Chi-Square Test for Independence

To test whether two categorical variables are independent

Yao Yao on June 9, 2015

总结自:

  • 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。


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.

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.



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