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Analysis Using R. 1.Finds the classical scaling solution. 2. Computes the Trace criterion & Magnitude criterion to determine the # of dimensions. (1*). R>voles_mds<- cmdscale(watervoles, k=13, eig=TRUE). R>voles_mds$eig

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Analysis using r

Analysis Using R

1.Finds the classical scaling solution.

2. Computes the Trace criterion & Magnitude

criterion to determine the # of dimensions.


R>voles_mds<- cmdscale(watervoles, k=13, eig=TRUE)


[1] 7.359910e-01 2.626003e-01 1.492622e-01 6.990457e-02 2.956972e-02

1.931184e-02 8.326673e-17 -1.139451e-02 -1.279569e-02 -2.849924e-02

[11] -4.251502e-02 -5.255450e-02 -7.406143e-02

Analysis using r

We have 13 eigenvalues, some positive & some negative.

R>sum(abs(voles_mds$eig[1:2]))/ sum (abs(voles_mds$eig))

[1] 0.6708889

This code satisfies Magnitude Criterion. (2*)

R>sum((voles_mds$eig[1:2])^2/ sum((voles_mds$eig)^2)

[1] 0.9391378

Which is the criteria suggested by Mardia et al.;

the above code is exactly:

k n

∑│λ│ / ∑ │λ│

i=1 i=1

0.6708889 & .9391378 are large enough to suggest that the original distances

Between water vole populations can be represented adequately

in two dimensions. (3*)

Analysis using r

R>x<- voles_mds$points[, 1] 1: Coordinate 1

R>y<-voles_mds$points[, 2] 2: Coordinate 2

R>plot(x,y,xlab= "Coordinate 1", ylab= "Coordinate 2", xlim= range(x) * 1.2,

type= "n")

R>text(x,y, labels= colnames(watervoles))

6 British populations are

close to populations living

in the Alps, Yugoslavia,

Germany, Norway, &

Pyrenees I. (Arvicola terrestris)

These British populations are

distant from the populations in

Pyrenees II, North Spain

and South Spain.

(Arvicola sapidus) (4*)

Analysis using r

Minimum Spanning Tree (MDS solution) highlight possible distortions. (5,6*)

Links of a minimum spanning tree of the proximity matrix may be plotted onto

the two-dimensional scaling representation to identify distortions (if any) produced

by scaling solutions.

These distortions look like nearby points on the plot that are not linked by an edge of

a tree. (7*)


R>st<- mst(watervoles)

R>plot(x,y,xlab= "Coordinate 1", ylab= "Coordinate 2", xlim= range (x) * 1.2,

type= "n")

R>for (i in 1:nrow(watervoles)) {

+ w1<- which(st[i, ] == 1)

+ segments(x[i], y[i], x[w1], y[w1]) }

R>text(x, y, labels= colnames(watervoles))

Analysis using r

The apparent closeness of distortions.

the populations, Germany &

Norway, by the points

suggested here in this MDS

solution, does not reflect

accurately their calculated

dissimilarity. (8*)

Analysis using r

Non-Metric Scaling distortions.

Example: Analyzing Voting Behavior among Republicans.

R Function isoMDS from package “MASS”


R>data("voting", package= "HSAUR")

R>voting_mds<- isoMDS(voting)

Analysis using r

Two-Dimensional Solution: distortions.

R>x<- voting_mds$points[, 1]

R>y<- voting_mds$points[, 2]

R>plot(x,y, xlab="Coordinate 1", ylab= "Coordinate 2",

+ xlim= range(voting_mds$points[, 1]) *1.2,

+ type= "n" )

R>text(x,y, labels= colnames(voting))

R>voting_sh<- Shepard(voting[ lower.tri(voting)],

+ voting_mds$points

Figure suggests that

there is more variation

in voting behavior

among Republicans. (*9)

Analysis using r

R>library("MASS") distortions.

R>voting_sh<- Shepard(voting[lower.tri(voting)],

+ voting_mds$points)

R>plot(voting_sh, pc= ".", xlab= "Dissimilarity",

+ ylab= "Distance", xlim= range(voting_sh$x),

+ ylim= range(voting_sh$x))

R>lines(voting_sh$x, voting_sh$yf, type= "S")

Quality of a multidimensional scaling can be assessed

informally by plotting the original dissimilarities & the

distances obtained from a multidimensional scaling

in a scatterplot.

Ideally, all the lines should fall on the bisecting line.