Contingency Tables — Analyzing Two Categorical Variables

🎯 Learning Objectives

• Construct contingency tables (two-way tables) from raw data
• Compute marginal distributions and conditional distributions
• Determine whether two categorical variables are associated
• Calculate and interpret joint probabilities and conditional probabilities
• Create and interpret mosaic plots and stacked bar charts

📋 Prerequisites

• Categorical Data (02-categorical-frequency) — frequency tables for single variables
• Data Types & Scales (01-data-types-scales) — understanding nominal vs ordinal
• Basic arithmetic: counting, division, percentages

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📖 Core Content

7.1 Intuition: When One Variable Isn't Enough

• Does gender relate to job promotion?
• Is there a relationship between smoking and lung cancer?
• Does education level affect voting preference? Contingency tables (also called two-way tables or cross-tabulations) are the primary tool for summarizing and analyzing two categorical variables simultaneously.

Everyday analogy: Imagine a restaurant wants to know if there's a relationship between the day of the week (weekday vs weekend) and customer satisfaction (satisfied vs dissatisfied). They'd create a table crossing "Day" with "Satisfaction" — that's a contingency table. 🔑 Key Insight: The question is always: "Does the distribution of one variable change depending on the other variable?" If yes, the variables are associated.

7.2 Anatomy of a Contingency Table

Example: Gender and Product Preference

• Cell frequencies: The 4 inner values (40, 60, 70, 30) — number of people in each combination
• Row totals: 100 men, 100 women
• Column totals: 110 prefer A, 90 prefer B
• Grand total: 200 (sum of all cells = sum of row totals = sum of column totals)

7.3 Joint Distribution

7.4 Marginal Distribution

• Male: 100/200 = 50%
• Female: 100/200 = 50% Marginal distribution of Product (Column totals):
• Product A: 110/200 = 55%
• Product B: 90/200 = 45% Key insight: The "margins" of the table give us the univariate distributions.

7.5 Conditional Distribution

• Product A: 40/100 = 40%
• Product B: 60/100 = 60% Among Females:
• Product A: 70/100 = 70%
• Product B: 30/100 = 30% Interpretation: Males prefer Product B (60% vs 40%), while Females prefer Product A (70% vs 30%). This is a clear association between gender and product preference.

🔑 Check for association: If the conditional distributions are the same for all values of the other variable, there's no association. If they differ, there's an association.

7.6 Independence vs Association

• Males: 55% A, 45% B
• Females: 55% A, 45% B Identical! Gender and product preference are independent — knowing gender tells you nothing about product preference.

7.7 Simpson's Paradox

7.8 Mosaic Plots

• The width of each column represents the marginal frequency of that category
• The height of each rectangle represents the conditional frequency
• The area of each rectangle represents the joint frequency (Diagram) Interpretation: If the variables were independent, the rectangles would align perfectly (no "break" in the pattern). The more the rectangles shift, the stronger the association.

7.9 Measures of Association for Categorical Variables

Contingency Coefficient

Relative Risk and Odds Ratio

$$\text{RR} = \frac{\text{Risk in exposed group}}{\text{Risk in unexposed group}}$$ $$\text{OR} = \frac{\text{Odds of exposure in cases}}{\text{Odds of exposure in controls}}$$

• Smokers: 60/100 = 0.60
• Non-smokers: 15/100 = 0.15 Relative Risk: 0.60/0.15 = 4.0 Interpretation: Smokers are 4 times more likely to develop lung cancer than non-smokers.

7.10 Worked Examples

Example 1: Constructing a Contingency Table (Easy)

• Regular exercise, Good health: 40
• Regular exercise, Poor health: 10
• Sometimes exercise, Good health: 35
• Sometimes exercise, Poor health: 15
• Never exercise, Good health: 20
• Never exercise, Poor health: 30 Construct the contingency table and find marginal distributions. Solution: Step 1 — Fill the table:

• All row totals are 50 — equal thirds. Each exercise level has 33.3%. Step 3 — Marginal distribution of health:
• Good: 95/150 = 63.3%
• Poor: 55/150 = 36.7% Step 4 — Conditional distributions: Among Regular exercisers:
• Good: 40/50 = 80%
• Poor: 10/50 = 20% Among Sometimes exercisers:
• Good: 35/50 = 70%
• Poor: 15/50 = 30% Among Never exercisers:
• Good: 20/50 = 40%
• Poor: 30/50 = 60% Interpretation: There's a clear association: more frequent exercise is associated with better health. Among regular exercisers, 80% have good health. Among those who never exercise, only 40% have good health.

