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。

What is a chi-square testPermalink

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?Permalink

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 IndependencePermalink

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 HypothesesPermalink

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 DataPermalink

• Degrees of freedom: $DF=(r-1)\ast (c-1)$
• Expected frequencies: $E_{r,c}=(n_r\ast n_c)/n$
  • $E_{r,c}$ 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

ExamplePermalink

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.