Pearson's Chi-Square Test of Independence

Independence creates expected counts

For row $r$ and column $c$, independence implies

Pearson's statistic adds standardized squared deviations (Pearson, 1900).

A two-by-two table

Every expected count is 25. Each cell contributes $(5^2)/25=1$, so the statistic equals 4. The degrees of freedom are $(2-1)(2-1)=1$.

Assumptions and implementation

Observations must be independent. Expected counts should support the large-sample approximation. Sparse tables can require exact or simulation methods.

Continuity correction, likelihood-ratio statistics, and missing-category handling can change software output. A signed-residual heatmap locates cells with large values of $(O-E)/\sqrt E$.

From a global statistic to cell patterns

The Pearson statistic summarizes all departures from independence. Each term $(O_{rc}-E_{rc})^2/E_{rc}$ is nonnegative, so cells with large standardized gaps drive the total. Report the contingency table with row or column percentages. Then show adjusted residuals when readers need to locate the association. Cramér's $V$ gives a scale-free summary of association strength, while the chi-square statistic also grows with sample size.

Suppose one cell contains 30 observations and its independence expectation is 20. That cell contributes $(30-20)^2/20=5$ before contributions from the other cells are added. Its positive residual means the category combination occurs more often than independence predicts.

Conditions, sparse tables, and coding

Each sampled unit should contribute to one table cell. Repeated observations from the same person violate the independent-count model. Survey clusters and sampling weights also require design-based procedures. Inspect expected counts, not observed counts, when assessing the usual reference distribution. Sparse tables can give a poor chi-square approximation. Combine categories only when the scientific definition supports the combination. Exact or Monte Carlo procedures can address small tables without changing category meaning.

Build expected counts directly from the margins and verify that they reproduce every row and column total. Preserve structural zeros, which represent impossible combinations, separately from sampled zeros. Report missing-category handling because listwise deletion changes both margins. For ordered categories, a general independence test discards order. A trend test may answer a more focused question. (Pearson, 1900)

Further reading

Pearson introduces the chi-square goodness-of-fit framework (Pearson, 1900). Agresti develops contingency-table analysis and sparse-table cautions (Agresti, 2013).