Example 2: Determining Independence (Medium)

• Black and Brown hair: same distribution (66.7% brown eyes) — consistent between these two
• Blonde hair: different (44.4% brown eyes, 55.6% blue eyes) Conclusion: Hair color and eye color are not independent. Blonde-haired people are more likely to have blue eyes than people with black or brown hair.

Example 3: Simpson's Paradox (Harder)

• Women are more likely to apply to Program A (high admit rate, 80%)
• Men are more likely to apply to Program B (low admit rate, 20%)
• The overall rate is confounded by program choice Lesson: The apparent overall gender disparity is NOT due to gender discrimination — it's due to different application patterns. This is Simpson's Paradox.

7.11 Edge Cases & Gotchas

Small Cell Counts

More Than Two Categories

Zero Rows or Columns

7.12 Why This Matters

• Medicine: Treatment vs outcome tables
• Marketing: Customer segment vs purchase behavior
• Social science: Demographics vs voting patterns
• Quality control: Defect type vs production shift They also form the foundation for:
• BSMA1004 (Stats 2): Chi-square tests of independence
• BSMS2002 (Business Analytics): Customer segmentation and A/B testing
• BSMA3012 (Linear Stat Models): Logistic regression uses odds ratios

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📐 Key Formulas / Concepts

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⚠️ Common Pitfalls

Pitfall 1: Confusing Conditional and Joint Distributions

• Joint: "X% of all respondents" (denominator = grand total)
• Conditional: "X% of [group]" (denominator = row total for that group)

Pitfall 2: Claiming Causation from Association

Pitfall 3: Ignoring Simpson's Paradox

Pitfall 4: Computing Percentages in the Wrong Direction

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📝 Practice Questions

Q1: Building a Contingency Table

200 people are surveyed: 120 women, 80 men. 90 people prefer online shopping, 110 prefer in-store. Among online shoppers, 60 are women. Build the contingency table.

<details> <strong>Solution</strong>

Known:

• Women: 120, Men: 80
• Online: 90, In-store: 110
• Women who prefer online: 60

Check: 60+60+30+50 = 200 ✅

</details> > **Q2: Conditional Distributions** > > Using the table from Q1, compute: > > a) % of women who prefer online shopping b) % of men who prefer online shopping c) % of online shoppers who are women d) Is there an association between gender and shopping preference? > > <details> <strong>Solution</strong> > > a) **Women online:** 60/120 = 50% b) **Men online:** 30/80 = 37.5% c) **Online shoppers who are women:** 60/90 = 66.7% > > d) **Association?** Yes. Women are more likely to prefer online shopping (50%) than men (37.5%). The conditional distributions differ. </details> > **Q3: Independence Check** > >

Are studying and passing independent?

<details> <strong>Solution</strong>

Conditional distributions:

Among those who studied: Pass = 75/100 = 75%, Fail = 25/100 = 25%

Among those who didn't study: Pass = 15/50 = 30%, Fail = 35/50 = 70%

The distributions are very different (75% pass vs 30% pass). Therefore, studying and passing are NOT independent — there's a strong association.

(As we'd expect — studying helps!)

</details> > **Q4: Joint Probabilities** > >

Find the joint probabilities for all four cells.

<details> <strong>Solution</strong>

Joint probability = cell / grand total

• P(Morning AND Coffee) = 30/100 = 0.30
• P(Morning AND Tea) = 20/100 = 0.20
• P(Afternoon AND Coffee) = 15/100 = 0.15
• P(Afternoon AND Tea) = 35/100 = 0.35

Check: 0.30 + 0.20 + 0.15 + 0.35 = 1.00 ✅

</details> > **Q5: Simpson's Paradox** > > A university has two colleges. The combined admission data shows: > >

But within each college, admit rates are equal (50% for both genders). Explain how this is possible.

