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Generalized Procrustes Analysis

Generalized Procrustes Analysis (GPA) is a method for determining the degree of agreement, or consensus, among data matrices. For instance, consider 20 judges who rate four brands of coffee on 10 attributes (e.g., bitterness, richness, smoothness). The ratings for each judge can be recorded in a two-dimensional (10 × 4) matrix. GPA then can be used to determine the extent to which the judges agree in their views of the four brands of coffee. To the extent the 20 judges do not agree, individual differences in the patterns of ratings can also be examined with GPA. At the heart of the analysis is a consensus configuration, which is derived through a process of scaling, rotating, and averaging the original rating matrices. Each judge's ratings can be compared to this consensus configuration, and an overall consensus proportion can be computed that indicates the degree of similarity among the judge's views of the four coffees. GPA is also extremely flexible and can accommodate any number of matrices of varying dimensions. Qualitative judgments or quantitative data can be analyzed, and the matrices must be matched on only a single dimension. For instance, each of the 20 judges could rate the four brands of coffee using a different set of attributes as well as a different number of attributes. The flexibility of GPA can also be seen in the variety of studies in which it has been employed. Researchers have examined individuals' perceptions of food, products, medical treatments, genetic engineering, and personality traits using this technique.

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.

Table 1 Managers' Ratings of Six Employees
Manager 1 Manager 2
John Bob Amy Jan Fred Jill John Bob Amy Jan Fred Jill
Extraverted 1 1 5 4 5 3 Blunt 2 3 1 3 4 1
Sharing 5 5 2 1 2 3 Patient 5 1 4 2 2 3
Motivated 2 2 4 3 4 1 Creative 5 2 3 4 1 3
Funny 1 2 2 2 4 1 Outgoing 2 4 1 4 3 3
Loud 1 2 2 1 4 1
Manager 3 Manager 4
Outgoing 2 4 2 4 5 3 Easygoing 2 3 1 3 4 3
Carefree 2 4 3 4 4 2 Outgoing 1 3 1 5 5 1
Generous 3 2 4 2 2 5 Nurturing 3 3 5 5 1 5
Trusting 3 2 4 3 2 4 Calm 5 2 4 1 2 5
Organized 4 3 3 2 3 3 Intelligent 5 3 5 2 3 3
Athletic 3 2 2 2 4 2

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.

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