Help: Categorical × Categorical — Chi-Square Test

Cat × Cat — Chi-Square Test

Detailed analysis modal for the association between two categorical variables, including the chi-square test of independence, effect sizes, and optional exact tests for small samples.

Overview

When you click a Cat × Cat cell in the association matrix, Statulator opens a modal with four tabs:

Statistics — Observed frequency table with row and column totals, expected frequency table (cells below 5 highlighted in red), Pearson chi-square statistic with degrees of freedom and p-value, Cramér’s V effect size, and an assumption check for minimum expected cell counts. For 2 × 2 tables, Fisher’s exact test, odds ratio, and relative risk with 95 % CIs are also displayed.
Graph — A grouped bar chart or mosaic plot visualising the joint distribution of the two categories.
Diagnostics — Diagnostic views including standardised residual plots to identify which cells contribute most to the chi-square statistic.
Interpretation — Auto-generated plain-language summary covering the test result, effect size, assumption status, and practical recommendations.

You can download the complete analysis as a PDF report or export the graph as a PNG image using the buttons in the modal footer.

Worked Example

Scenario: Treatment Outcome by Drug Group

A clinical trial has 400 patients assigned to Drug A, Drug B, or Placebo. The outcome is recorded as Improved or Not Improved. The researcher wants to know whether improvement rates differ across treatment groups.

Steps in Statulator:

1 Load the CSV on the Dataset Analysis page.

2 Click Select Variables, confirm both DrugGroup and Outcome are detected as Categorical, and save.

3 Click Stat Analysis to generate the matrix.

4 Click the cell at the intersection of DrugGroup and Outcome.

What you will see:

The Observed Frequencies table shows a 3 × 2 contingency table with row totals (per drug group) and column totals (Improved / Not Improved). The Expected Frequencies table shows what counts you would expect if treatment and outcome were independent.

The chi-square statistic is χ² = 18.5, df = 2, p < 0.001 — a highly significant association. Cramér’s V = 0.22 indicates a small-to-moderate effect size. The assumption check confirms all expected cell counts are above 5.

Since this is a 3 × 2 table (not 2 × 2), Fisher’s exact test and odds ratio are not shown — these appear only for 2 × 2 tables.

Interpretation Guide

Observed vs Expected Frequencies

The observed frequency table shows the actual count of observations in each combination of categories. The expected frequency table shows the counts you would see if the two variables were completely independent. Large discrepancies between observed and expected counts drive a large chi-square statistic.

Pearson Chi-Square Test

This tests the null hypothesis that the two variables are independent. A small p-value (typically < 0.05) leads to rejecting independence and concluding that the variables are associated. The chi-square statistic itself is not directly interpretable as an effect size — use Cramér’s V for that.

Cramér’s V

A standardised effect size ranging from 0 (no association) to 1 (perfect association). Common benchmarks: V < 0.1 = negligible, 0.1–0.3 = small, 0.3–0.5 = medium, > 0.5 = large. These benchmarks depend on the table dimensions; for large tables, even moderate V values may be substantively meaningful.

Fisher’s Exact Test (2 × 2 only)

When the contingency table is exactly 2 × 2, Statulator also provides Fisher’s exact test. This is preferred over chi-square when expected cell counts are small (below 5). The exact p-value does not rely on the chi-square approximation.

Odds Ratio & Relative Risk (2 × 2 only)

For 2 × 2 tables, the odds ratio (OR) compares the odds of the outcome between the two groups, while the relative risk (RR) compares probabilities. Both are shown with 95 % confidence intervals. An OR or RR of 1.0 indicates no association; values above 1 indicate higher odds/risk in the first group.

Assumption Check: Expected Cell Counts

The chi-square approximation is reliable when all expected cell counts are at least 5. If any expected count falls below 5, it is highlighted in red in the expected frequency table, and the assumption check will show a warning. In this case, Fisher’s exact test (for 2 × 2 tables) is recommended, or consider collapsing sparse categories.

