# Introduction to MDS

**
Michael Friendly **

Psychology 6140

### 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.
- Similarity or dissimilarity can be measured directly (e.g., by
ratings) or indirectly (e.g., by confusion, substitutability,
etc.).
- Dissimilarity can be considered as a measure of distance in a
psychological space.
- From interitem distances, one can recover the locations of points
in space.

### 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.
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.
- Direct judgments
- Incomplete designs
- Stimulus confusion data
- Sorting and clustering techniques

### Internal analyses (same data)

- Interpret axes or directions in the MDS space
- Map clusters onto MDS solution
- Network model: Pathfinder algorithm

### External analyses (additional data)

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

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).
- Model M1 Metric scaling or replicated MDS
- Model M2 Subject's dissimilarities are a power function of
group distances
- Model M3 Individual differences a la INDSCAL
- 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.
### Cluster analysis

- Hierarchical clustering (Proc CLUSTER)
- Additive, overlapping clusters (Proc OVERCLUS)

### Correspondence analysis

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