<details> <strong>Solution</strong>

This is Simpson's Paradox. It happens when:

• Women disproportionately apply to the more competitive college (lower admit rate)
• Men disproportionately apply to the less competitive college (higher admit rate)

Even though each college admits genders equally, the overall rate favors men because of the different application patterns.

Example:

• College X (competitive, 30% admit): 90% of women apply here, 10% of men
• College Y (easy, 70% admit): 10% of women apply here, 90% of men

Women: 0.90(30%) + 0.10(70%) = 34% overall Men: 0.10(30%) + 0.90(70%) = 66% overall

The overall disparity is entirely due to application patterns, not discrimination.

</details> > **Q6: Relative Risk** > >

Compute the relative risk. Interpret.

<details> <strong>Solution</strong>

Risk in exposed: 50/(50+150) = 50/200 = 0.25 Risk in unexposed: 10/(10+190) = 10/200 = 0.05

Relative Risk: 0.25/0.05 = 5.0

Interpretation: Exposed individuals are 5 times more likely to develop the disease compared to unexposed individuals.

$\boxed{\text{RR} = 5.0}$

</details> > **Q7: Building from Raw Data** > > Raw data (20 people): > > - Male, Urban: 5 > - Male, Rural: 3 > - Female, Urban: 8 > - Female, Rural: 4 > > Create the contingency table and compute the conditional distribution of location for each gender. > > <details> <strong>Solution</strong> > > **Contingency table:** > >

Conditional distributions:

Males: Urban = 5/8 = 62.5%, Rural = 3/8 = 37.5% Females: Urban = 8/12 = 66.7%, Rural = 4/12 = 33.3%

Conclusion: The distributions are very similar (62.5% vs 66.7% urban). There's little to no association between gender and urban/rural location in this sample.

</details> > **Q8: Application** > > You're analyzing a marketing campaign. You send an email to 2,000 customers (1,000 get Version A, 1,000 get Version B). Version A gets 80 clicks (8%), Version B gets 120 clicks (12%). Among those who clicked, 60% made a purchase for Version A vs 40% for Version B. > > a) Create a contingency table for email version × clicked (Yes/No). b) Create a contingency table for email version × purchased (Yes/No) among those who clicked. c) Which version has a higher click rate? Higher conversion rate? d) What would you recommend? > > <details> <strong>Solution</strong> > > **a) Version × Clicked:** > >

b) Among clickers, Version × Purchased:

Version A clickers: 80, of which 60% = 48 purchased Version B clickers: 120, of which 40% = 48 purchased

c) Comparison:

• Click rate: Version B (12%) > Version A (8%)
• Conversion rate (click→purchase): Version A (60%) > Version B (40%)
• Total purchases: Identical (48 each)

d) Recommendation:

• If the goal is clicks (e.g., ad revenue per click), choose Version B
• If the goal is purchases, they're equal (48 each)
• But Version A has a better quality of click (higher conversion rate)
• Optimal choice: Consider an A/B test with a combined metric (total revenue). If the goal is purchases, the versions are equivalent.

</details> * * * ## 🔗 Cross-References - **Next topic:** [Correlation](/notes/01-foundation-bsma1002-stats-1-week04-07-correlation) — measuring relationships between numerical variables - **Previous:** [Data Visualization](/notes/01-foundation-bsma1002-stats-1-week03-05-data-visualization) — visualizing relationships - **Week 7 (Probability):** Conditional probability from contingency tables - **BSMA1004 (Stats 2):** Chi-square test of independence - **BSMS2002 (Business Analytics):** Market segmentation with contingency tables [Join Discord](https://discord.gg/gE2m4Qrdqv) [Previous**Data Visualization**](/notes/01-foundation-bsma1002-stats-1-week03-05-data-visualization)[Next**Correlation**](/notes/01-foundation-bsma1002-stats-1-week04-07-correlation)