Introduction to MDS

Michael Friendly
Psychology 6140

1. Basic Ideas of Scaling


Deriving a quantitative scale to represent an internal, psychological response or reaction to stimuli. E.g., preference, liking, perceived brightness or seriousness of crimes, brand preference, voting.

Unidimensional scaling:

Assume all stimuli lie along a single dimension. Determine the locations of stimuli on this dimension from observed data (e.g., ratings, rank orders) by fitting a model (e.g., law of categorical judgment). Determine goodness of fit.

Similarity scaling:

Rather than specifying a dimension, obtain data on the similarity or dissimilarity between stimuli. Use this data to determine the number of dimensions necessary to fit the similarity data, and the locations of points in a multidimensional space.

2. Metric vs. Non-Metric MDS

Metric MDS (Torgerson's method)

Interpoint distances (ratio scaled data) can be converted to a matrix of "scalar products" which can be factored to give stimulus coordinates directly. If distance data are only on an interval scale, an "additive constant" can be estimated to convert to ratio-scaled distances.

Non-metric MDS

Uses only the ordinal (rank order) properties of the data, and recovers the "psychophysical function" relating observed (dis)similarity to distance in psychological space. Nonmetric methods make very weak assumptions about the data, but work because ordinal information on interpoint distances provide a large number of constraints.

3. Individual Differences MDS

A variety of methods are available for studying the nature of individual differences in MDS. They require one (dis)similarity matrix for each subject.

Replicated MDS

All subjects are assumed to have the same underlying configuration for the stimuli. Subjects may be allowed to differ in their psychophysical function (AlSCAL: CONDITON=MATRIX).

Weighted MDS (INDSCAL Model)

Subjects are assumed to use the same dimensions, but each subject may weight the dimensions differently. A subject's weights on the dimensions indicate how important or salient the dimensions are for him/her.

4. Methods for obtaining similarity judgments & data

  1. Direct judgments
  2. Incomplete designs
  3. Stimulus confusion data
  4. Sorting and clustering techniques

5. Techniques for interpreting MDS Solutions

Internal analyses (same data)

  1. Interpret axes or directions in the MDS space
  2. Map clusters onto MDS solution
  3. Network model: Pathfinder algorithm

External analyses (additional data)

Use ratings of the objects on properties, attributes, or preference to help interpret the dimensions.
  1. Vector model: Regard each property as a "vector" in the multidimensional space. Fit by multiple regression.
  2. Ideal point model: Regard each property as having an "ideal point" in the multidimensional space.
  3. Canonical correlations: Find linear combinations of the properties which relate most highly to the dimensions.

6. Confirmatory MDS

A recent model for MDS which provides statistical tests of hypotheses, comparisons among submodels, and estimation of individual standard errors for stimulus points. Requires metric assumptions about the data. Implemented in Ramsay's MULTISCALE or PROC MLSCALE (SAS V5.18).
  1. Model M1 Metric scaling or replicated MDS
  2. Model M2 Subject's dissimilarities are a power function of group distances
  3. Model M3 Individual differences a la INDSCAL
  4. Model M4 Allows unequal standard errors for each stimulus point

Hubert's Quadratic Assignment approach

An alternative way to test a hypothesis about the multidimensional structure is to specify the hypothesis as a (lower triangular) matrix of hypothesized distances. The goodness of fit of the hypothesis is measured by the correlation between entries in the data and hypothesis matrices, and Hubert provides a permutation test of significance.

7. Related techinques

Cluster analysis

  1. Hierarchical clustering (Proc CLUSTER)
  2. Additive, overlapping clusters (Proc OVERCLUS)

Correspondence analysis

Analysis of cross-classified frequency data to display pattern of association among row and column objects.