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

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.
• 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.