Fisher's Exact Test

When sample sizes are small and expected cell counts fall below 5, chi-square approximation is unreliable. Fisher's exact test calculates the exact probability of the observed table (and more extreme tables) under the null hypothesis of independence. Licklider detects the small-sample condition automatically and switches to Fisher's exact test without requiring researcher input.

STEP 1 — The Scenario

Researcher Profile

A clinical researcher conducting a safety analysis of a Phase I trial. They are analyzing whether a specific adverse event (hepatotoxicity) occurred more frequently in the treated arm vs. placebo. Due to the rare nature of the event, counts are small.

Research Question

"Is hepatotoxicity significantly more frequent in the treatment arm compared to placebo?"

Dataset

Treatment: n=24, hepatotoxicity n=6 (25%)
Placebo: n=26, hepatotoxicity n=1 (3.8%)
2×2 table: Arm (Treatment/Placebo) × Event (Yes/No)
Expected cell count for Placebo + Event = (27×26)/50 = ~14.0 … but actual count = 1

STEP 2 — Licklider Analysis Path

1. Data Upload & Table Construction

Licklider accepts individual-level data (PatientID, Arm, Event columns) or a pre-formatted 2×2 table. The contingency table is constructed automatically with row and column totals.

2. Expected Cell Frequency Check & Test Selection

Licklider checks all expected cell counts: Treatment+No: large, Treatment+Yes: ~3.5, Placebo+No: large, Placebo+Yes: ~1.4. Since expected count <5 in at least one cell, Fisher's exact test runs alongside the chi-square workflow and the Inspector highlights the small-cell advisory: "Expected cell count <5 detected — review Fisher's exact result alongside chi-square."

3. Fisher's Exact Test

The exact p-value is calculated from the hypergeometric distribution. Both the two-sided and one-sided p-values are reported. For a safety analysis where the prior hypothesis is directional (treatment increases hepatotoxicity), the one-sided p-value may be appropriate — but Licklider requires a recorded justification if one-sided is selected (One-Sided Test Lock, 4.4.3).

4. Effect Size: Odds Ratio

For a 2×2 table, the Odds Ratio with 95% CI (Fisher's exact method, or Woolf method) is reported alongside the exact p-value. For rare events, OR and RR converge — but Licklider notes this and reports both.

5. Figure Generation

A simple bar chart showing event rates (%) for each arm with 95% CI (Wilson score interval for proportions). The exact p-value and OR are annotated. Individual patient counts are shown in the figure for complete transparency.

STEP 3 — Guards & Disclosures Activated

Expected Cell Frequency Guard (Auto-trigger)

The guard surfaces Fisher's exact test alongside chi-square when any expected cell count <5. Expected cell counts are always disclosed in the statistical report, along with the small-cell advisory that prompts review of the Fisher result.

One-Sided Test Lock (4.4.3)

If the researcher selects a one-sided p-value for the safety analysis, Licklider requires a written justification: "Which direction was pre-specified as the hypothesis of interest, and when was this specified (before or after data collection)?" The justification and the two-sided p-value are both disclosed in the Methods text.

Effect Size + CI + N Mandatory (3.1.2)

The OR and 95% CI are required alongside the exact p-value. Event counts and percentages per arm are also mandatory in the Methods text or figure legend.

N Disclosure (1.2.1)

Total n enrolled, n per arm, event count per arm, and event rate (%) per arm are all disclosed. For safety analyses, the disclosure of the absolute event rate difference (ARD) is also required.

STEP 4 — Export Package

Figure: Event rate bar chart with Wilson CI error bars — 300 dpi TIFF/PDF
Statistical Report: 2×2 table (observed + expected counts); Fisher's exact p (two-sided); OR with 95% CI; event rates per arm; ARD; NNT (if clinically relevant)
Methods Text: "The association between treatment arm and hepatotoxicity incidence was assessed using Fisher's exact test (selected because expected cell count <5 for Placebo + Event). Effect size is reported as OR with 95% CI (Fisher's exact method). The two-sided p-value is reported. Treatment: n=24, events=6 (25%); Placebo: n=26, events=1 (3.8%)."
Figure Legend: Event rates, CI method, exact p, OR, CI, n per arm
Preprocessing Audit Log: Expected cell check result; test selection rationale; one-sided test justification (if applicable)