Homogeneity or Independence?
Introduced| English |
|---|
| homogeneity/ˈhɒməʊdʒneɪti/ |
| independence/ˌɪndɪˈpendəns/ |
Homogeneity vs. independence
- Both use a two-way table and the same chi-square math — the design tells them apart.
- Homogeneity 齐性: several groups are sampled separately; do they have the same distribution of one variable?
- Independence 独立性: one sample, two variables recorded; are the two variables related?
- Same computation, different question and hypotheses.
Test of homogeneity
- Used when you take separate samples from different populations or groups.
- $H_0$: the distribution of the categorical variable is the same across all groups.
- $H_a$: the distributions differ for at least one group.
- Example: is the favorite-sport breakdown the same across three schools?
Test of independence
- Used when you take one sample and record two categorical variables.
- $H_0$: the two variables are independent (no association).
- $H_a$: the two variables are associated.
- Example: in one survey, are gender and handedness related?
Conditions and expected counts
- Same conditions for both: random, 10%, and every expected count $\ge 5$.
- Expected counts use the same $\frac{(\text{row})(\text{col})}{\text{grand total}}$ formula.
- Degrees of freedom: $(\text{rows}-1)(\text{columns}-1)$, as in 8.4.
- Only the wording of $H_0/H_a$ changes between the two tests.
Homogeneity vs. independence is decided by the DESIGN, not the math. Several separately-sampled groups compared on one variable → homogeneity. One sample classified by two variables → independence. The chi-square statistic, expected counts, and $df$ are identical — only the hypotheses and conclusion wording differ.
Two studies, same-looking $2\times3$ table:
- Sampled $100$ boys and $100$ girls separately, recorded favorite subject → homogeneity (same distribution across the two groups?).
- Surveyed $200$ people once, recording gender and subject → independence (are the two related?).
Chi-square tests of homogeneity and independence share the same math but differ by design: homogeneity compares one variable's distribution across several separate samples; independence checks whether two variables are related in one sample. Both check random, 10%, and every expected count $\ge 5$.
One group's category distribution
Homogeneity asks if this distribution is the same across groups.
You sample 100 boys and 100 girls SEPARATELY and compare favorite subject. This is a test of...
Several separate samples compared on one variable → homogeneity.
You survey ONE sample of 200 people, recording gender AND handedness. This is a test of...
One sample, two variables → independence.
Homogeneity and independence tests use the same chi-square statistic and expected-count formula.
Only the hypotheses/design differ; the math is identical.
What decides whether a two-way chi-square test is homogeneity or independence?
Design decides it — several samples vs. one sample.
A test of ___ checks whether two variables in one sample are related (one word).
Independence: one sample, two variables.