## Chi-Square Test for Categorical Association ### Scenario Breakdown - **Two categorical variables**: smoking status (binary) and COPD status (binary) - **Cross-sectional design**: no temporal relationship - **Large sample**: n = 500 (adequate for chi-square) - **Research question**: Is there an association between two categorical variables? ### Why Chi-Square Test? **Key Point:** The chi-square test of independence is the standard test for examining the association between two categorical variables in a 2×2 contingency table or larger. **High-Yield:** The chi-square statistic is calculated as: $$\chi^2 = \sum \frac{(O - E)^2}{E}$$ where O = observed frequency and E = expected frequency under the null hypothesis of independence. ### Contingency Table Structure | | COPD Yes | COPD No | Total | | --- | --- | --- | --- | | **Smoker** | a | b | a+b | | **Non-smoker** | c | d | c+d | | **Total** | a+c | b+d | n | Expected frequency for each cell: $E = \frac{(\text{row total}) \times (\text{column total})}{n}$ ### Test Selection Algorithm ```mermaid flowchart TD A[Research question about association?]:::decision --> B{Data type?}:::decision B -->|Both categorical| C{Sample size adequate?}:::decision B -->|One continuous, one categorical| D[t-test or ANOVA]:::action B -->|Both continuous| E[Correlation/Regression]:::action C -->|Yes, all expected freq ≥5| F[Chi-square test]:::outcome C -->|No, expected freq <5| G[Fisher's exact test]:::action ``` ### Comparison: When to Use Which Test | Test | Variable 1 | Variable 2 | Sample Size | Use Case | | --- | --- | --- | --- | --- | | **Chi-square** | Categorical | Categorical | Large (n>30) | Association between two categorical variables | | Fisher's exact | Categorical | Categorical | Small (n<30) or expected freq <5 | 2×2 table with small numbers | | Unpaired t-test | Categorical (2 groups) | Continuous | Any | Compare means between two groups | | Pearson correlation | Continuous | Continuous | Any | Linear relationship between two continuous variables | | Paired t-test | Categorical (paired) | Continuous | Any | Compare means in matched/paired design | **Mnemonic:** **CACA** — **C**ategorical + **C**ategorical = **A**ssociation test = Chi-square (or Fisher's if small n) **Clinical Pearl:** In epidemiological studies, chi-square is routinely used to test associations between exposure (e.g., smoking) and outcome (e.g., COPD) in cross-sectional and case-control designs. **Warning:** Chi-square assumes independence of observations and requires expected frequencies ≥5 in at least 80% of cells. If this assumption is violated, use Fisher's exact test instead. [cite:Park 26e Ch 10; Bland JM & Altman DG, BMJ 1994]
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