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Tests on asymmetry for ordered categorical variables.

Ingo Klein1, Monika Doll1

  • 1Department of Statistics and Econometrics, University of Erlangen-Nürnberg, Nürnberg, Germany.

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This study introduces new skewness measures for ordered categorical data, common in social sciences. These novel statistical methods offer improved analysis for behavioral and educational research.

Keywords:
62Ordered categorical variablesmaximal invariantsskewness analysisskewness ordering

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Area of Science:

  • Statistics
  • Social Sciences
  • Behavioral Sciences

Background:

  • Skewness is a standard statistical concept for continuous and discrete variables.
  • Limited literature exists on skewness measures for ordered categorical variables, despite their prevalence in social and behavioral sciences.

Purpose of the Study:

  • To propose new skewness functionals for ordered categorical variables.
  • To ensure these measures are invariant under strictly increasing transformations.
  • To develop a new class of skewness tests with good power behavior.

Main Methods:

  • Developed skewness functionals based on maximal-invariants for ordered categorical data.
  • Demonstrated that the proposed functionals preserve skewness ordering.
  • Derived the asymptotic distribution for the new skewness statistic.
  • Evaluated the power of the corresponding skewness tests.

Main Results:

  • A new class of skewness functionals for ordered categorical variables was proposed.
  • The proposed functionals maintain a meaningful order of skewness.
  • The asymptotic distribution of the skewness statistic was derived.
  • Skewness tests demonstrated good power performance.

Conclusions:

  • The new skewness measures provide a valuable tool for analyzing ordered categorical data in behavioral, educational, and social sciences.
  • The developed skewness tests are effective for detecting skewness in this data type.
  • The findings contribute to a more robust statistical analysis of common data structures in applied research.