8. Associations and Causation

a. Comparing Two Categorical Variables

Yurk

Where We’ve Been

So far, we’ve examined variables one at a time:

  • Frequency tables and bar plots for a single categorical variable
  • Summary statistics and histograms for a single numerical variable
  • Hypothesis tests and CIs for a single proportion

Today’s question: how do we summarize and visualize the relationship between two categorical variables — and what does it mean to say they’re associated?

Association and Independence

When we examine two variables together:

  • Associated: there is a relationship
    • Knowing the value of one variable tells you something about the other
  • Independent: there is no relationship
    • Knowing one variable tells you nothing about the other

Our goal: decide whether two categorical variables are associated.

Comic Characters

  • 15,126 characters from DC and Marvel comics
  • Two key categorical variables:
    • Alignment: good, bad, neutral
    • Identity type: secret, public, no dual, unknown
  • Research question: is alignment associated with identity type?

Contingency Tables

Coffee Tea Total
Morning 25 15 40
Night 35 25 60
Total 60 40 100

Example: 100 students surveyed about drink preference (coffee/tea) and morning/night habit.

A contingency table displays two categorical variables together.

  • Each cell holds the count for that combination of values
  • Row totals: counts for each row category
  • Column totals: counts for each column category
  • Grand total: the sample size — often called n

Comics Contingency Table

align No Dual Public Secret Unknown Total
Bad 453 2,106 4,352 7 6,918
Good 640 2,905 2,430 0 5,975
Neutral 377 946 908 2 2,233
Total 1,470 5,957 7,690 9 15,126

Raw counts alone don’t reveal patterns clearly — the row totals are very different.

Conditional Proportions

Conditional proportion: the proportion within a row or column — calculated relative to that group’s total, not the grand total.

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

\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}}

Conditional proportions reveal patterns that raw counts hide.

Comics Conditional Proportions

align No Dual Public Secret Unknown
Bad 0.065 0.304 0.629 0.001
Good 0.107 0.486 0.407 0.000
Neutral 0.169 0.424 0.407 0.001

Key finding: Bad characters are more likely to have secret identities (62.9%) than good characters (40.7%) — the rows differ, suggesting alignment and identity type are associated.

Caffeine and Sleep

A campus survey of 200 students asked about caffeine intake and typical sleep duration:

Under 7 hrs sleep 7+ hrs sleep Total
High caffeine 75 45 120
Low caffeine 20 60 80
Total 95 105 200

  1. What proportion of high-caffeine students sleep under 7 hours? 75 / 120 \approx 0.625 — about 62.5% of high-caffeine students sleep under 7 hours.

  2. What proportion of low-caffeine students sleep under 7 hours? 20 / 80 = 0.25 — 25% of low-caffeine students sleep under 7 hours.

  3. Do caffeine intake and sleep duration appear associated? Yes. The proportion sleeping under 7 hours is much higher in the high-caffeine group (62.5%) than the low-caffeine group (25%). Knowing one tells you something about the other.

Visualizing Two Categorical Variables

Three bar-plot variants — same comics data, three views:

Dodged

Compare counts across groups

Stacked

Show composition of each group

Standardized

Compare proportions within groups

Standardized is the workhorse for spotting associations — every group on the same vertical scale.

Comics Standardized Bar Plot

Standardized bar plot of identity proportions by alignment

The Secret bands aren’t the same size across alignments — visual evidence of an association.

Summary

Summarizing two categorical variables:

  • Contingency table: counts for each combination
  • Conditional proportions: proportions within a row (or column) — reveal patterns hidden by raw counts
  • Standardized bar plot: visualize the conditional proportions side-by-side

Key concepts:

  • Associated: the conditional proportions differ across groups
  • Independent: the conditional proportions are (essentially) the same

Up next (8b): when we can and can’t conclude that one variable causes changes in the other.

References

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