Direct answer
Chi-square independence uses an asymptotic table discrepancy for independent observations, Fisher’s exact test conditions on fixed 2×2 margins, and McNemar’s test analyzes discordant outcomes from paired binary observations.
Visual explanation
The same cells can represent different designs
What this calculation tells you
Categorical tests compare observed counts with counts expected under a specific null structure. The same four cells can represent independent groups or paired transitions, leading to different tests.
Identify the sampling unit, pairing, table dimensions, marginal constraints, count sparsity, and whether the question concerns independence, goodness of fit, or paired change.
Where it is used
Experiments
Compare entered response counts under independent or paired designs.
Quality
Analyze pass/fail categories across documented groups or paired methods.
Surveys
Describe category association while preserving complex-design limitations.
Education
Connect table geometry to the null hypothesis and sampling unit.
Common situations
- Analyzing an r×c contingency table.
- Handling a sparse 2×2 table.
- Comparing paired binary outcomes.
- Choosing an effect measure after a categorical test.
Start with the statistical question
Identify the sampling unit, pairing, table dimensions, marginal constraints, count sparsity, and whether the question concerns independence, goodness of fit, or paired change.
Three 2×2 tables share cell positions but use different arrows: independent row groups, fixed-margin rearrangements, and paired off-diagonal transitions.
Worked example
For paired before/after binary outcomes, only subjects who changed contribute to McNemar’s core comparison. A chi-square independence test would ignore that each row pair belongs to one person.
Assumptions that carry the result
Counts must represent observations rather than percentages copied without denominators. Chi-square approximations need suitable expected counts; Fisher and McNemar have their own conditional and pairing conventions.
Interpret the result without overreaching
A significant association does not establish direction, causation, or practical size. Report counts and an effect measure or interval appropriate to the design.
- Using percentages without original counts.
- Applying independent-table tests to paired outcomes.
- Treating Fisher’s exact test as merely a small-sample chi-square replacement with identical conditioning.
Practical questions
Frequently asked questions
When should I use Fisher’s exact test?
It is a defined exact conditional procedure for 2×2 tables, especially useful when asymptotic expected-count conditions are weak.
Can McNemar test independent groups?
No. It is built for paired or matched binary outcomes and focuses on discordant pairs.
Does chi-square show which cells matter?
The overall statistic does not by itself provide a complete multiplicity-aware cell interpretation; residuals and planned contrasts need care.
Further reading
Authoritative sources
Use these primary and professional resources to check definitions, conventions, or requirements that may extend beyond this guide.