Discriminant DiSTATIS: A Multi-Way Discriminant Analysis for Distance Matrices, Illustrations with the Sorting Task

The sorting task is commonly used in many sub-fields of psychology to study how people perceive and mentally represent their environment. Compared to related experiments, the sorting task is easy to administer, requires minimal training, and has minimal language requirements—features that make the sorting task particularly well-suited to study novice perception, and to compare perspectives of disparate populations.

The current standard approach to analyze sorting data is to use a method called DiSTATIS—a method that optimally maps the consensus of how participants perceived the stimuli. However, experimenters often intend to investigate not how participants perceived stimuli, but how groups perceived hypothesized categories. Because current methods cannot directly investigate such research questions, experimenters must tinker with available methods in order to address their research questions.

Using current methods, a common approach is to separately analyze each group, and to visually compare the groups’ stimulus maps. However, this approach is problematic for two reasons. First, a stimulus map maximizes differences between stimuli, not between categories, and so a stimulus map is only indirectly relevant to the hypothesized categories. I call this the one-table problem. Second, separate stimulus maps can be visually compared, but not statistically compared. I call this the multi-table problem. In short, for simple ANOVA-like research questions, such as how do the groups differently perceive the categories, current methods cannot appropriately map, quantify, or test the differences between the categories or the differences between the groups.

This inability to appropriately investigate basic research questions on sorting data calls for the development of a new statistical method. To address such research questions requires a method that maximizes the differences between the categories and also the differences between the groups—a multi-way discriminant analysis for distance matrices. In this disser-tation, I developed such a method, called DiDiSTATIS.

In order to develop DiDiSTATIS, I sub-divide the dissertation into three Parts. In Part I, I develop DiMDS to solve the one-table problem: were the hypothesized categories perceived?Instead of mapping the stimuli, DiMDS maps the hypothesized categories, and how the stimuli vary about their categories. DiMDS then quantifies the between-category and within-category variability in order to test how well the hypothesized categories explain the observed sorting pattern.

In Part II, I develop HiDiSTATIS to solve the multi-table problem: did the groups differ?Instead of mapping the participants, HiDiSTATIS maps the groups, and how the participants vary about the groups. HiDiSTATIS then quantifies the between-group and within-group variability in order to test whether the groups differently perceived the stimuli.

In Part III, I integrate DiMDS and HiDiSTATIS to give the general method, discriminant DiSTATIS (DiDiSTATIS), which solves the general problem: did the groups differently per-ceive the hypothesized categories? DiDiSTATIS maps, quantifies, and tests these ANOVA-like group and category effects.

I then provide two examples, in music cognition and sensory evaluation. I also provide a software package written in the statistical language, R, to reproduce the results shown here, and to enable future use of these new methods.