What We’ve Done vs. What We’ll Do

Previously: Examined variables individually

• Frequency tables for single categorical variables
• Bar plots for single categorical variables
• Summary statistics for single numerical variables
• Inference for a single proportion

Now: Examine relationships between two variables

• Research questions often ask about relationships
• “Does treatment affect outcome?”
• “Do different groups show different patterns?”

Association and Independence

When we examine two variables together:

• Associated: There IS a relationship between the two variables
  • Knowing the value of one variable tells you something about the other
• Independent: There is NO relationship between the two variables
  • Knowing one variable tells you nothing about the other

Our goal: Determine if two categorical variables are associated

Contingency Tables

A contingency table summarizes the relationship between two categorical variables:

• Each cell shows the count for that combination of values
• Row totals: frequency of each category of the row variable
• Column totals: frequency of each category of the column variable
• Grand total: total number of observations

Also called a two-way table or cross-tabulation

Conditional Proportions

Conditional proportions: Proportions calculated within groups

• Most useful when conditioned on the grouping variable
• Reveals patterns that raw counts may hide

Formula (conditioned on rows): (count in cell) / (row total)

Example:

\[\begin{array}{c} \text{Proportion of} \\ \text{bad characters with} \\ \text{secret identities} \end{array} = \frac{\text{Bad characters with secret identity}}{\text{Total bad characters}}\]

Visualizing Two Categorical Variables

Three types of bar plots for two categorical variables:

• Dodged bar plot: Bars side-by-side, compare counts across groups
• Stacked bar plot: Bars stacked, show composition of each group
• Standardized bar plot: Bars stacked to same height, compare proportions within groups

Standardized bar plots are most useful for detecting associations

Explanatory and Response Variables

When we suspect one variable may affect another:

• Explanatory variable: The variable that might influence or explain the other
  • Also called: independent variable, predictor
• Response variable: The variable that might be affected or respond
  • Also called: dependent variable, outcome

Not all studies have clear explanatory/response roles

Two Scope of Inference Questions

Recall from before: The first scope of inference question

1. Generalization: Can we apply findings to the broader population?
   • Answered by: How we sampled (random vs. convenience)

NEW: The second scope of inference question

2. Causation: Can we conclude cause-and-effect?
   • Answered by: How we conducted the study

The Complete Picture

These are TWO SEPARATE QUESTIONS:

A study can:

• Generalize but not establish causation
• Establish causation but not generalize
• Do both, or neither

Observational Study vs. Experiment

Observational study: Researchers observe without manipulation

• Cases observed “as they are”
• No control over explanatory variable
• Example: Survey asking about diet and health

Experiment: Researchers deliberately manipulate explanatory variable

• Cases assigned to treatment conditions
• Control over explanatory variable
• Example: Randomly assigning patients to receive drug or placebo

Confounding Variables

A confounding variable is associated with BOTH the explanatory and response variables

flowchart TD
    A[Temperature<br/>confounder] --> B[Ice cream sales]
    A --> C[Drowning deaths]

    style A fill:#ffcccc,stroke:#cc0000,stroke-width:3px
    style B fill:#cce5ff,stroke:#0066cc,stroke-width:2px
    style C fill:#cce5ff,stroke:#0066cc,stroke-width:2px

Problem: Creates alternative explanations for observed associations

• In observational studies, we can’t rule out confounders
• We can only say variables are associated, not that one causes the other

Random Assignment

The solution to confounding: Random assignment

Random assignment: Cases are randomly assigned to treatment groups

• Researchers control who receives which treatment
• Assignment made by chance (like flipping a coin)
• Not based on patient characteristics, preferences, or doctor’s choice

Key benefit: Ensures treatment groups are similar on average with respect to all possible confounders

Why Experiments Can Establish Causation

Random assignment breaks the confounder link:

In the actual randomized stent study:

1. Researchers controlled treatment assignment (not patients/doctors)
2. Random assignment balanced severity of stenosis (and other potential confounders) between groups
3. Any difference in stroke rates must be due to the stent itself

Without random assignment, we couldn’t distinguish:

• Does the stent cause more strokes?
• Or do sicker patients (who would get stents) just have more strokes anyway?

Random Assignment vs. Random Sampling

These are different concepts that address different questions!

Summary: Two Categorical Variables

Summarizing relationships:

• Contingency tables: Counts for each combination
• Conditional proportions: Proportions within groups (reveal patterns)
• Standardized bar plots: Visualize proportional differences

Key concepts:

• Associated: Variables are related
• Independent: No relationship between variables

Summary: Scope of Inference

Two separate questions about what we can conclude:

• Generalization → How did we sample?
• Causation → How did we conduct the study?

References

• Introduction to Modern Statistics (2e) textbook by Mine Çetinkaya-Rundel and Johanna Hardin
• Sections 1.1, 1.2, 2.2, 2.3, 2.4, 4.1, 4.2, 4.3