Matrices with a given number of violations of Regular Minimality |
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Authors: | Ehtibar N. Dzhafarov,Ali Ü nlü ,Hans Colonius |
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Affiliation: | a Department of Psychological Sciences, Purdue University, 703 Third Street West Lafayette, IN 47907, USAb Faculty of Statistics, Dortmund Technical University, Germanyc Department of Psychology, Oldenburg University, Germany |
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Abstract: | A row (or column) of an n×n matrix complies with Regular Minimality (RM) if it has a unique minimum entry which is also a unique minimum entry in its column (respectively, row). The number of violations of RM in a matrix is defined as the number of rows (equivalently, columns) that do not comply with RM. We derive a formula for the proportion of n×n matrices with a given number of violations of RM among all n×n matrices with no tied entries. The proportion of matrices with no more than a given number of violations can be treated as the p-value of a permutation test whose null hypothesis states that all permutations of the entries of a matrix without ties are equiprobable, and the alternative hypothesis states that RM violations occur with lower probability than predicted by the null hypothesis. A matrix with ties is treated as being represented by all matrices without ties that have the same set of strict inequalities among their entries. |
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Keywords: | Discriminability Rank order matrices Permutation test Regular Minimality Tied ranks |
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