Measures of Association

Chapter 13: Measures of Association

13.1 Getting Ready

  • In this chapter, the focus is on measuring the strength of relationships between two variables in a contingency table.

  • This foundational material supports subsequent chapters, all dedicated to different methods for assessing relationships between numeric variables.

  • Recommended Materials: Load the anes20 data set in R, and the DescTools and descr libraries. It’s also advised to use a calculator or R for calculations.

13.2 Going Beyond Chi-squared

  • Chi-squared test: A tool for measuring statistical significance, but inadequate for assessing relationship strength or effect size.

  • Strength of the relationship: Refers to how much the dependent variable's values depend on the independent variable's values.

    • If chi-squared is insignificant, the dependent variable’s outcomes vary randomly around expected outcomes (null hypothesis).

    • If significant, it confirms a relationship exists but not the extent of dependency between the variables.

  • Column percentages can provide insight but lack precision and uniform standards for interpretation.

    • Example: Difference in religious importance by education.

    • Education Data:

      • Rows indicate education levels and columns indicate the religious importance assigned by these educational categories.

      • 25.5% of respondents without a high school diploma assign low importance to religion, vs. 41% among graduates, resulting in a 15.5 percentage point increase.

      • Moderate importance has a 7 percentage point decrease from low to high education.

13.3 Measures of Association for Crosstabs

13.3.1 Cramer’s V

  • Measures of association summarize strength between two variables.

  • Cramer’s V is especially useful for nominal-level variables and is derived from the chi-squared statistic.

    • Formula:
      Cramer’s V = rac{ ext{sqrt}( ext{Chi-squared})}{N imes ext{min}( ext{rows} - 1, ext{columns} - 1)}

    • Interpretation: Ranges from 0 (no relationship) to 1 (perfect relationship).

    • Example Calculation: For religious importance by education with ase chi-squared value 108.2, N=8129,
      Cramer’s V = ext{sqrt}(108.2) / (8129 imes 2) = 0.082

13.3.2 Lambda

  • Lambda ( ) is a proportional reduction in error statistic, representing the error reduction from prediction using the independent variable.

    • Formula:
      ext{Lambda} = rac{E1 - E2}{E1}

    • Where:

    • E1 = Error from guessing without an independent variable.

    • E2 = Error from guessing with an independent variable.

  • Example: For region and religious importance, you can guess correctly more often with lambda calculations indicating approximately a 3.1% reduction in guessing error.

13.4 Ordinal Measures of Association

  • Cramer’s V and Lambda fail to incorporate directional context.

  • Example: Discussion of educational levels affecting religious importance has a negative trend, while age increases suggest a positive relationship.

13.4.1 Gamma

  • Gamma (γ): Measures positive versus negative pair rankings in a crosstab.

    • Formula:
      ext{Gamma} = rac{N_ ext{similar} - N_ ext{different}}{N_ ext{similar} + N_ ext{different}}

  • A positive gamma indicates that similar rankings dominate, while negative indicates the opposite.

  • Applying this to prior examples allows for quantifying relationships concretely with gamma showing an inherent bias towards overstating relationships due to disregarding tied observations.

13.4.2 Tau-b and Tau-c

  • Both tau-b and tau-c correct for tied pairs in the calculation of ordinal association, ensuring more accurate relationship assessments.

    • Suitable for square (tau-b) vs. rectangular (tau-c) tables, both scale from -1 (perfect negative) to +1 (perfect positive).

13.5 Revisiting the Gender Gap in Abortion Attitudes

  • Previous chapter examined gender gaps in attitudes towards abortion using chi-square tests, finding no significant differences in illegal views but minor differences in beliefs about choice.

  • By using crosstabs, a continuum from illegal to choice can reveal nuanced understandings of sexual preferences and implications of less significant statistical associations.

13.6 Next Steps

  • Subsequent chapters shift towards numeric variable relationships, utilizing scatter plots and correlation coefficients.

13.7 Exercises

13.7.1 Concepts and Calculations

  • Analyze relationships, including percentages of political identification separated by sex.

  • Examine and calculate Cramer’s V, Lambda, and Gamma for further insights.

13.7.2 R Problems

  • Experiment with independent variables influencing attitudes towards the Supreme Court's abortion stance, leveraging crosstab functions clearly outlined.