Generalized Procrustes Analysis

Abbreviated Example

Four managers from a department store freely describe and then rate six of their employees on 5-point scales constructed from their individual descriptive adjectives. A high score on the rating scale indicates that a particular adjective is an accurate description of the employee. The data are reported in Table 1.

The goal of GPA is to assess the degree of similarity among the managers' views of the employees. More specifically, the goal is to determine if the patterns, or profiles, of the six employees are similar across the four managers. Because the focus is on the profiles of the employees, the analysis does not require a fixed set of attributes.

A number of prescaling methods are typically recommended when conducting GPA. As with any scaling method, the investigator must consider the impact of controlling statistical differences in the data. It is well known, for example, that the mean and standard deviation of z scores are equal to zero and one, respectively. Converting any two variables to z scores will thus equate the two variables on their means and standard deviations. With GPA, three scaling methods are recommended: centering, dimensional, and isotropic.

Centering rescales each manager's ratings such that the mean of each attribute is equal to zero. Isotropic scaling “shrinks” or “expands” each manager's ratings to remove individual differences in scale usage. The ratings for those managers who use relatively few scale values will be expanded with a multiplicative constant greater than one, and the ratings for those managers who use relatively more extreme scale values will be shrunken with a multiplicative constant less than one. The isotropic scaling values for the four managers (.75, 1.01, 1.48, and .94, respectively) indicate that the overall range in the third manager's ratings was less than the range in the other managers' ratings. In fact, it can be seen in Table 1 that the third manager did not use the full range of scale values (1–5). Finally, dimensional scaling adjusts for differences in the number of attributes in the managers' matrices. The magnitudes of the original values are uniformly increased or decreased depending on the size of the matrix. In this way, matrices with large numbers of attributes will not spuriously influence the results.

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