Comparing Categorical Groups
Introduced| English |
|---|
| conditional distributions/kənˈdɪʃənl ˌdɪstrɪˈbjuːʃnz/ |
| independent/ˌɪndɪˈpendənt/ |
| causation/kɔːˈseɪʃn/ |
| confounding variable/kənˈfaʊndɪŋ ˈveərɪəbl/ |
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.
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.
One group's conditional distribution
Within one group, the slices are its conditional distribution.
Two categorical variables appear independent when their conditional distributions are...
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?
70 − 45 = 25 percentage points.
You should compare raw counts, not proportions, to judge association between categorical variables.
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.
A confounding variable offers an alternative explanation.
Which statements about association vs. causation are correct?
Even strong association never by itself proves causation.