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 ork-means cluster analysis. To Obtain a Multidimensional Scaling Analysis
1. From the menus choose:
Analyze>Scale>Multidimensional Scaling... 2. Select at least four numeric variables for analysis.
3. In the Distances group, select eitherData are distancesorCreate 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:
v Specify the shape of the distance matrix when data are distances. v Specify the distance measure to use when creating distances from data.
Multidimensional Scaling Shape of Data
If your active dataset represents distances among a set of objects or represents distances between two sets of objects, specify the shape of your data matrix in order to get the correct results.
Note: You cannot selectSquare symmetricif the Model dialog box specifies row conditionality.
Multidimensional Scaling Create Measure
Multidimensional scaling uses dissimilarity data to create a scaling solution. If your data are multivariate data (values of measured variables), you must create dissimilarity data in order to compute a
multidimensional scaling solution. You can specify the details of creating dissimilarity measures from your data.
Measure.Allows you to specify the dissimilarity measure for your analysis. Select one alternative from the Measure group corresponding to your type of data, and then choose one of the measures from the drop-down list corresponding to that type of measure. Available alternatives are:
v Interval. Euclidean distance, Squared Euclidean distance, Chebychev, Block, Minkowski, or Customized.
v Counts. Chi-square measure or Phi-square measure.
v Binary. Euclidean distance, Squared Euclidean distance, Size difference, Pattern difference, Variance, or Lance and Williams.
Create Distance Matrix.Allows you to choose the unit of analysis. Alternatives are Between variables or Between cases.
Transform Values.In certain cases, such as when variables are measured on very different scales, you may want to standardize values before computing proximities (not applicable to binary data). Choose a standardization method from the Standardize drop-down list. If no standardization is required, choose None.
Multidimensional Scaling Model
Correct estimation of a multidimensional scaling model depends on aspects of the data and the model itself.
Level of Measurement.Allows you to specify the level of your data. Alternatives are Ordinal, Interval, or Ratio. If your variables are ordinal, selectingUntie tied observations requests that the variables be treated as continuous variables, so that ties (equal values for different cases) are resolved optimally. Conditionality.Allows you to specify which comparisons are meaningful. Alternatives are Matrix, Row, or Unconditional.
Dimensions.Allows you to specify the dimensionality of the scaling solution(s). One solution is calculated for each number in the range. Specify integers between 1 and 6; a minimum of 1 is allowed only if you selectEuclidean distanceas the scaling model. For a single solution, specify the same number for minimum and maximum.
Scaling Model. Allows you to specify the assumptions by which the scaling is performed. Available alternatives are Euclidean distance or Individual differences Euclidean distance (also known as INDSCAL). For the Individual differences Euclidean distance model, you can selectAllow negative subject weights, if appropriate for your data.
Multidimensional Scaling Options
You can specify options for your multidimensional scaling analysis.
Display.Allows you to select various types of output. Available options are Group plots, Individual subject plots, Data matrix, and Model and options summary.
Criteria.Allows you to determine when iteration should stop. To change the defaults, enter values for S-stress convergence,Minimum s-stress value, andMaximum iterations.
Treat distances less than n as missing.Distances that are less than this value are excluded from the analysis.
ALSCAL Command Additional Features
The command syntax language also allows you to:v Use three additional model types, known as ASCAL, AINDS, and GEMSCAL in the literature about multidimensional scaling.
v Carry out polynomial transformations on interval and ratio data. v Analyze similarities (rather than distances) with ordinal data. v Analyze nominal data.
v Save various coordinate and weight matrices into files and read them back in for analysis. v Constrain multidimensional unfolding.