Comparing Categorical Groups
Introduced| English | Meaning |
|---|---|
| conditional distributions/kənˈdɪʃənl ˌdɪstrɪˈbjuːʃnz/ | The same across groups, the variables appear independent. |
| independent/ˌɪndɪˈpendənt/ | Knowing whether one event occurs does not change the other's probability. |
| causation/kɔːˈseɪʃn/ | A relationship in which a change or event helps bring about another change or event. |
| confounding variable/kənˈfaʊndɪŋ ˈveərɪəbl/ | A variable associated with both an explanatory variable and a response, making their effects difficult to separate. |
Comparing groups fairly
- To see if two categorical variables are related, compare conditional distributions 条件分布 across groups.
- A conditional distribution is the set of proportions for one variable within a single group.
- "What does the subject breakdown look like for girls vs for boys?"
- Using proportions (not raw counts) makes the comparison fair when groups differ in size.
Association vs. independence
- If the conditional distributions are the same across groups, the variables appear independent 独立.
- If they differ, the variables show an association — knowing one tells you something about the other.
- The bigger the difference between groups, the stronger the association.
- "Same proportions everywhere" is the picture of no relationship.
A segmented bar chart compares the conditional distribution in each group
Differences in proportions
- Quantify the association with a difference in proportions: e.g. $60\%$ of girls vs $40\%$ of boys chose science.
- That $20$-percentage-point gap is the evidence of a relationship, stated in context.
- Always report which group is higher and by how much.
- A tiny difference may just be sampling noise; a large one signals real association.
Association is still not causation
- Even a strong association between two categorical variables does not prove causation 因果关系.
- A hidden confounding variable 混杂变量 could drive both.
- Observational data can reveal association but rarely establishes cause.
- Only a well-designed experiment (Unit 3) can support a causal claim.
To judge association, always compare conditional proportions, never raw counts — a group with more people will have bigger counts everywhere even with no relationship. And "$60\%$ vs $40\%$" is an association, not proof that being a girl causes choosing science: a confounding variable could explain the gap.
Do exercise habits relate to sleeping well? Survey results:
- Of exercisers: $70\%$ sleep well. Of non-exercisers: $45\%$ sleep well.
- The conditional distributions differ ($70\%$ vs $45\%$) → an association.
- But maybe healthy people both exercise and sleep well — a confounder. No causation proven.
Compare conditional distributions across groups: if they're the same, the variables look independent; if they differ, there's an association, measured by a difference in proportions (stated in context). Association still does not prove causation — a confounding variable may be at work.
Write from age as a confounder
- A treatment group is mostly older than the comparison group.
- The success rates differ by 15 percentage points. The lesson gap of 25 points is another study.
Why is the 15 point gap hard to credit to the treatment?
A confounder is mixed up with the factor you want to study.
What is age in this comparison?
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Age and treatment changed together.
One group's conditional distribution
Within one group, the slices are its conditional distribution.
Two categorical variables appear independent when their conditional distributions are...
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Same conditional distributions → no association → independence.
70% of exercisers vs 45% of non-exercisers sleep well. What is the difference in proportions, in percentage points?
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70 − 45 = 25 percentage points.
You should compare raw counts, not proportions, to judge association between categorical variables.
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Compare conditional proportions — counts are unfair when groups differ in size.
A hidden variable that could drive both variables and explain an association is a ___ variable.
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A confounding variable offers an alternative explanation.
Which statements about association vs. causation are correct?
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Even strong association never by itself proves causation.