# Multidimensional scaling

Thu, 14 Jan 1999 22:39:08 +0300

>Dear A.,
>
>Could you educate me and the list about what INDSCAL is or other names
>it goes by ? Likewise, could you possibly provide a introductory review
>or reference or Stat's package that performs this type of analysis ?
>Many thanks in advance.
>
>All the best,
>
>Howard
>****************************************
>Howard Gershenfeld, M.D., Ph.D.
>Dept. of Psychiatry
>Univ. of Texas Southwestern Medical Center
>Dallas,Tx. 75235-8898
>214-648-8000
>214-648-8030 (Fax)
>E-mail: hgersh@mednet.swmed.edu
>

Multidimensional scaling (MDS) is a family of multivariate techniques
designed to represent similarity data as a spatial (usually two or
three-dimensional) map. Input data matrix (or several matrices) for MDS
consists of coefficients that show degree of (dis)similarity (closeness,
distances) between column and row elements (this feature makes MDS similar
to cluster analysis).
Elements after analysis must be represented by points of multidimensional
space and coordinates of these points are chosen so distances between them
as much as possible are close to the original data (this feature relates MDS
to factor analysis). There exist several indices of fit that measure the
degree of correspondence between actual similarities and distances
calculated from coordinates of elements.

Similarity data can be gathered in a variety of ways:
- By direct pairwise comparisons: subject judges similarity between pairs of
elements and expresses it as a number (category, rank etc.). Physical
proximity of objects (for instance, real distances between clients during
group therapy) may be used as well;
- By calculating probabilities of co-occurrence or conditional
probabilities. For instance, probability of transition from one
psychological state to another, frequency of social interactions between
persons, errors of misperception, probability of two elements to be placed
together into common groups during sorting by subject(s);
- By computing indices of (dis)similarity for binary, frequency or ratio
variables from usual rectangular data matrix, similarity of profiles;

Several unique features make MDS very attractive to researcher:
1. Several data matrices can be analyzed simultaneously.
(a) Replicated MDS assumes that behind each matrix lies essentially the same
configuration of stimuli, and differences between matrices results from
measurement errors (response bias). For instance, subject can judge
similarity of elements using different sets of construct or from different
perspectives and researcher can extract configuration that does not depend
from surface wording;
(b) Weighted MDS (or Individual Differences Scaling - INDSCAL) introduces
concepts of group stimulus space and weight space. Group space describe set
of dimensions that are common to several similarity matrices (i.e. different
persons, or different occasions of testing for a person), whereas weights
describe the importance each person attaches to each dimension. The more
weight is close to 1, the more important is a dimension for a given person.
Vector of weights describes individual space of meaning. Special index of
weirdness shows how unusual is individual space comparing to average set of
weights.
2. With MDS can be analyzed symmetrical and asymmetrical matrices, with
various patterns of missing values, of different conditionality, measurement
level (metrical, nonmetrical) and shape. Rectangular matrices for me is of
special interest: using this type of experiment subject judges similarity of
stimuli from two different sets (for instance, some ideal figures or
literature personages and real persons).

Procedures for conducting MDS can be found in major statistical packages
(SPSS, SYSTAT, BMDP, Statistica, SAS, see, for example,
http://www.spss.com, http://www.statsoft.com. BTW, on site
http://www.statsoft.com there is an excellent downloadable book on
statistics in HTML format, devoted to various multivariate methods). From
site http://www.xlstat.com you can download freeware copy of XLSTAT - add-on
to MS Excel that includes MDS among other multivariate techniques). And, at
last, free program ALSCAL (this algorithm is used in commercial statistical
packages) by Forrest Young - executable file for PC, bibliography and even
source code in Fortran - you will find at
http://forrest.psych.unc.edu/research/ALSCAL.html

Sincerely,