Consistent Small-Sample Variances for Six Gamma-Family Measures of Ordinal Association

Area of Science:

• Statistics
• Psychometrics
• Ordinal Data Analysis

Background:

• Gamma-family measures are bivariate ordinal correlation coefficients.
• Accurate confidence intervals (CIs) are crucial for interpreting these measures.
• The Cliff consistent (CC) variance estimator previously offered superior CI accuracy but was limited to a subset of gamma-family measures.

Purpose of the Study:

• To derive the Cliff consistent (CC) variance estimator for the remaining six gamma-family measures.
• To compare the accuracy of confidence intervals constructed using the CC variance with those from other available variance estimators.
• To provide accessible R computer code for calculating all 10 gamma-family measures and their CC variances.

Main Methods:

• Derivation of the CC variance for six underrepresented gamma-family measures.
• Comparative analysis of CIs generated by different variance estimators via simulations.
• Illustration of methods using real-world data from the Schedule for Nonadaptive and Adaptive Personality (Disinhibition and Avoidance scales).

Main Results:

• The CC variance was successfully derived for the remaining six gamma-family measures.
• Simulations indicated that CIs constructed with the CC variance generally demonstrated improved accuracy compared to other estimators.
• The study provides a comprehensive set of tools for analyzing all 10 gamma-family measures.

Conclusions:

• The extension of the CC variance to all gamma-family measures enhances their utility in statistical analysis.
• Researchers can now obtain more reliable confidence intervals for a wider range of ordinal correlation measures.
• The availability of R code facilitates the practical application of these advanced statistical methods.