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Overview
In spite of being designed for judgment data, MDS can be used to analyze any correlation matrix, treating correlation as a type of similarity measure. That is, the higher the correlation of two variables, the closer they will be located in the map created by MDS. Though it is possible to use MDS with objective distance data and with quantitative variables in general, it is more common to use factor analysis to group such variables, or to use Q-mode factor analysis or cluster analysis when grouping cases, when dimensions are objective and measurable. Nonetheless, because MDS does not require assumptions of linearity, metricity, or multivariate normality, sometimes it is preferred over factor analysis for these reasons even for objective data. On the other hand, MDS does not take account of control relationships as factor analysis does. Pros and cons of MDS vs. factor analysis are discussed below. MDS is popular in marketing research for brand comparisons, and in psychology, where it has been used to study the dimensionality of personality traits. Other uses include analysis of particular academic disciplines using citation data (Small, 1999) and any application involving ratings, rankings, differences in perceptions, or voting.
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The models supported by the SPSS ALSCAL module are:
ALSCAL VARIABLES = V1 TO Vn /SHAPE = ASYMMETRIC /CONDITION = ROW /MODEL = GEMSCAL /CRITERIA = DIM(4) DIRECTIONS(4)
The SPSS syntax for the example is:
ALSCAL
VARIABLES=Grief Savor Surprise Love Exhaustion Wrong Anger Pulling Meets Revulsion Pain KnowFear Sleep
/SHAPE=SYMMETRIC
/LEVEL=INTERVAL
/CONDITION=MATRIX
/MODEL=EUCLID
/CRITERIA=CONVERGE(0.001) STRESSMIN(0.005) ITER(30) CUTOFF(0) DIMENS(2,3)
/PLOT=DEFAULT ALL
/PRINT=DATA HEADER
Iteration history for the 3 dimensional solution (in squared distances)
Young's S-stress formula 1 is used.
Iteration S-stress Improvement
1 .14137
2 .12308 .01829
3 .12218 .00089
Iterations stopped because
S-stress improvement is less than .001000
Stress and squared correlation (RSQ) in distances
RSQ values are the proportion of variance of the scaled data (disparities)
in the partition (row, matrix, or entire data) which
is accounted for by their corresponding distances.
Stress values are Kruskal's stress formula 1.
For matrix
Stress = .10148 RSQ = .91731
Configuration derived in 3 dimensions
Stimulus Coordinates
Dimension
Stimulus Stimulus 1 2 3
Number Name
1 Grief .7614 -.8057 -.2675
2 Savor -.9389 .1779 .6737
3 Surprise -1.9272 1.4084 -.1769
4 Love -1.4629 -.2019 -.0737
5 Exhausti .1238 -1.2525 .5999
6 Wrong .9674 -.1513 1.0382
7 Anger 2.3366 .4615 .3991
8 Pulling -.8300 .9047 -.5853
9 Meets -1.6104 .3462 .6658
10 Revulsio .8550 -.6452 -.8355
11 Pain .7070 -.1818 -1.1268
12 KnowFear 1.6198 1.8280 -.0883
13 Sleep -.6014 -1.8883 -.2228
Here it can be seen there is a love-savor-meets cluster as well as a grief-revulsion-pain-wrong cluster. Anger is closer to the latter cluster than the former. Additional observations might be made on the basis of clustering. The axes are more difficult to interpret than the clusters, but it might be said there are two axes: the horizontal love vs. anger axis, and a vertical sleep vs. alertness axis (inferring that fear of plane crash equates to alertness). However, there is subjectivity and ambiguity. One might use multiple expert interpreters to validate a modal interpretation. Note also, the higher the stress for the solution, the less reliable the location of objects in MDS space and hence the less reliable the interpretation.
For comparison, here is the three-dimensional solution for the same dataset, graphically reflecting the coordinates above:
The three-dimensional map is harder to read. Looking at the table of stimulus coordinates aids in the interpretation. The clusters and first two dimensions remain largely the same. Dimension 1 is still love-surprise-meets on the negative end to anger on the positive pole. Likewise, dimension 2 is still sleep-exhaustion on the negative pole to knowfear-surprise on the positive pole. The third dimension is very difficult to interpret (suggesting the two-dimensional solution, being more interpretable while yielding the same clusters, may be better). It goes from pain on the negative pole to wrong on the positive pole, with smaller coordinate values and less well differentiated poles
It is also possible in the input data table to have one or more sourceid variables (ex., to set up groups for men vs. women). Thus DATA LIST / r_id c_id men women. would be followed by four columns of data: the cell row id, the column row id, the proximity score for that cell for men, and the proximity score for that cell for women. Thus one would be entering two data matrices. The SPSS manual describes other data entry options.
In SPSS, select Analyze, Scale, Multidimensional Scaling (PROXSCAL)(note you must have purchased and installed the SPSS Categories add-on to see this menu choice); in the Multidimensional Scaling: Data Format dialog box which opens, specify the your data type as illustrated below (the illustration shows default selections). Note that like ALSCAL, PROXSCAL can create proximities from raw data. Note also that INDSCAL models can be implemented by specifying multiple sources in the Data Format dialog.
It may be desirable to run the analysis once specifying proximities as interval and once as ordinal. The run with the lower stress is the better model. If stress is similar for both runs, the proximity data can be said to approach being metric.
The "Apply transformations" section applies only for multiple data sources as in INDSCAL models, where "Across all sources simultaneously" is selected for global rather than local analysis. Global, or unconditional, analysis is appropriate when there are multiple matrices which are similar in nature.
Note that PROXSCAL supports four alternative scaling models:
PROXSCAL VARIABLES=Grief Savor Surprise Love Exhaustion Wrong Anger Pulling Meets Revulsion Pain
KnowFear Sleep
/SHAPE=LOWER
/INITIAL=SIMPLEX
/TRANSFORMATION=RATIO
/PROXIMITIES=DISSIMILARITIES
/ACCELERATION=NONE
/CRITERIA=DIMENSIONS(2,3) MAXITER(100) DIFFSTRESS(.0001) MINSTRESS(.0001)
/PRINT=COMMON DISTANCES TRANSFORMATIONS INPUT HISTORY STRESS DECOMPOSITION
/PLOT=STRESS COMMON.