Measures of Association 8-9

Quantitative Research Methods in Political Science

• This lecture covers measures of association for nominal and ordinal variables in political science research.
• Measures of association quantify the strength and direction of relationships between variables.
• They are crucial for assessing the importance of relationships and validating theories.
• Cause-and-effect relationships are effectively documented with these measures.
• Predictions can be formulated because if variables are related, a score on one variable can predict a score on another.

Bivariate Association

• Two variables are associated if the distribution of one changes under different categories or scores of the other.
• A bivariate table (example provided) illustrates the relationship between job satisfaction (X) and productivity (Y) for 173 factory workers.
• Examining columns in the table reveals the pattern of scores on Y for each score on X.
• This example illustrates that 30 of 60 workers with low job satisfaction reported low productivity while 25 of 61 workers with moderate job satisfaction reported moderate productivity.
• The relationship between these variables reveals the conditional distributions of Y (productivity), which are affected by X (job satisfaction).

Questions to Ask

• To fully analyze the relationship between two variables, one must ask:
  • Does association exist?
  • If association exists, how strong is it?
  • If association exists, what is the pattern and/or the direction of the association?

Does Association Exist?

• Chi-square testing can establish association.
• Chi-square values above 0.00 represent association, however, this doesn't always indicate statistical significance.
• To better understand the relationships, it is advisable to use column percentages to compute and compare.

Does Association Exist? Cont'd

• This is further illustrated by using percentages from a bivariate table as shown in the slides, to detect conditional distributions of Y (e.g., productivity).
• This makes it clearer to see the relationship between job satisfaction and productivity.

Does Association Exist? Cont'd

• Another example of this using productivity and age group.
• If an association did not exist the conditional distributions of Y across columns would be similar for each group.

How Strong is Association?

• Association strength ranges from perfect non-association to perfect association.
• Perfect non-association is when the conditional distributions of Y do not change across columns.
• Perfect association is when each value of Y corresponds to only one value of X.
• The likelihood of perfect association is low, which makes measures of association essential to understand the strength of the relationship between variables.

How Strong is Association? Cont'd

• Perfect associations suggest a causal relationship – where variation in one variable causes variation in another.
• This is important because a measure of association describes the strength of the relationship between two variables.

What is the Pattern and/or Direction of Association?

• Analyzing the direction and pattern of the association involves determining which values or categories of one variable are linked with those of another.
• Examples show positive and negative associations. Positive associations occur when high scores on one variable are aligned with high scores on the other (or low with low), while negative ones indicate variation in opposite directions (high with low, or low with high).

What is the Pattern and/or Direction of Association? Cont'd

• Another example where an increase in education is related to a decrease in television viewership.

Measures of Association

• Measures of association detail the strength and direction of bivariate relationships, often displayed as a single numerical value.
• The type of measure used depends on the level of measurement of the variables (e.g., nominal, ordinal, and interval/ratio).
• Examples including Phi and Cramer's V apply to nominal level data.
• Gamma, Somers' d, and Kendall's tau-b and tau-c apply to ordinal data.

Chi Square-Based Measures of Association

• Researchers use chi-square based measures of association to assess relationships between nominal level variables.
• Chi-square values reflect the level of association; however, this does not indicate statistical significance.
• The strength of association can be assessed using transformations of chi-square values.

Phi

• Phi (φ) is a measure of association used for 2x2 tables with nominal variables.
• Its formula includes the chi-square value (χ2) and the sample size (n).
• Phi values range from 0.00 (no association) to 1.00 (perfect association).

Cramer's V

• Cramer's V is a more general measure of association for nominal variables when the variable table is larger than 2x2.
• Its formula considers the size of the table along with the chi-square value and sample size.

Measures of Association for Ordinal Level Variables

• These variables (e.g., collapsed ordinal variables with few categories), need different measures of association compared to nominal data.
• The key measures are Gamma (G), Somers' d, Kendall's tau-b, and Kendall's tau-c, which also provide the direction of the relationship.

The Logic of Pairs

• Measures of association for ordinal variables, such as Gamma, Somers' d, and Kendall's Tau-b and Tau-c, work by comparing pairs of respondents.
• Pairs are classified as similar, dissimilar, or tied based on their rankings on the independent and dependent variables.

Analyzing Measures of Association for Ordinal Level Variables

• Measures of association, like Gamma, Somers' d, tau-b, and tau-c, consider the number of "similar" versus "dissimilar" pairs when analyzing ordinal variables. These differ in how they handle tied pairs.
• When similar and dissimilar pairs are equal, the statistic is 0.00 and no association exists.
• As the number of similar pairs greatly exceeds dissimilar pairs, the value approaches 1.00 (perfect association).

Interpreting Strength for Ordinal Level Measures

• The strength of ordinal association is interpreted based on the value of these measures.
• This interpretation is similar to what Phi and Cramer's V used, with a value between 0.00 and 0.10 indicating a weak relationship, 0.11 and 0.30 indicating a moderate relationship, and greater than 0.30 indicating a strong relationship.

Direction of Relationship

• Nominal variables (using phi or Cramer's V) only determine association strength; ordinal variables (using Gamma, Somers' d, Kendall's tau-b, tau-c) also identify directional trends (positive or negative).
• A positive relationship indicates that both variables increase or decrease together, whereas a negative relationship indicates that as one variable increases, the other decreases.

Positive Versus Negative Relationships

Visualized

• This provides a visualization of positive and negative relationships. This can be depicted as a positive relationship shows values increasing in the same direction (values on X increase and values on Y increase).
• A negative relationship demonstrates an opposite trend (values on X increase as values on Y decrease).

Gamma

• Gamma (G) gauges association strength by counting similar pairs against dissimilar pairs as a proportion.

Gamma Cont'd

• Gamma measures the positive trend of a relationship (as opposed to Kendall's Tau that measures the strength of the relationship independent of direction).

Limitations of Gamma

• Gamma overlooks tied pairs (those with identical rankings), which might exaggerate the association strength. More sophisticated measures (Somers' d, Kendall's tau-b, Kendall's tau-c) address such ties by looking beyond the basic count approach.

Kendall's Tau-b

• Kendall's tau-b is a measure of the strength and direction of monotonic association between two ordinal variables.
• Its calculation involves considering the frequency of similar and dissimilar pairs.
• Tau-b is valid for a 2x2 tables but becomes more detailed with larger tables.

Kendall's Tau-c

• Tau-c is a measure of association similar to Kendall's tau-b, allowing for unequal categories.

Somers' d

• Somers' d is an asymmetric measure of ordinal association.
• It measures the association between two variables where one is considered the independent variable (predictor) and the other the dependent variable (predicted).
• It provides a more focused approach to assessing the predictiveness for every variable, and its result changes when each is treated as either the predictor or the predicted.