#### Date of Award

12-1997

#### Level of Access

Open-Access Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Psychology

#### Advisor

Colin Martindale

#### Second Committee Member

Joel A. Gold

#### Third Committee Member

William Halteman

#### Abstract

In a square asymmetric matrix, the relationships among objects in the lower triangular half-matrix, differ from the relationships among the same objects in the upper triangular half. Square, asymmetric matrices can arise in similarity and preference data, when the direction of comparison is important. An asymmetric matrix can be rendered symmetric by averaging corresponding entries above and below the main diagonal. The difference between the original and the symmetric matrix is purely asymmetric, or skew-symmetric. The symmetric and skew-symmetric pans are orthogonal. An eigenvector-eigenvalue decomposition analyses the asymmetries into rank 2 skew-symmetric matrices, having an optimum least squares fit to the asymmetries (Gower, 1977).

In this dissertation I derive an alternating least squares, nonmetric analogue of the canonical decomposition of asymmetry, suitable for ordinal-level data. In simulation studies, the nonmetric version gives better metric and nonmetric recovery, than does the canonical decomposition, when the asymmetries have been distorted by a range-compressing monotonic transform. The nonmetric technique appears to out-perform the canonical decomposition in detecting simplexes, and possibly in recovering multiplicative bias coefficients. However, canonical decomposition gives superior recovery after range-expanding monotonic transforms, and in the presence of error.

An eigenvalue ratio test is proposed for determining the number of eigenvectors to extract in the canonical decomposition. The test quantifies changes in the slope of the log eigenvalue plot. In simulation studies the test appears to maintain its anticipated Type I error rate. The test is "under-powered", which may help it to extract only well-identified eigenvectors.

Finally, directional similarity judgments were collected for all possible pairs of exemplars of two semantic categories. The exemplars differed in typicality. After Tversky (1977) this should produce asymmetries related to the typicality. No asymmetries were found, however. Power analysis indicated that a correlation ratio for the asymmetries of .05 could have been detected 90% of the time. An extreme groups analysis also did not indicate asymmetry. The first eigenvector underlying the symmetric data, however, was highly correlated with typicality. Hence, Tversky's model was not supported.

#### Recommended Citation

Borkum, Jonathan M., "Making Sense of Direction: Proximity and Order in Asymmetric Paired Comparison Data" (1997). *Electronic Theses and Dissertations*. 2329.

https://digitalcommons.library.umaine.edu/etd/2329