Categorical Variable Relationships: Independence, Association, and Segmented Bar Charts

Associations in Variables: Variables can exhibit associations in many different ways and to varying degrees. Understanding these associations is critical for interpreting data sets, such as the historical records of the Titanic.
Defining Independence: - In the context of a contingency table, variables are considered independent when the distribution of one variable remains identical across all categories of another variable. - If independence is established, there is no association between the variables in question.
The Logical Equivalence of Terms: - To say that variables are "independent" is equivalent to saying they are "not associated." - To say that variables are "not independent" is equivalent to saying they are "associated."
Visual and Formal Checks: While formal mathematical methods to check for independence exist, a primary method involves comparing distributions visually or through conditional tables.

Conditional Distributions: These are used to look closer at variable relationships by conditioning one variable on the values of another.
The Titanic Dataset Example (Table 3.7): This conditional table displays the distribution of ticket Class (First, Second, Third, and Crew) conditioned on Survival status (Alive or Dead).
Conditional Table Data for Survival: - Condition: Alive (Survivors) - First Class: $203$ individuals, representing $28.6\%$ - Second Class: $118$ individuals, representing $16.6\%$ - Third Class: $178$ individuals, representing $25.0\%$ - Crew: $212$ individuals, representing $29.8\%$ - Total Alive: $711$ individuals, representing $100\%$ - Condition: Dead (Non-survivors) - First Class: $122$ individuals, representing $8.2\%$ - Second Class: $167$ individuals, representing $11.2\%$ - Third Class: $528$ (transcript notes $354$ but calculations show $35.4\% \times 1490 \approx 527.46$ ; the table lists $35.4\%$ - Crew: $673$ individuals, representing $45.2\%$ - Total Dead: $1490$ individuals, representing $100\%$
Observing Associations: The margins of the contingency table are vital because they show the actual numbers of people involved. By comparing the percentages across the "Alive" and "Dead" rows, it becomes clear that the distribution of classes is not the same for both groups, implying an association between ticket Class and Survival.

Definition of a Segmented Bar Chart (Figure 3.9): This display treats each individual bar as a "whole" (representing $100\%$ ). It divides the bar proportionally into segments corresponding to the percentage of each category within that group.
Key Features of Figure 3.9: - Y-Axis: Percent, ranging from $0\%$ to $100\%$ . Each bar always totals $100\%$ - X-Axis Categories: Survival status, specifically "Alive" and "Dead." - Color/Segment Coding: Identifies the ticket Class (First, Second, Third, and Crew). - Comparison of Bar Heights: Even though the raw totals for survivors ( $711$ ) and non-survivors ( $1490$ ) are significantly different, the bars in the chart are the same height because the data has been converted into percentages.
Interpreting the Chart: - If the segments within the bars look different across the categories (e.g., the "First Class" segment is much larger in the "Alive" bar than in the "Dead" bar), then the variables are associated. - In the Titanic case, the visible differences in distributions across the categories of survival indicate that survival was not independent of ticket class.
Comparison with Other Visuals: The segmented bar chart is often used as an alternative to side-by-side pie charts (referenced as Figure 3.6 in the text) to display the same type of data.

Requirement for Proper Conclusion: When tasked with answering questions regarding association or independence between two categorical variables, there is a specific procedural requirement: - You must create a segmented bar chart. - You must interpret the chart by describing the differences or similarities in the distributions.
Identifying Association: If the segments of the bars vary significantly between the categories being compared, you must conclude there is an association (the variables are not independent).