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Related Experiment Videos

Goodman and Kruskal's Gamma Coefficient for Ordinalized Bivariate Normal Distributions.

Alessandro Barbiero1, Asmerilda Hitaj2

  • 1Department of Economics, Management, and Quantitative Methods, Università degli Studi di Milano, Via Conservatorio, 7, 20122, Milan, Italy. alessandro.barbiero@unimi.it.

Psychometrika
|October 27, 2020
PubMed
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Discretizing a bivariate normal distribution affects ordinal association measures. Goodman and Kruskal

Area of Science:

  • Statistics
  • Probability Theory
  • Ordinal Data Analysis

Background:

  • Bivariate normal distributions are fundamental in statistical modeling.
  • Ordinal association measures like Goodman and Kruskal's gamma are crucial for analyzing ranked data.
  • Discretization transforms continuous variables into ordinal categories, potentially altering association metrics.

Purpose of the Study:

  • To investigate the impact of discretization on ordinal association measures derived from a bivariate normal distribution.
  • To compare Goodman and Kruskal's gamma with Kendall's rank correlation before and after discretization.
  • To propose a method for constructing bivariate ordinal variables with specific marginal distributions and association levels.

Main Methods:

  • Considered a bivariate normal distribution with linear correlation.
Keywords:
Bivariate normal distributionDiscretizationGamma coefficientLatent variableOrdinal association

Related Experiment Videos

  • Discretized the continuous components using assigned sets of thresholds.
  • Calculated Goodman and Kruskal's gamma and Kendall's rank correlation for the resulting ordinal variables.
  • Explored various experimental settings by varying thresholds and marginal distributions.

    • Main Results:

    • The absolute value of Goodman and Kruskal's gamma was consistently higher than Kendall's rank correlation after discretization.
    • This discrepancy decreased with an increased number of categories.
    • Equally probable categories also reduced the difference between the two association measures.

    Conclusions:

    • Discretization systematically inflates the measured ordinal association compared to the continuous case.
    • The extent of this inflation depends on the number of categories and the distribution of thresholds.
    • A method is proposed for creating bivariate ordinal variables with controlled marginals and association by ordinalizing a bivariate normal distribution.