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Factoring a 2 x 2 contingency table.

Stanley Luck1

  • 1Science, Technology and Research Institute of Delaware, Wilmington, DE, United States of America.

Plos One
|October 26, 2019
PubMed
Summary
This summary is machine-generated.

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A new two-component proportional representation framework explains 2x2 contingency tables. This reveals that the choice between odds ratios and relative risk depends on application-specific trade-offs, not just mathematical properties.

Area of Science:

  • Statistics
  • Data Analysis
  • Contingency Table Analysis

Background:

  • 2x2 contingency tables are fundamental in statistical analysis.
  • Existing measures for proportional variation have limitations.
  • Understanding effect size measures like odds ratio and relative risk is crucial.

Purpose of the Study:

  • To introduce a two-component proportional representation for 2x2 contingency tables.
  • To analyze the relationship between different effect size measures.
  • To investigate the limitations of measures like Gini information gain.

Main Methods:

  • Factorization of contingency tables into proportion and diagonal matrices.
  • Geometric interpretation using point vectors in a one-simplex.

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  • Derivation of row and column sum invariant measures.
  • Analysis of perspective functions for effect size mapping.
  • Monte Carlo simulations for stochastic effects and confidence intervals.

    • Main Results:

    • A novel two-component proportional representation framework is established.
    • Row and column sum invariant measures for proportional variation are derived.
    • The choice between odds ratio and relative risk is application-dependent.
    • Gini information gain (IGG) is shown to be equivalent to phi-squared (ϕ²) in CART, but can be misleading due to marginal sum dependence.
    • Relationships between various effect size measures are detailed.

    Conclusions:

    • Pure mathematics cannot universally determine the superiority of one effect size measure over another.
    • The proposed framework offers a unified perspective on effect size measures.
    • Caution is advised when using Gini information gain due to its sensitivity to marginal sums.
    • Monte Carlo methods provide a practical approach for estimating confidence intervals for effect sizes.