Formulas

Chi-Square Test of Independence

\[ \chi^2 = \sum_{i}\sum_{j} \frac{(O_{ij} - E_{ij})^2}{E_{ij}}, \quad E_{ij} = \frac{n_{i\cdot}\,n_{\cdot j}}{N} \]

\(O_{ij}\) = observed count in row \(i\), column \(j\).
\(E_{ij}\) = expected count; \(\text{df} = (r-1)(c-1)\).

Cramér’s V

\[ V = \sqrt{\frac{\chi^2}{N\,(\min(r,c)-1)}} \]

\(r\) = number of rows; \(c\) = number of columns; \(N\) = total observations.

Odds Ratio & Relative Risk (2 × 2)

\[ \text{OR} = \frac{a \cdot d}{b \cdot c}, \quad \text{RR} = \frac{a/(a+b)}{c/(c+d)} \]

95 % CI for OR via log-transform:

\[ \ln(\text{OR}) \pm 1.96\sqrt{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}} \]

Fisher’s Exact Test (2 × 2)

\[ p = \frac{\binom{a+b}{a}\binom{c+d}{c}}{\binom{N}{a+c}} \]

Two-sided p-value is the sum of all table probabilities ≤ the observed probability.

Assumptions & Requirements

Independence: Each observation should be independent (no repeated measures or clustering). Violations can inflate the chi-square statistic.
Expected cell counts: All expected counts should be ≥ 5 for the chi-square approximation to be valid. When this is violated, cells are flagged in red and Fisher’s exact test is recommended for 2 × 2 tables.
Mutually exclusive categories: Each observation should fall into exactly one category for each variable.
Sample size: The test requires at least 1 observation in every cell. Very sparse tables (many zero cells) may produce unreliable results; consider collapsing categories.
Missing data: Only observations with non-missing values for both variables are included (complete cases).

Textbook Examples

Medicine

Association between Smoking Status (Never, Former, Current) and Lung Cancer (Yes, No) in 1,200 hospital patients.

Results: χ² = 42.3, df = 2, p < 0.001, Cramér's V = 0.19. Expected counts all above 5.
Interpretation: A significant association exists between smoking status and lung cancer, though V = 0.19 indicates a small effect. Current smokers had the highest observed-to-expected ratio.

Education

Relationship between Teaching Method (Lecture, Active Learning) and Pass/Fail outcome in 300 students.

Results: χ² = 6.4, df = 1, p = 0.011, OR = 1.85 [1.15, 2.98], Fisher's exact p = 0.013.
Interpretation: Active learning students had 85 % higher odds of passing. Both chi-square and Fisher’s exact test agree. The 2 × 2 table allows odds ratio and relative risk reporting.

Agriculture

Association between Soil Type (Clay, Loam, Sand) and Disease Presence (Diseased, Healthy) across 500 plants.

Results: χ² = 28.1, df = 2, p < 0.001, V = 0.24.
Interpretation: Disease incidence differs significantly across soil types. Sand soil had a higher-than-expected disease rate, suggesting soil drainage may play a role.

Social Science

Gender (Male, Female) vs preferred news source (TV, Online, Print, Radio) in 2,000 survey respondents.

Results: χ² = 15.7, df = 3, p = 0.001, V = 0.09. One cell (Female × Radio) had expected count 4.2.
Interpretation: A statistically significant but very small association. The low expected count in one cell is flagged; for this 2 × 4 table, the chi-square is still reasonably valid as only one cell falls below 5.

References

Pearson, K. (1900). On the criterion that a given system of deviations can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 50(302), 157–175.
Fisher, R. A. (1922). On the interpretation of χ² from contingency tables. Journal of the Royal Statistical Society, 85(1), 87–94.
Fisher, R. A. (1935). The logic of inductive inference (with discussion). Journal of the Royal Statistical Society, 98(1), 39–82.
Cramér, H. (1946). Mathematical Methods of Statistics. Princeton University Press.
Agresti, A. (2013). Categorical Data Analysis (3rd ed.). Wiley.
Cochran, W. G. (1954). Some methods for strengthening the common χ² tests. Biometrics, 10(4), 417–451.

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