Chi-Square Tests With Expected Counts and Residuals

The chi-square statistic measures the overall discrepancy between observed and expected frequencies. It sums the squared differences between observed and expected values, divided by the expected values: χ² = Σ(O-E)²/E. A larger χ² indicates a greater deviation from what was expected under the null hypothesis. The value itself has no inherent meaning until compared to the chi-square distribution with the appropriate degrees of freedom.

The rule that expected frequencies should be ≥ 5 is a guideline to ensure the chi-square approximation is valid. When expected counts are too low, the chi-square distribution doesn't accurately model the test statistic. If you have expected values < 5, consider: (1) combining adjacent categories to increase expected counts, (2) using Fisher's exact test for 2×2 tables, or (3) using simulation-based methods. Some sources accept 80% of cells having E ≥ 5 with none below 1.

A standardized residual is (Observed - Expected) / √Expected, showing how much each cell contributes to the overall χ² statistic. Residuals > 2 or < -2 indicate cells with notable deviations from expectation. Positive residuals mean observed counts exceed expected; negative means fewer than expected. Examining residuals helps identify which specific categories or cells are driving a significant result.

Independence means two variables are not related—knowing one tells you nothing about the other. The chi-square test of independence tests whether the row and column variables in a contingency table are independent. If we reject independence, we conclude there's an association (relationship) between the variables. However, association doesn't imply causation—it just means the variables vary together in some systematic way.

Use goodness-of-fit when you have ONE categorical variable and want to compare observed frequencies to a specific expected distribution (e.g., testing if a die is fair). Use the test of independence when you have TWO categorical variables and want to test if they're related (e.g., is there an association between education level and voting preference). The goodness-of-fit compares to a theoretical distribution; independence compares to what you'd expect if variables were unrelated.

The chi-square test is always right-tailed because the χ² statistic can only be positive (it's a sum of squared terms). Large χ² values indicate large deviations from expected frequencies, which is evidence against the null hypothesis. Small χ² values (near zero) indicate observed frequencies are close to expected, supporting the null hypothesis. There's no such thing as 'too good' a fit in the left tail.

The chi-square test is designed for categorical (count) data, not continuous measurements. If you have continuous data, you could: (1) categorize it into bins and use chi-square, though this loses information, (2) use parametric tests like t-tests or ANOVA for comparing means, or (3) use non-parametric tests like Mann-Whitney or Kruskal-Wallis. Converting continuous to categorical should be done thoughtfully, as bin choice affects results.

Chi-square tests for association between categorical variables, while correlation (like Pearson's r) measures linear relationships between continuous variables. For ordinal categorical data, you might use Spearman's correlation or Kendall's tau instead. A significant chi-square tells you variables are associated but doesn't measure strength or direction. For 2×2 tables, Cramér's V or phi coefficient can measure association strength after a significant chi-square.

For 2×2 tables, the standard chi-square test works but has special considerations: (1) With small expected counts (< 5), use Fisher's exact test instead, (2) Some apply Yates' continuity correction to reduce chi-square slightly, though this is debated, (3) The degrees of freedom is always 1 for 2×2 tables. You can also use odds ratios or relative risk to describe the association. McNemar's test is used for paired 2×2 data.

A complete chi-square report should include: (1) χ² value and degrees of freedom: χ²(df) = value, (2) p-value, (3) sample size (N), (4) a description of what was tested, (5) whether the result was significant at your chosen α level, (6) for independence tests, consider reporting an effect size like Cramér's V, and (7) mention any cells with low expected frequencies. For example: 'χ²(2) = 8.45, p = 0.015, N = 150. There was a significant association between variables.'