Measures of Association in Statistics

Measures of Association

• Nominal Scale: Measures association between nominal variables
  • Odds Ratio Based: Yule's Q and Yule's Y
  • Chi-Square Based: Pearson's phi, Pearson's contingency, Cramer's V
  • Proportional Reduction in Error (PRE) Based: Lambda, Goodman and Kruskal's tau, Uncertainty Coefficient

Nominal (Chi-Square Based) Measures of Association

• Pearson's phi (f)

  • Used for two variables with two levels each.
  • Formula: f = (c*C^2) / n, where c is the chi-square statistic, C is the contingency coefficient, and n is the sample size.
  • Range: 0 ≤ f ≤ √(q-1), where q is the minimum of the number of levels in each variable.

• Pearson's Contingency (C)

  • Used for two variables with more than two levels.
  • Formula: C = (cC^2) / (og), where c is the chi-square statistic, C is the contingency coefficient, o is the number of rows, and g is the number of columns.
  • Range: 0 ≤ C ≤ 1.
  • Interpretation:
    • Positive C indicates a positive monotonic association.
    • Negative C indicates a negative monotonic association.
  • Strength of association:
    • 0.01 to 0.09: weak association
    • 0.10 to 0.29: moderate association
    • 0.30 and above: strong association
  • Note that the interpretation of the strength of association may vary across disciplines.

Ordinal Measures of Association

• Goodman and Kruskal's Gamma (g)
  • Measures the strength and direction of the monotonic association between two ordinal variables.
  • Example: A p-value of 0.0404 for Gamma indicates significant monotonic association between wage classification and job satisfaction. This positive association is strong.

Tests of Independence for Ordinal Variables

• Linear Trend Tests
  • Treat ordinal variables as quantitative, providing greater power for analyzing ordinal data.
  • Examples:
    • Mantel-Haenszel test for linear association (I x J)
    • Cochran-Armitage Trend test (I x 2)
    • Wilcoxon Rank Sum or Mann-Whitney test (2 x J)

Mantel-Haenszel Test for Linear Association

• Usage:
  • Applies to I x J contingency tables.
  • Measures the strength of the linear association between two ordinal variables.
• Formula: Mc^2 = (n-1) * r^2, where r is the Pearson product-moment correlation between the row and column variables.
• Interpretation:
  • The test statistic has an asymptotic chi-square distribution with 1 degree of freedom.
  • The square root of the test statistic follows a standard normal distribution.
  • The test can be used for one-tailed tests (Ha: ⍴ > 0).
• Strengths:
  • Treats both variables as ordinal.
  • Can be used when one variable is nominal with only two categories.
• Symmetrical nature:
  • Like Pearson's Chi-Square and Likelihood Ratio Chi-Square tests, the Mantel-Haenszel test does not distinguish between response and explanatory variables.

Three-Way Contingency Tables

• Conditional Association
  • The effect of X on Y, holding Z constant at a specific level.
  • Example: The percentage of admission to graduate school is higher for male applicants when they choose natural sciences or social sciences, illustrating the conditional association.
• Simpson's Paradox
  • Occurs when the marginal association direction differs from the conditional association direction.
  • Example: Ignoring discipline, the percentage of admission to graduate school is lower for males than females. However, for each discipline (natural sciences and social sciences), males have a higher admission percentage.

Three-Way Contingency Tables (Cont.)

• Conditional Odds Ratio
  • The odds ratio between X and Y when Z is fixed at a specific level.
  • Example: The odds of admission to graduate school for female applicants are lower than those for male applicants, both when natural sciences or social sciences are considered.
• Marginal Odds Ratio
  • The odds ratio between X and Y, ignoring Z.
  • Example: The odds of admission to graduate school were 45% higher for females than males when ignoring the discipline chosen.
• Mantel-Haenszel Chi-square Test
  • Tests for conditional independence between two variables, controlling for a third variable.
  • The test statistic has a chi-square distribution with one degree of freedom.
  • Example: The Mantel-Haenszel test indicates a statistically significant association between a mother's alcohol consumption and a child's malformations, after controlling for other factors.

Statistical Inference in Three-Way Tables

• Estimate of the Common Odds Ratio (θ)
  • Useful when a homogenous association exists.
  • Can be used to summarize K conditional (partial) associations as long as they share the same direction.
• Breslow-Day Test
  • Used to test for homogenous association in three-way tables.
  • Compares observed counts to estimated expected frequencies under the assumption of a common odds ratio.
  • Assumes:
    • Sufficiently large samples for each group (level of variable Z).
    • At least 80% of expected cell counts greater than 5.
  • Example: The Breslow-Day test can be used to determine if the odds ratio of admission to graduate school for male vs. female applicants is consistent across different chosen disciplines.

R Software:

• R Commander
  • Pearson Chi-Square test/Fisher's Exact Test: Accessed through 'Statistics > Contingency Tables > Two-way table...' or by entering and analyzing a two-way table.
  • Likelihood Ratio Chi-Square Test:
    • Requires loading the 'MASS' library.
    • Use the 'data' function to input your data.

Explore the various measures of association used in statistics, particularly focusing on nominal scales. This quiz covers concepts such as odds ratios, chi-square based measures, and proportional reduction in error. Test your understanding of Pearson's phi and contingency coefficients while enhancing your statistical knowledge.