Chi-Square Test for Independence

总结自：

• Stat Trek: Chi-Square Test for Independence
• Wikipedia: Chi-squared test

• Chi: [kaɪ]

其实还有 Chi-square test for variance in a normal population 以及 Chi-squared distribution，这里不涉及。

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What is a chi-square test

A chi-square test is also referred to as $ \chi^2 $ test ($ \chi $ 这个符号在 latex 里就是 \chi).

The test is applied when you have two categorical variables from a single population. It is used to determine whether these two categorical variables are independent.

Digress: What is a categorical variable?

Variables can be classified as categorical (aka, qualitative) or quantitative (aka, numerical).

• Categorical variables take on values that are names or labels. E.g.
  • the color of a ball (red, green, blue, etc.)
  • the breed of a dog (collie, shepherd, terrier, etc.)
• Quantitative variables represent a measurable quantity. E.g.
  • the population of a city

When to Use Chi-Square Test for Independence

The test procedure is appropriate when the following conditions are met:

• The sampling method is simple random sampling.
• The variables under study are each categorical.
• If sample data are displayed in a contingency table, the expected frequency count for each cell of the table is at least 5.

State the Hypotheses

Given variable $ A $ (which has $ r $ levels), and variable $ B $ (which has $ c $ levels),

• $ H_0 $: variable $ A $ and variable $ B $ are independent.
• $ H_a $: variable $ A $ and variable $ B $ are not independent.

Analyze Sample Data

• Degrees of freedom: $ DF = (r - 1) * (c - 1) $
• Expected frequencies: $ E_{r,c} = (n_r * n_c) / n $
  • $ E_{r,c} $ is the expected frequency count for level $ r $ of variable $ A $ and level $ c $ of variable $ B $
  • $ n_r $ is the total number of sample observations at level $ r $ of variable $ A $
  • $ n_c $ is the total number of sample observations at level $ c $ of variable $ B $
  • $ n $ is the total sample size
• Test statistic: $ \chi^2 = \sum{\left [ \frac{(O_{r,c} - E_{r,c})^2}{E_{r,c}} \right ]} $
  • $ O_{r,c} $ is the observed frequency count for level $ r $ of variable $ A $ and level $ c $ of variable $ B $
• p-value: 计算时需要 $ DF $ 和 $ \chi^2 $ 两个值，可以使用 Chi-Square Calculator: Online Statistical Table

Example

Question: Is there a gender gap? Do the men’s voting preferences differ significantly from the women’s preferences?

• $ H_0 $: “Gender” and “Voting Preference” are independent.
• $ H_a $: “Gender” and “Voting Preference” are not independent.

\[\begin{align} DF &= (r - 1) * (c - 1) = (2 - 1) * (3 - 1) = 2 \newline E_{1,1} &= (400 * 450) / 1000 = 180000/1000 = 180 \newline E_{1,2} &= (400 * 450) / 1000 = 180000/1000 = 180 \newline E_{1,3} &= (400 * 100) / 1000 = 40000/1000 = 40 \newline E_{2,1} &= (600 * 450) / 1000 = 270000/1000 = 270 \newline E_{2,2} &= (600 * 450) / 1000 = 270000/1000 = 270 \newline E_{2,3} &= (600 * 100) / 1000 = 60000/1000 = 60 \newline \chi^2 &= (200 - 180)^2/180 + (150 - 180)^2/180 + (50 - 40)^2/40 + \newline & \phantom{\{\}=1} (250 - 270)^2/270 + (300 - 270)^2/270 + (50 - 60)^2/60 \newline &= 2.22 + 5.00 + 2.50 + 1.48 + 3.33 + 1.67 = 16.2 \end{align}\]

查表得 $ P(DF=3, \chi^2>16.2) = 0.0003 $.

Since the p-value (0.0003) is less than the significance level (0.05), we cannot accept the null hypothesis. Thus, we conclude that there is a relationship between gender and voting preference.