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Measures of Association in Statistics
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Measures of Association in Statistics

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Questions and Answers

Which of the following measures assesses the association for two variables with two levels each?

  • Yule's Q (correct)
  • Pearson's contingency, C
  • Cramer’s V
  • Goodman and Kruskal’s tau, t

    • A positive value of Goodman and Kruskal’s Gamma indicates a negative monotonic association.

    False

    What is the primary purpose of using measures based on the odds ratio?

    To assess associations between categorical variables.

    The coefficient Cramer’s V is used for variables that have __________ levels.

    <p>more than 2</p> Signup and view all the answers

    What does a C value of 0.30 or above characterize?

    <p>Strong association</p> Signup and view all the answers

    Match the following coefficients with their type of association:

    <p>Yule’s Q = Nominal Goodman and Kruskal’s tau = Ordinal Pearson’s phi = Nominal Cramer’s V = Nominal</p> Signup and view all the answers

    Lambda, or Guttman's Coefficient of Predictability, is a measure based on odds ratio.

    <p>False</p> Signup and view all the answers

    Which test is specifically mentioned for linear association in ordinal data?

    <p>Mantel-Haenszel test</p> Signup and view all the answers

    What does a p-value of 0.0404 indicate in the context of wage classification and job satisfaction?

    <p>There is sufficient evidence of a significant monotonic association.</p> Signup and view all the answers

    Ordinal variables treat data as qualitative rather than quantitative.

    <p>False</p> Signup and view all the answers

    What is the primary advantage of using linear trend tests for ordinal data?

    <p>Greater power for ordinal variables</p> Signup and view all the answers

    The formula for Mc2 is calculated using the variable n and r, where Mc2 = ( n - 1) ______ 2.

    <p>r</p> Signup and view all the answers

    Match the following tests with their respective designs:

    <p>Mantel-Haenszel test = I x J Cochran-Armitage Trend test = I x 2 Wilcoxon Rank Sum test = 2 x J Mann-Whitney test = 2 x J</p> Signup and view all the answers

    As the level of variable X increases, what trend is typically observed in responses on variable Y?

    <p>Responses on Y tends to increase</p> Signup and view all the answers

    The Wilcoxon Rank Sum test is designed for analyzing two groups against a nominal variable.

    <p>False</p> Signup and view all the answers

    What does 'r' represent in the formula used for Mc2?

    <p>Correlation coefficient</p> Signup and view all the answers

    Which statistical test is used to determine if there is an association between a mother's alcohol consumption and child's malformation?

    <p>Mantel-Haenszel test</p> Signup and view all the answers

    The Mantel-Haenszel test differentiates between response and explanatory variables.

    <p>False</p> Signup and view all the answers

    What distribution does the square root of the Mantel-Haenszel test follow?

    <p>Standard normal distribution</p> Signup and view all the answers

    Fisher’s Exact test is typically used for __________ tables.

    <p>2 x 2</p> Signup and view all the answers

    Which of the following variables can be treated with the Mantel-Haenszel test?

    <p>Ordinal variables</p> Signup and view all the answers

    Match the following tests with their appropriate usage:

    <p>Fisher's Exact test = Used for small sample sizes in 2 x 2 tables Mantel-Haenszel test = Assess linear association between ordinal variables Pearson Chi-Square test = Used for testing independence between categorical variables Cochran-Armitage Trend test = Applicable for analyzing trends in proportions</p> Signup and view all the answers

    The p-value of 0.01037 in the Mantel-Haenszel test indicates strong evidence against the null hypothesis.

    <p>True</p> Signup and view all the answers

    The null hypothesis in the context of the Mantel-Haenszel test states that there is no __________ between the two variables.

    <p>association</p> Signup and view all the answers

    What is the value of the common odds ratio in the Mantel-Haenszel chi-squared test example?

    <p>0.4118627</p> Signup and view all the answers

    The Breslow-Day test is effective for small sample sizes.

    <p>False</p> Signup and view all the answers

    What is the primary effect studied in conditional association?

    <p>The effect of X on Y while fixing Z at some level</p> Signup and view all the answers

    Simpson’s Paradox indicates that the direction of the marginal association can be different from the conditional association.

    <p>True</p> Signup and view all the answers

    What hypothesis is tested using the Breslow-Day test?

    <p>homogeneous association</p> Signup and view all the answers

    The Mantel-Haenszel test provides evidence for the _______ independence between the two variables.

    <p>conditional</p> Signup and view all the answers

    What happens to the odds of admission for females compared to males when controlling for applicant’s chosen discipline in natural sciences?

    <p>The odds of admission are less likely for females than males.</p> Signup and view all the answers

    The percentage of admission was __________ for males than for females when controlling for the applicant’s chosen discipline.

    <p>higher</p> Signup and view all the answers

    Which of the following assumptions must be satisfied for the Breslow-Day test?

    <p>At least 80% of expected cell counts should be greater than 5</p> Signup and view all the answers

    What does a p-value of 0.02881 indicate in the context of the Mantel-Haenszel chi-squared test?

    <p>significant evidence against null hypothesis</p> Signup and view all the answers

    In the marginal table, what was the percentage of admission for males?

    <p>7.85%</p> Signup and view all the answers

    Match the following tests with their descriptions:

    <p>Mantel-Haenszel Test = Tests for association controlling for other variables Breslow-Day Test = Tests for homogeneous association Pearson Chi-Square Test = Tests independence in contingency tables Fisher’s Exact Test = Used for small sample sizes in categorical data</p> Signup and view all the answers

    Match the terms with their definitions:

    <p>Conditional Association = Effect of X on Y fixed at Z Simpson’s Paradox = Difference in associations based on condition Conditional Odds Ratio = Odds of X and Y at fixed Z levels Marginal Odds Ratio = Odds of X and Y ignoring Z</p> Signup and view all the answers

    To analyze two-way tables in R Commander, one must first choose 'Statistics > _______'.

    <p>Contingency Tables</p> Signup and view all the answers

    The odds ratio is always the same regardless of whether Z is controlled or ignored.

    <p>False</p> Signup and view all the answers

    What illustrates Simpson’s Paradox in the context of admissions?

    <p>The sample odds of admission to graduate school were 45% higher for females than males.</p> Signup and view all the answers

    Study Notes

    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.

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    Description

    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.

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