Homogeneity or Independence? · 齐性还是独立性？
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| homogeneity/ˈhɒməʊdʒneɪti/ | 齐性 | qí xìng |
| independence/ˌɪndɪˈpendəns/ | 独立性 | dú lì xìng |
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.
齐性对独立性
- 两者都用双向表和相同的卡方计算——设计把它们区分开。
- 齐性：****几个组被分别抽样；它们在一个变量上是否有相同的分布？
- 独立性：**一个样本，记录两个变量；这两个变量是否相关**？
- 相同的计算，不同的问题和假设。
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?
齐性检验
- 当你从不同总体或组中取分开的样本时使用。
- $H_0$：该分类变量的分布在所有组中相同。
- $H_a$：至少一个组的分布不同。
- 例如：三所学校的最爱运动构成是否相同？
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?
独立性检验
- 当你取一个样本并记录两个分类变量时使用。
- $H_0$：两个变量独立（无关联）。
- $H_a$：两个变量相关联。
- 例如：在一次调查中，性别和惯用手是否相关？
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.
条件与期望计数
- 两者条件相同：随机、10%、每个期望计数 $\ge 5$。
- 期望计数用同一个 $\frac{(\text{row})(\text{column})}{\text{grand total}}$ 公式。
- 自由度：$(\text{rows}-1)(\text{columns}-1)$，与 8.4 相同。
- 两种检验之间只有 $H_0/H_a$ 的措辞改变。
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.
齐性对独立性由设计决定，而非数学。几个分别抽样的组在一个变量上比较 → 齐性。一个样本按两个变量分类 → 独立性。卡方统计量、期望计数和 $df$ 完全相同——只有假设和结论措辞不同。
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?).
两项研究，看起来相同的 $2\times3$ 表：
- 分别抽样 $100$ 个男生和 $100$ 个女生，记录最爱科目 → 齐性（两组分布是否相同？）。
- 一次性调查 $200$ 人，记录性别和科目 → 独立性（两者是否相关？）。
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 $\ge 5$.
卡方的齐性和独立性检验共享相同的数学，但按设计区分：齐性在几个分开的样本间比较一个变量的分布；独立性在一个样本中检查两个变量是否相关。两者都检查随机、10%、每个期望计数 $\ge 5$。
One group's category distribution · 一个组的类别分布
Homogeneity asks if this distribution is the same across groups. · 齐性问这个分布在各组间是否相同。
You sample 100 boys and 100 girls SEPARATELY and compare favorite subject. This is a test of... · 你分别抽样 100 个男生和 100 个女生并比较最爱科目。这是……检验。
Several separate samples compared on one variable → homogeneity. · 几个分开的样本在一个变量上比较 → 齐性。
You survey ONE sample of 200 people, recording gender AND handedness. This is a test of... · 你调查一个 200 人的样本，记录性别和惯用手。这是……检验。
One sample, two variables → independence. · 一个样本、两个变量 → 独立性。
Homogeneity and independence tests use the same chi-square statistic and expected-count formula. · 齐性和独立性检验使用相同的卡方统计量和期望计数公式。
Only the hypotheses/design differ; the math is identical. · 只有假设/设计不同；数学完全相同。
What decides whether a two-way chi-square test is homogeneity or independence? · 是什么决定双向卡方检验是齐性还是独立性？
Design decides it — several samples vs. one sample. · 设计决定它——几个样本对一个样本。
A test of ___ checks whether two variables in one sample are related (one word). · ___ 检验检查一个样本中两个变量是否相关（填英文一词 independence）。
Independence: one sample, two variables. · 独立性：一个样本，两个变量。