WK11: Statistical inference: Two categorical variables: Two-way contingency table

Categorical variables are variables whose values fall into groups or categories.
Examples:
- Gender (categorical by nature)
- Age (grouped into classes, e.g., child 0-18, adult 19-100)

Used for preliminary investigation of the relationship between two categorical variables.
Displays counts for combinations of categories.
Example: Investigating the relationship between living arrangement and age.
- Living arrangements (rows): Parent's home, another person's home, own place, group quarters, other.
- Age (columns): 19, 20, 21, 22 years old.
Each cell contains the number of individuals belonging to a specific age and living arrangement combination.

Counts for each category of a single variable, regardless of the other variable.
Calculated by summing up counts across all categories of the other variable.
Example: Marginal distribution for living arrangement:
- Count of individuals living in their parent's home, regardless of age.
Marginal distribution is represented by percentages or proportions.
Proportion = Count of individuals in a category / Total number of individuals.
Example: 1,357 individuals live in their parents' home out of a total of 2,984.
- Proportion = $1357 / 2984 = 0.455$ (45.5%)
Check that percentages sum up to 100% or proportions sum up to 1.

Distribution of values or categories of one variable, considering only individuals belonging to a particular category in the other variable.
Used to investigate the relationship between two categorical variables.
If conditional distributions are similar, there is likely no relationship.
If conditional distributions are different, there might be a relationship.
To determine whether living arrangement depends on age, obtain conditional distributions of living arrangements for each age category and compare.
Proportion calculation: Count of individuals in a living arrangement category / Total number of individuals in that age category.
Example: For individuals who are 19, 324 live in their parents' home out of 540 total.
- Proportion = $324 / 540 = 0.6$ (60%)
Check that percentages in each conditional distribution sum up to 100% or proportions sum up to 1.
Visual comparison of conditional distributions can be done using plots.

When doing statistical inference, it is important to watch out for lurking variables.
Lurking variables explain changes in the response, but are not known prior to the study.
Example: Simpson's Paradox
- Relationship between two quantitative variables (X and Y) appears decreasing.
- By considering a categorical variable, the relationship between X and Y is actually increasing.
- If the effect of the lurking variable is not taken into account, misleading conclusions may result.

Relationship between survival of a victim in an accident and mode of transport to the hospital.
Conditional distributions of survival conditioned on mode of transport:
- Helicopter: 32% death, 68% survival.
- Road: 23.6% death, 76.4% survival.
Helicopter transport appears to have a lower survival rate.
Lurking variable: Seriousness of the accident (less serious, more serious).
Data separated by seriousness of the accident:
- Serious Accidents:
  - Helicopter: 52% survival.
  - Road: 40% survival.
- Less Serious Accidents:
  - Helicopter: 84% survival.
  - Road: 80% survival.
In both cases, helicopter transport yields a higher survival rate.
This example shows how a lurking variable can reverse the overall interpretation of the data.