An investigation of curvature in psychological spaces

Many psychological theories and models that posit psychological spaces assume these spaces are geometrically flat, such as distances measured in a plane. Many models rely on multidimensional scaling (MDS) to provide instantiations for these spaces. MDS is a method that associates distances with similarities for a set of stimuli, producing a representational space in which the stimuli are points. An early step in preparing data for scaling is the addition of a constant to each distance measurement. We demonstrate that the optimal choice of additive constant can minimize curvature that may be present in the data, and therefore distort the derived configuration. We discuss three new tests for uncovering curvature in psychological proximity data prior to scaling. In the first test, data are assumed to lie on a ratio scale, and can be applied to any such data. In the second test, data are assumed to lie on a ratio scale and it is further assumed that some subsets of points are collinear with and between pairs of points. In the third test, data are assumed to lie on an ordinal scale, with additional assumptions concerning the configuration of points. The performance and power of each of these tests with simulated noisy data sets are assayed, revealing the tests maintain sufficient power in the presence of moderate noise. The tests are applied to data from three experiments.