# Multidimensional Scaling (PROXSCAL)

Multidimensional scaling attempts to find the structure in a set of proximity measures between objects. This process is accomplished by assigning observations to specific locations in a conceptual low-dimensional space such that the distances between points in the space match the given (dis)similarities as closely as possible. The result is a least-squares representation of the objects in that low-dimensional space, which, in many cases, will help you to further understand your data.

**Example.** Multidimensional scaling can be very useful in
determining perceptual relationships. For example, when considering
your product image, you can conduct a survey to obtain a dataset that
describes the perceived similarity (or proximity) of your product
to those of your competitors. Using these proximities and independent
variables (such as price), you can try to determine which variables
are important to how people view these products, and you can adjust
your image accordingly.

**Statistics and plots.** Iteration history, stress measures,
stress decomposition, coordinates of the common space, object distances
within the final configuration, individual space weights, individual
spaces, transformed proximities, transformed independent variables,
stress plots, common space scatterplots, individual space weight scatterplots,
individual spaces scatterplots, transformation plots, Shepard residual
plots, and independent variables transformation plots.

Multidimensional Scaling Data Considerations

**Data.** Data can be supplied in the form of proximity matrices
or variables that are converted into proximity matrices. The matrices
can be formatted in columns or across columns. The proximities can
be treated on the ratio, interval, ordinal, or spline scaling levels.

**Assumptions.** At least three variables must be specified.
The number of dimensions cannot exceed the number of objects minus
one. Dimensionality reduction is omitted if combined with multiple
random starts. If only one source is specified, all models are equivalent
to the identity model; therefore, the analysis defaults to the identity
model.

**Related procedures.** Scaling all variables at the numerical
level corresponds to standard multidimensional scaling analysis.

To Obtain a Multidimensional Scaling

This feature requires the Categories option.

- From the menus choose:
This opens the Data Format dialog box.

- Specify the format of your data:
**Data Format.**Specify whether your data consist of proximity measures or you want to create proximities from the data.**Number of Sources.**If your data are proximities, specify whether you have a single source or multiple sources of proximity measures.**One Source.**If there is one source of proximities, specify whether your dataset is formatted with the proximities in a matrix across the columns or in a single column with two separate variables to identify the row and column of each proximity.- The proximities are in a matrix across columns. The proximity matrix is spread across a number of columns equal to the number of objects. This leads to the Proximities in Matrices across Columns dialog box.
- The proximities are in a single column. The proximity matrix is collapsed into a single column, or variable. Two additional variables, identifying the row and column for each cell, are necessary. This leads to the Proximities in One Column dialog box.

**Multiple Sources.**If there are multiple sources of proximities, specify whether the dataset is formatted with the proximities in stacked matrices across columns, in multiple columns with one source per column, or in a single column.- The proximities are in stacked matrices across columns. The proximity matrices are spread across a number of columns equal to the number of objects and are stacked above one another across a number of rows equal to the number of objects times the number of sources. This leads to the Proximities in Matrices across Columns dialog box.
- The proximities are in columns, one source per column. The proximity matrices are collapsed into multiple columns, or variables. Two additional variables, identifying the row and column for each cell, are necessary. This leads to the Proximities in Columns dialog box.
- The proximites are stacked in a single column. The proximity matrices are collapsed into a single column, or variable. Three additional variables, identifying the row, column, and source for each cell, are necessary. This leads to the Proximities in One Column dialog box.

- Click Define.

This procedure pastes PROXSCAL command syntax.