Introduction

Multidimensional scaling aims to reduce the dimensionality of data while preserving the original relationships between observations. This paper focuses on the theoretical features of the methods and highlights the key factors to consider when conducting sensitivity analysis on MDS results.

Input Dissimilarity Metrics

One of the primary factors influencing the stability of MDS solutions is the choice of dissimilarity metric used to measure the distances between data points. Commonly used metrics include Euclidean distance, Manhattan distance, and correlation-based distances, among others. The choice of metric may significantly impact the MDS output, so sensitivity analysis should involve comparing results across different distance measures.

#  Calculating dissimilarity matrix using different metrics
library(proxy)
data_matrix <- as.matrix(iris[, 1:4]) # Using the iris dataset as an example
euclidean_dissimilarity <- dist(data_matrix, method = "euclidean")
manhattan_dissimilarity <- dist(data_matrix, method = "manhattan")
correlation_dissimilarity <- as.dist(1 - cor(data_matrix))

Choice of MDS Algorithms

Various MDS algorithms are available, such as classical MDS, metric MDS, and non-metric MDS. Each algorithm has its own assumptions, strengths, and weaknesses. Sensitivity analysis should involve comparing the stability of MDS solutions generated by different algorithms to assess their appropriateness for the data at hand.

# Applying different MDS algorithms
library(MASS)
library(smacof)
cmds_solution <- cmdscale(euclidean_dissimilarity, k = 2)
mds_solution <- smacof::mds(euclidean_dissimilarity, type = "interval", ndim = 2)$conf
nmds_solution <- smacof::mds(euclidean_dissimilarity, type = "ordinal", ndim = 2)$conf

Dimensionality

Determining the appropriate number of dimensions to represent the data is crucial for MDS analysis. Sensitivity analysis should explore the effect of varying the number of dimensions on the stability of the MDS solution. This can be accomplished using various approaches, such as the scree plot method, stress values, or parallel analysis.

#  Assessing dimensionality using the scree plot method
stress_values <- c()
for (k in 1:10) {
  mds_solution <- smacof::mds(euclidean_dissimilarity, type = "interval", ndim = k)
  stress_values <- c(stress_values, mds_solution$stress)
}
plot(1:10, stress_values, type = "b", xlab = "Number of dimensions", ylab = "Stress")

Assessment Criteria

Assessing the robustness of MDS solutions requires the use of assessment criteria to quantify the stability of the results. Commonly used criteria include Procrustes analysis, stress values, and goodness-of-fit measures like the Shepard diagram. Sensitivity analysis should compare the MDS results based on these assessment criteria to determine the most stable solution.

Conclusion

In conclusion, sensitivity analysis is an essential step in evaluating the robustness of MDS solutions. This paper has highlighted the key factors to consider when performing such an analysis, including input dissimilarity metrics, choice of MDS algorithms, dimensionality, and assessment criteria. By considering these factors and employing the appropriate R markdown code examples, researchers can ensure they select the most stable and appropriate MDS solution for their data analysis needs.

References

Borg, I., & Groenen, P. J. (2005). Modern Multidimensional Scaling: Theory and Applications. Springer Science & Business Media.

Cox, T. F., & Cox, M. A. A. (2000). Multidimensional Scaling. Chapman and Hall/CRC.

Kruskal, J. B., & Wish, M. (1978). Multidimensional Scaling. Sage Publications.