regression (see Barlow et al., 1972). In essence the aim is to representthe fitted distances, d ij , as d ij = ˆd ij + ε ij where the disparities ˆd ij aremonotonic with the observed proximities and, subject to this constraint, resemblethe d ij as closely as possible. Algorithms to achieve this are describedin Kruskal (1964b). For a given set of disparities the required coordinates canbe found by minimising some function of the squared differences between theobserved proximities and the derived disparities (generally known as stress inthis context). The procedure then iterates until some convergence criterion issatisfied. Again for details see Kruskal (1964b).17.3 Analysis Using RWe can apply classical scaling to the distance matrix for populations of watervoles using the R function cmdscale. The following code finds the classicalscaling solution and computes the two criteria for assessing the required numberof dimensions as described above.R> data("watervoles", package = "HSAUR2")R> voles_mds voles_mds$eig[1] 7.359910e-01 2.626003e-01 1.492622e-01 6.990457e-02[5] 2.956972e-02 1.931184e-02 8.326673e-17 -1.139451e-02[9] -1.279569e-02 -2.849924e-02 -4.251502e-02 -5.255450e-02[13] -7.406143e-02Note that some of the eigenvalues are negative. The criterion P 2 can be computedbyR> sum(abs(voles_mds$eig[1:2]))/sum(abs(voles_mds$eig))[1] 0.6708889and the criterion suggested by Mardia et al. (1979) isR> sum((voles_mds$eig[1:2])^2)/sum((voles_mds$eig)^2)[1] 0.9391378The two criteria for judging number of dimensions differ considerably, but bothvalues are reasonably large, suggesting that the original distances between thewater vole populations can be represented adequately in two dimensions. Thetwo-dimensional solution can be plotted by extracting the coordinates fromthe points element of the voles_mds object; the plot is shown in Figure 17.1.It appears that the six British populations are close to populations livingin the Alps, Yugoslavia, Germany, Norway and Pyrenees I (consisting of the© 2010 by Taylor and Francis Group, LLC