Number of Dimensions

These figures show some of the initial output for the categorical principal components analysis. After the iteration history of the algorithm, the model summary, including the eigenvalues of each dimension, is displayed. These eigenvalues are equivalent to those of classical principal components analysis. They are measures of how much variance is accounted for by each dimension.

Figure 1. Iteration history
Iteration history
Figure 2. Model summary
Table showing Cronbach's alpha and variance accounted for in dimensions 1 and 2

The eigenvalues can be used as an indication of how many dimensions are needed. In this example, the default number of dimensions, 2, was used. Is this the right number? As a general rule, when all variables are either single nominal, ordinal, or numerical, the eigenvalue for a dimension should be larger than 1. Since the two-dimensional solution accounts for 94.52% of the variance, a third dimension probably would not add much more information.

For multiple nominal variables, there is no easy rule of thumb to determine the appropriate number of dimensions. If the number of variables is replaced by the total number of categories minus the number of variables, the above rule still holds. But this rule alone would probably allow more dimensions than are needed. When choosing the number of dimensions, the most useful guideline is to keep the number small enough so that meaningful interpretations are possible. The model summary table also shows Cronbach’s alpha (a measure of reliability), which is maximized by the procedure.

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