Chi-Square Test

The chi-square test of independence assesses whether two categorical variables are associated. Licklider checks the expected cell frequency assumption, reports Cramér's V as the effect size, and enforces that the test result cannot be exported without disclosing whether the chi-square approximation is valid.

STEP 1 — The Scenario

Researcher Profile

An epidemiologist studying whether APOE genotype (ε2/ε3, ε3/ε3, ε3/ε4 — 3 levels) is associated with Alzheimer's disease status (Affected vs. Unaffected — 2 levels) in a case-control sample.

Research Question

"Is APOE genotype distribution significantly different between Alzheimer's-affected and unaffected individuals?"

Dataset

Total n=320: 160 affected, 160 unaffected
Genotype distribution (observed): ε2/ε3 (n=52), ε3/ε3 (n=144), ε3/ε4 (n=124)
Contingency table: 2 rows (status) × 3 columns (genotype)

STEP 2 — Licklider Analysis Path

1. Data Upload & Contingency Table Construction

Licklider accepts either: (a) individual-level data with one row per participant (status + genotype columns), or (b) a pre-formatted contingency table. It automatically constructs the 2×3 table with row and column totals.

2. Expected Cell Frequency Check

Before running the chi-square test, Licklider calculates expected frequencies for each cell (E = row total × column total / grand total). It reports: "All expected cell counts are ≥5 — chi-square approximation is valid." If any cell has E < 5, Fisher's exact test is recommended instead.

3. Chi-Square Test

Pearson's chi-square test of independence: χ², df, and p-value are calculated. For a 2×3 table: df = (rows−1) × (columns−1) = 1×2 = 2.

4. Effect Size: Cramér's V

Cramér's V is calculated: V = √(χ²/(n × min(rows−1, cols−1))). For a 2×3 table, min(1,2)=1, so V = √(χ²/n). Bias-corrected Cramér's V is offered for small samples. Interpretation guidance: V≈0.10 small, V≈0.30 medium, V≈0.50 large.

5. Post-Hoc Cell-Level Analysis

If the overall test is significant, Licklider calculates standardized residuals for each cell to identify which genotype-status combinations drive the association. Cells with |standardized residual| > 2 are highlighted as the primary contributors.

6. Figure Generation

A stacked bar chart showing genotype proportions by disease status, with observed counts annotated in each segment. Alternatively, a grouped bar chart is offered. Cramér's V and p-value are annotated.

STEP 3 — Guards & Disclosures Activated

Expected Cell Frequency Guard

If any expected cell count is < 5, Licklider blocks the chi-square test and requires the researcher to use Fisher's exact test, merge categories, or acknowledge the limitation. The expected cell counts are always disclosed in the statistical report.

Effect Size + CI + N Mandatory (3.1.2)

The chi-square test result cannot be exported without Cramér's V. Star annotations or a p-value alone are not permitted. The 95% CI for Cramér's V is calculated via bootstrap (n=1000).

Multiple Comparison Compliance (4.1.5)

If cell-level post-hoc analyses are performed (standardized residuals interpreted as significance), the multiple comparison burden (6 cells in a 2×3 table) must be acknowledged. Licklider applies Bonferroni or Holm correction to the cell-level p-values derived from standardized residuals.

STEP 4 — Export Package

Figure: Stacked or grouped bar chart — genotype proportions by disease status — 300 dpi TIFF/PDF
Statistical Report: Observed and expected frequency table; χ², df, p-value; Cramér's V (95% CI); standardized residuals table (if post-hoc requested); expected cell frequency check summary
Methods Text: "Association between APOE genotype and Alzheimer's disease status was assessed using Pearson's chi-square test of independence. All expected cell frequencies were ≥5 (chi-square approximation valid). Effect size is reported as Cramér's V (95% CI via bootstrap). n=320."
Figure Legend: Bar color coding, p-value, Cramér's V, n per group
Preprocessing Audit Log: Expected cell frequency check result; post-hoc correction method (if applied)