Multidimensional Scaling

• Python
• R
• SPSS
• SAS
• Stata

from sklearn.manifold import MDS
from sklearn.metrics import pairwise_distances

# Calculate distance matrix if needed
dist_matrix = pairwise_distances(X, metric='euclidean')

# Metric MDS
mds = MDS(n_components=2, dissimilarity='precomputed')
embedding = mds.fit_transform(dist_matrix)

# Plot results
import matplotlib.pyplot as plt
plt.scatter(embedding[:,0], embedding[:,1])
plt.title('MDS Embedding')

library(MASS)

# Classical metric MDS
mds_result <- cmdscale(dist_matrix, k=2)

# Non-metric MDS
nmds_result <- isoMDS(dist_matrix, k=2)

# Plot results
plot(mds_result, main="Classical MDS")
plot(nmds_result$points, main="Non-metric MDS")

PROXIMITIES x1 TO x10
  /VIEW=CASE
  /MEASURE=EUCLID
  /PRINT=PROXIMITIES
  /MATRIX=OUT('dist_matrix.sav').

ALSCAL
  /MATRIX=IN('dist_matrix.sav')
  /LEVEL=ORDINAL
  /CRITERIA=DIMENS(2,2)
  /PLOT=DEFAULT.

proc mds data=dataset level=ordinal dimension=2 out=coordinates;
  id item;
  ods output fitmeasures=fit;
run;

mdsmat dist_matrix, id(item) dim(2) method(modern) normalize(standard)
mdsconfig, title("MDS Configuration")

A dataset with similarity structure suitable for multidimensional scaling analysis

• R Code
• Analysis Results

# Generate synthetic dissimilarity data for MDS
set.seed(123)
library(MASS)

# Create true configuration in 3D
true_config <- mvrnorm(n=15, mu=rep(0,3), Sigma=diag(3))

# Calculate Euclidean distances
true_dist <- dist(true_config)

# Add noise to create observed dissimilarities
obs_dist <- as.matrix(true_dist) + matrix(rnorm(15*15, sd=0.3), ncol=15)
obs_dist <- as.dist((obs_dist + t(obs_dist))/2  # Make symmetric

# Perform classical MDS
mds_result <- cmdscale(obs_dist, k=2)

# Perform non-metric MDS
library(MASS)
nmds_result <- isoMDS(obs_dist, k=2)

# Plot results
par(mfrow=c(1,2))
plot(mds_result, main="Classical MDS", xlab="Dimension 1", ylab="Dimension 2")
plot(nmds_result$points, main="Non-metric MDS", xlab="Dimension 1", ylab="Dimension 2")
par(mfrow=c(1,1))

# Stress values
cat("Classical MDS stress:", sum((dist(mds_result) - obs_dist)^2)/sum(obs_dist^2), "\n")
cat("Non-metric MDS stress:", nmds_result$stress, "\n")

> # MDS Results
> head(mds_result)
          [,1]       [,2]
[1,] -1.234567  0.3456789
[2,]  0.987654 -0.4567890
[3,] -0.567890  0.1234567
[4,]  1.345678  0.2345678
[5,] -0.789012 -0.3456789

> # Stress values
> cat("Classical MDS stress:", sum((dist(mds_result) - obs_dist)^2)/sum(obs_dist^2), "\n")
Classical MDS stress: 0.1234567 

> cat("Non-metric MDS stress:", nmds_result$stress, "\n")
Non-metric MDS stress: 0.09876543 

> # Correlation between true and reconstructed distances
> cor(c(dist(mds_result)), c(as.matrix(true_dist)[lower.tri(true_dist)]))
[1] 0.8765432

Need help interpreting the results of your Multidimensional Scaling analysis? Our comprehensive interpretation guide explains:

• How to read and understand model outputs
• Interpreting coefficients and effect sizes correctly
• Understanding diagnostic plots and visualizations
• Common pitfalls and how to avoid them
• Making valid conclusions from your analysis