Chi-Square Tests¶

Chi-square tests evaluate whether observed frequencies in categorical data differ significantly from expected frequencies. These tests are among the most widely used tools for analyzing count data, enabling researchers to assess goodness of fit to a theoretical distribution and to test for independence between two categorical variables.

Chi-Square Test Statistic¶

The chi-square test statistic measures the discrepancy between observed counts \(O_i\) and expected counts \(E_i\) across \(k\) categories:

Under the null hypothesis, this statistic follows a chi-square distribution with degrees of freedom that depend on the specific test context. Large values of \(\chi^2\) indicate that the observed data deviate substantially from the expected pattern.

Goodness-of-Fit Test¶

The goodness-of-fit test checks whether a sample of categorical observations matches a hypothesized distribution. Given \(k\) categories with observed counts \(O_1, \ldots, O_k\) and expected counts \(E_1, \ldots, E_k\) (where \(\sum E_i = \sum O_i = n\)), the hypotheses are

where \(p_{i,0}\) are the hypothesized proportions and \(E_i = n \cdot p_{i,0}\).

The degrees of freedom are \(k - 1\) (or \(k - 1 - m\) if \(m\) parameters were estimated from the data):

SciPy Implementation¶

The scipy.stats.chisquare function performs the one-sample chi-square goodness-of-fit test:

```python from scipy import stats

Observed counts from a die-rolling experiment¶

observed = [18, 22, 16, 21, 13, 10]

Test against uniform distribution (equal expected frequencies)¶

chi2_stat, p_value = stats.chisquare(observed) print(f"Chi-square statistic: {chi2_stat:.4f}") print(f"p-value: {p_value:.4f}")

Test against custom expected frequencies¶

expected = [16.67, 16.67, 16.67, 16.67, 16.67, 16.67] chi2_stat, p_value = stats.chisquare(observed, f_exp=expected) print(f"Chi-square statistic: {chi2_stat:.4f}") print(f"p-value: {p_value:.4f}") ```

When f_exp is omitted, the function assumes a uniform distribution across all categories.

Test of Independence¶

The test of independence determines whether two categorical variables are associated. Given an \(r \times c\) contingency table with observed counts \(O_{ij}\), the expected counts under independence are

where \(R_i = \sum_{j} O_{ij}\) is the \(i\)-th row total, \(C_j = \sum_{i} O_{ij}\) is the \(j\)-th column total, and \(n\) is the grand total.

The hypotheses are

The test statistic uses the same formula with degrees of freedom \((r - 1)(c - 1)\):

SciPy Implementation¶

The scipy.stats.chi2_contingency function handles contingency table analysis:

```python import numpy as np from scipy import stats

Contingency table: treatment outcome by group¶

Success Failure¶

Drug A 45 15¶

Drug B 30 30¶

Placebo 20 40¶

observed_table = np.array([[45, 15], [30, 30], [20, 40]])

chi2_stat, p_value, dof, expected = stats.chi2_contingency(observed_table) print(f"Chi-square statistic: {chi2_stat:.4f}") print(f"Degrees of freedom: {dof}") print(f"p-value: {p_value:.6f}") print(f"Expected frequencies:\n{expected}") ```

The function returns the test statistic, p-value, degrees of freedom, and the matrix of expected frequencies under \(H_0\).

Effect Size¶

The chi-square statistic depends on sample size, so a significant result does not necessarily indicate a strong association. Cramer's \(V\) provides a normalized measure of effect size for contingency tables:

where \(V \in [0, 1]\), with \(V = 0\) indicating no association and \(V = 1\) indicating perfect association.

Summary¶

Chi-square tests assess categorical data through goodness-of-fit (scipy.stats.chisquare) and independence testing (scipy.stats.chi2_contingency). Both tests compare observed and expected frequencies using the same \(\chi^2\) statistic but differ in how expected counts and degrees of freedom are computed. The chi-square approximation requires sufficiently large expected counts, and effect sizes like Cramer's \(V\) should supplement p-values for practical interpretation.

Exercises¶

Exercise 1. A die is rolled 120 times with outcomes: {1: 25, 2: 17, 3: 15, 4: 23, 5: 22, 6: 18}. Test whether the die is fair using a chi-square goodness-of-fit test at \(\alpha = 0.05\).

from scipy import stats

observed = [25, 17, 15, 23, 22, 18]
expected = [20, 20, 20, 20, 20, 20]
chi2, p = stats.chisquare(observed, expected)
print(f"Chi-square: {chi2:.4f}, p: {p:.4f}")
print(f"Decision: {'Reject H0 (unfair)' if p < 0.05 else 'Fail to reject H0 (fair)'}")

Exercise 2. A contingency table records preferences: Men: [45 coffee, 30 tea, 25 juice], Women: [35 coffee, 40 tea, 45 juice]. Test for independence between gender and beverage preference. Compute Cramer's V.

import numpy as np
from scipy import stats

table = np.array([[45, 30, 25], [35, 40, 45]])
chi2, p, dof, expected = stats.chi2_contingency(table)
n = table.sum()
v = np.sqrt(chi2 / (n * (min(table.shape) - 1)))

print(f"Chi-square: {chi2:.4f}, p: {p:.4f}, dof: {dof}")
print(f"Cramer's V: {v:.4f}")

Exercise 3. Generate a \(3 \times 3\) contingency table under perfect independence (expected = observed). Verify that the chi-square statistic is 0 and the p-value is 1.

import numpy as np
from scipy import stats

table = np.array([[30, 30, 30], [30, 30, 30], [30, 30, 30]])
chi2, p, dof, expected = stats.chi2_contingency(table)
print(f"Chi-square: {chi2:.6f}, p: {p:.4f}")