Homogeneity or Independence?

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?

Homogeneity compares populations on one variable; independence asks about two variables in one sample

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.

Write from independence in one sample

• One random sample of people is asked about two categorical variables.
• The question is whether the variables are associated.