Chi-Square Test

Formula

\[ \chi^2 = \sum_{i} \frac{(O_i - E_i)^2}{E_i} \]

Parameters

• \(O_i\): observed counts
• \(E_i\): expected counts under the null

What it means

Compares observed counts to expected counts under a null model.

What it's used for

• Goodness-of-fit tests for categorical distributions.
• Independence tests in contingency tables.

Key properties

• Under \(H_0\), the statistic is approximately \(\chi^2\)-distributed.
• Degrees of freedom depend on the number of categories.

Common gotchas

• Expected counts should not be too small (rule of thumb: \(E_i\ge 5\)).
• For small samples, consider exact tests.

Example

Observed \([18, 22]\), expected \([20, 20]\): \(\chi^2=(18-20)^2/20 + (22-20)^2/20=0.4\).

How to Compute (Pseudocode)

Input: data, null hypothesis H0, test statistic T
Output: test statistic and p-value decision summary

compute the observed test statistic T_obs from the data
obtain the null distribution (analytic approximation or exact table, depending on the test)
compute the p-value from the null distribution and tail convention
compare p-value to alpha (if making a decision)
return T_obs and p-value

Complexity

• Time: Depends on the specific test (summary-statistic computation is often linear in sample size; p-value computation may be constant-time with a CDF call or more expensive if resampling is used)
• Space: Depends on whether intermediate summaries or resampled/null distributions are materialized
• Assumptions: Test-specific assumptions (independence, variance structure, distributional assumptions) determine validity and exact computation details