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Questions and Answers
Which of the following measures assesses the association for two variables with two levels each?
Which of the following measures assesses the association for two variables with two levels each?
A positive value of Goodman and Kruskal’s Gamma indicates a negative monotonic association.
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?
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.
The coefficient Cramer’s V is used for variables that have __________ levels.
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What does a C value of 0.30 or above characterize?
What does a C value of 0.30 or above characterize?
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Match the following coefficients with their type of association:
Match the following coefficients with their type of association:
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Lambda, or Guttman's Coefficient of Predictability, is a measure based on odds ratio.
Lambda, or Guttman's Coefficient of Predictability, is a measure based on odds ratio.
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Which test is specifically mentioned for linear association in ordinal data?
Which test is specifically mentioned for linear association in ordinal data?
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What does a p-value of 0.0404 indicate in the context of wage classification and job satisfaction?
What does a p-value of 0.0404 indicate in the context of wage classification and job satisfaction?
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Ordinal variables treat data as qualitative rather than quantitative.
Ordinal variables treat data as qualitative rather than quantitative.
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What is the primary advantage of using linear trend tests for ordinal data?
What is the primary advantage of using linear trend tests for ordinal data?
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The formula for Mc2 is calculated using the variable n and r, where Mc2 = ( n - 1) ______ 2.
The formula for Mc2 is calculated using the variable n and r, where Mc2 = ( n - 1) ______ 2.
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Match the following tests with their respective designs:
Match the following tests with their respective designs:
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As the level of variable X increases, what trend is typically observed in responses on variable Y?
As the level of variable X increases, what trend is typically observed in responses on variable Y?
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The Wilcoxon Rank Sum test is designed for analyzing two groups against a nominal variable.
The Wilcoxon Rank Sum test is designed for analyzing two groups against a nominal variable.
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What does 'r' represent in the formula used for Mc2?
What does 'r' represent in the formula used for Mc2?
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Which statistical test is used to determine if there is an association between a mother's alcohol consumption and child's malformation?
Which statistical test is used to determine if there is an association between a mother's alcohol consumption and child's malformation?
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The Mantel-Haenszel test differentiates between response and explanatory variables.
The Mantel-Haenszel test differentiates between response and explanatory variables.
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What distribution does the square root of the Mantel-Haenszel test follow?
What distribution does the square root of the Mantel-Haenszel test follow?
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Fisher’s Exact test is typically used for __________ tables.
Fisher’s Exact test is typically used for __________ tables.
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Which of the following variables can be treated with the Mantel-Haenszel test?
Which of the following variables can be treated with the Mantel-Haenszel test?
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Match the following tests with their appropriate usage:
Match the following tests with their appropriate usage:
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The p-value of 0.01037 in the Mantel-Haenszel test indicates strong evidence against the null hypothesis.
The p-value of 0.01037 in the Mantel-Haenszel test indicates strong evidence against the null hypothesis.
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The null hypothesis in the context of the Mantel-Haenszel test states that there is no __________ between the two variables.
The null hypothesis in the context of the Mantel-Haenszel test states that there is no __________ between the two variables.
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What is the value of the common odds ratio in the Mantel-Haenszel chi-squared test example?
What is the value of the common odds ratio in the Mantel-Haenszel chi-squared test example?
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The Breslow-Day test is effective for small sample sizes.
The Breslow-Day test is effective for small sample sizes.
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What is the primary effect studied in conditional association?
What is the primary effect studied in conditional association?
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Simpson’s Paradox indicates that the direction of the marginal association can be different from the conditional association.
Simpson’s Paradox indicates that the direction of the marginal association can be different from the conditional association.
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What hypothesis is tested using the Breslow-Day test?
What hypothesis is tested using the Breslow-Day test?
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The Mantel-Haenszel test provides evidence for the _______ independence between the two variables.
The Mantel-Haenszel test provides evidence for the _______ independence between the two variables.
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What happens to the odds of admission for females compared to males when controlling for applicant’s chosen discipline in natural sciences?
What happens to the odds of admission for females compared to males when controlling for applicant’s chosen discipline in natural sciences?
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The percentage of admission was __________ for males than for females when controlling for the applicant’s chosen discipline.
The percentage of admission was __________ for males than for females when controlling for the applicant’s chosen discipline.
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Which of the following assumptions must be satisfied for the Breslow-Day test?
Which of the following assumptions must be satisfied for the Breslow-Day test?
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What does a p-value of 0.02881 indicate in the context of the Mantel-Haenszel chi-squared test?
What does a p-value of 0.02881 indicate in the context of the Mantel-Haenszel chi-squared test?
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In the marginal table, what was the percentage of admission for males?
In the marginal table, what was the percentage of admission for males?
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Match the following tests with their descriptions:
Match the following tests with their descriptions:
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Match the terms with their definitions:
Match the terms with their definitions:
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To analyze two-way tables in R Commander, one must first choose 'Statistics > _______'.
To analyze two-way tables in R Commander, one must first choose 'Statistics > _______'.
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The odds ratio is always the same regardless of whether Z is controlled or ignored.
The odds ratio is always the same regardless of whether Z is controlled or ignored.
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What illustrates Simpson’s Paradox in the context of admissions?
What illustrates Simpson’s Paradox in the context of admissions?
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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.
