0

I currently self-study multidimensional scaling. Amongst others, I study Borg & Groenen (2005): Modern multidimensional scaling : theory and applications.

On page 10, they present a real-life data set reported by Wish (1971). Wish (1971) asked 18 students to rate the global similarity of different pairs of nations such as France and China on a 9-point rating scale ranging from 1 = very different to 9 = very similar. Since the data set is publicly available, I wanted to replicate the result in R for practice purposes. As a first step, I wanted to replicate the following configuration also presented in Borg & Groenen (2005, p. 10).

enter image description here

I proceeded as follows:

library(smacof)                       ### this package contains the data set
data(wish)                            ### that is the data set

Since the data sets contains similarity ratings, I applied non metric multidimensional scaling using the isoMDS command of the MASS package. Even though the textbook authors speak of a "two-dimensional MDS configuration", I also tried higher-dimensional solutions. I therefore coded a loop that carries out multidimensional scaling for configurations containing dimensions from 2 to 9. 

X <- c()
for (i in 2:9) {
  MDS <- isoMDS(wish, k = i)
  X <- c(X, MDS$stress)
  plot(MDS$points[,c(1,2)])
  text(MDS$points[, 1], MDS$points[, 2], colnames(as.matrix(wish)), cex=.6, 
  pos = 1)
}
plot(X, type = "b")                    ### this  allowed me to plot the stress levels associated with each configuration

None of the resulting plots resembled the one presented in Borg & Groenen (2005, p. 10). For example, the map for 2 dimensions is as follows:

enter image description here

I checked that the data set is identical to the one reported by Borg & Groenen (2005, p. 10). I also tried out metric scaling as follows:

for (i in 2:9) {
  plot(smacofSym(wish, ndim=i))
}

Again, I was not able to replicate the results reported by Borg & Groenen (2005,p. 10). However, I am not sure if I made any mistake while trying to replicate the results.

DomB
  • 217
  • 1
  • 4
  • 14

1 Answers1

1

Using the base R cmdscale, I get a similar result to Borg & Groenen.

If you look at the structure of wish you will see that it is simply a vector of 66 numbers. I interpret this as the lower triangluar similarity matrix. I convert this into a full dissimilarity matrix so that I can use cmdscale and plot. The positions roughly align with the positions from Borg & Groenen.

library(smacof)
data(wish)

## Construct distance matrix
SM = matrix(0, nrow=12, ncol=12)
SM[lower.tri(SM)] = wish
SM = SM + t(SM)
diag(SM) = 9
DM = 9-SM

## MDS & plotting
MDS = cmdscale(DM)
plot(MDS, pch=20, xlim=c(-4,4), ylim=c(-4,4))
text(MDS, labels = attr(wish, "Labels"), adj=c(0.5,-0.6), cex=0.8)
abline(0.5,0.3, lty=2)
abline(-1,-3.8, lty=2)

Plot of Multi-Dimensional Scaling

G5W
  • 36,531
  • 10
  • 47
  • 80
  • I am a complete idiot. The Wish dataset contains similarity ratings. I therefore should have converted the dataset into dissimilarities. This is what you have done: ('DM = 9-SM' (see above)). Once I incorporated this step I managed to replicate the results as well. – DomB Apr 08 '19 at 13:42
  • If this is the worst mistake that you make in your life, you are a genius. – G5W Apr 08 '19 at 13:44
  • thanks. BTW, I have a follow-up questions. You may be interested in it: https://stats.stackexchange.com/questions/401881/multidimensional-scaling-how-to-determine-stress-per-point-in-r – DomB Apr 08 '19 at 17:52