# Multidimensional Scaling

Multidimensional scaling attempts to find the structure in a set of distance measures between objects or cases. This task is accomplished by assigning observations to specific locations in a conceptual space (usually two- or three-dimensional) such that the distances between points in the space match the given dissimilarities as closely as possible. In many cases, the dimensions of this conceptual space can be interpreted and used to further understand your data.

If you have objectively measured variables, you can use multidimensional scaling as a data reduction technique (the Multidimensional Scaling procedure will compute distances from multivariate data for you, if necessary). Multidimensional scaling can also be applied to subjective ratings of dissimilarity between objects or concepts. Additionally, the Multidimensional Scaling procedure can handle dissimilarity data from multiple sources, as you might have with multiple raters or questionnaire respondents.

Example. How do people perceive relationships between different cars? If you have data from respondents indicating similarity ratings between different makes and models of cars, multidimensional scaling can be used to identify dimensions that describe consumers' perceptions. You might find, for example, that the price and size of a vehicle define a two-dimensional space, which accounts for the similarities that are reported by your respondents.

Statistics. For each model: data matrix, optimally scaled data matrix, S-stress (Young's), stress (Kruskal's), RSQ, stimulus coordinates, average stress and RSQ for each stimulus (RMDS models). For individual difference (INDSCAL) models: subject weights and weirdness index for each subject. For each matrix in replicated multidimensional scaling models: stress and RSQ for each stimulus. Plots: stimulus coordinates (two- or three-dimensional), scatterplot of disparities versus distances.

Multidimensional Scaling Data Considerations

Data. If your data are dissimilarity data, all dissimilarities should be quantitative and should be measured in the same metric. If your data are multivariate data, variables can be quantitative, binary, or count data. Scaling of variables is an important issue--differences in scaling may affect your solution. If your variables have large differences in scaling (for example, one variable is measured in dollars and the other variable is measured in years), consider standardizing them (this process can be done automatically by the Multidimensional Scaling procedure).

Assumptions. The Multidimensional Scaling procedure is relatively free of distributional assumptions. Be sure to select the appropriate measurement level (ordinal, interval, or ratio) in the Multidimensional Scaling Options dialog box so that the results are computed correctly.

Related procedures. If your goal is data reduction, an alternative method to consider is factor analysis, particularly if your variables are quantitative. If you want to identify groups of similar cases, consider supplementing your multidimensional scaling analysis with a hierarchical or k-means cluster analysis.

To Obtain a Multidimensional Scaling Analysis

This feature requires the Statistics Base option.

1. From the menus choose:

Analyze > Scale > Multidimensional Scaling...

2. Select at least four numeric variables for analysis.
3. In the Distances group, select either Data are distances or Create distances from data.
4. If you select Create distances from data, you can also select a grouping variable for individual matrices. The grouping variable can be numeric or string.

Optionally, you can also:

• Specify the shape of the distance matrix when data are distances.
• Specify the distance measure to use when creating distances from data.

This procedure pastes ALSCAL command syntax.