Chi-Square, Fisher's Exact Test, Odds Ratios, and Relative Risk
Comparing observed versus expected frequencies, checking Cochran's cell-size rule, and interpreting 95% confidence intervals around OR and RR.
In this lesson you will learn
- 1Calculate Expected cell frequencies (E = RowTotal × ColTotal / N) and apply Cochran's threshold (E < 5) to switch to Fisher's Exact Test.
- 2Distinguish Odds Ratio (OR = ad / bc) for Case-Control studies from Relative Risk (RR) for prospective Cohort and RCT designs.
- 3Evaluate statistical significance of OR and RR by checking whether the 95% Confidence Interval excludes 1.0, and apply McNemar's test to paired binary outcomes.
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Contingency Table Anatomy & Risk Ratios
This section defines Expected Frequency (E), Pearson's Chi-Square, Fisher's Exact Test, Odds Ratio (OR), Relative Risk (RR), and McNemar's paired test.
Reporting Relative Risk in a retrospective case-control study or reporting Pearson's Chi-square when an expected cell count is 2.1 are two of the most common categorical audit errors.
When both your exposure (e.g. Treatment vs Control) and your clinical endpoint (e.g. Remission vs Relapse) are categorical, observations are cross-tabulated in a contingency table:
| Method / Metric | Mathematical Formula | When Required |
|---|---|---|
| Expected Count (E) | E = (Row Total × Column Total) / Grand Total N |
Checked before choosing between Pearson χ² and Fisher's Exact test. |
| Fisher's Exact Test | Exact hypergeometric probability across marginal totals. | Required in 2×2 tables whenever any cell has E < 5 (Cochran's rule). |
| Odds Ratio (OR) | (a / b) ÷ (c / d) = (a × d) / (b × c) |
Retrospective Case-Control studies and Binary Logistic Regression. |
| Relative Risk (RR) | [a / (a + b)] ÷ [c / (c + d)] |
Prospective Cohort studies and Randomized Controlled Trials (incidence known). |
| McNemar's Test | χ² = (b − c)² / (b + c) on discordant pairs. |
Paired binary outcomes measured Pre and Post on the same participants. |
Clinical Example
A small surgical safety audit compares wound infection between Technique A (1 infection, 19 clean) and Technique B (7 infections, 13 clean); total N = 40.
Interpretation: Even though one observed cell is 7, the Expected count in both infection cells is 4.0 (< 5). Cochran's rule depends on Expected counts (not Observed counts), so Fisher's Exact Test (two-sided p = .044) is reported.
Common Pitfall
Checking Cochran's '< 5' rule against Observed cell counts (O) instead of Expected cell counts (E = RowTotal × ColTotal / N).
Included with Any Analysez AI Study Package
This deep interactive chapter (Chi-Square, Fisher's Exact Test, Odds Ratios, and Relative Risk) and its clinical simulation labs are unlocked for everyone who has bought any Analysez AI study package.