McNemar’s Paired Chi-Square Test (χ² = (b − c)²/(b + c)) & Exact Discordant Odds Calculator

Test paired nominal shifts via McNemar’s χ², continuity-corrected χ², and matched Odds Ratio.

McNemar’s Chi-Square Statistic (df = 1)
8.526 χ²
Edwards Continuity-Corrected χ² 7.605 χ²_c
Matched-Pair Discordant Odds Ratio (b/c) 2.8 OR
Net Proportion Shift (b − c) / N 12 %
Inputs
⚡ Scenarios:
cell b
Discordant pairs shifting Pre-Negative → Post-Positive (cell b).
cell c
Discordant pairs shifting Pre-Positive → Post-Negative (cell c).
a + d
Concordant pairs remaining unchanged (a + d).
Results & Graph
McNemar’s Chi-Square Statistic (df = 1)
8.526 χ²
Edwards Continuity-Corrected χ²
7.605 χ²_c
Matched-Pair Discordant Odds Ratio (b/c)
2.8 OR
Net Proportion Shift (b − c) / N
12 %

McNemar’s Chi-Square Statistic (df = 1)

Graph 1 • McNemar’s Paired χ² Test • Domain Architecture & Live Telemetry Maps Discordant Neg To Pos Cell B, Discordant Pos To Neg Cell C, Concordant Pairs A Plus D through McNemar’s Paired χ² Test Formula.
McNemar’s Paired Chi-Square Test (χ² = (b − c)²/(b + c)) & Exact Discordant Odds Calculator • Custom Domain Schematic (Non-Parametric & Hypothesis Testing) discordadiscordaMcNemar’s Paired χ² Te Discordant Neg To Pos Cell B: 28 cell b • Discordant Pos To Neg Cell C: 10 cell c MCNEMAR’S CHI-SQUARE STATI 8.526 χ² • Edwards Continuity-Cor: 7.605 χ²_c • Matched-Pair Discordan: 2.8 OR Eq: χ²_McNemar = (b − c)² / (b + c);
Graph 2 • McNemar’s Chi-Square Statistic (df = 1) Across Discordant Neg To Pos Cell B Sweep (40% → 160%) Evaluates χ²_McNemar = (b − c)² / (b + c); χ²_Edwards = (|b − c| − 1)² / (b + c) across 7 operating points (current input highlighted).
0 χ² 12.15 χ² 24.31 χ² 0.0680 11.20 1.73 16.80 4.75 22.40 8.53 28 12.77 33.60 17.33 39.20 22.10 44.80
Graph 3 • Joint Sensitivity: Discordant Neg To Pos Cell B × Discordant Pos To Neg Cell C Compares McNemar’s Chi-Square Statistic (df = 1) when Discordant Pos To Neg Cell C is shifted +25% (solid) vs. −25% (dashed).
0 χ² 10.46 χ² 20.93 χ² --- Discordant Pos To Neg Cell C @ 75% 1.57 -30% 3.52 -15% 5.93 +0% 8.68 +15% 11.68 +30%
Summary: With discordant pairs b=28 and c=10 (N=150), unadjusted McNemar χ² = 8.526, continuity-corrected χ² = 7.605, and matched Odds Ratio = 2.8.
Summary: With discordant pairs b=28 and c=10 (N=150), unadjusted McNemar χ² = 8.526, continuity-corrected χ² = 7.605, and matched Odds Ratio = 2.8.
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: McNemar’s Paired χ² Test Formula

χ²McNemar = (b − c)² / (b + c)
χ²Edwards = (|b − c| − 1)² / (b + c)
ORmatched = b / c
χ²_McNemar = (b − c)² / (b + c); χ²_Edwards = (|b − c| − 1)² / (b + c); OR_matched = b / c
  • discordantNegToPosCellB: Discordant Neg To Pos Cell B (cell b)
  • discordantPosToNegCellC: Discordant Pos To Neg Cell C (cell c)
  • concordantPairsAPlusD: Concordant Pairs A Plus D (a + d)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: McNemar’s Paired χ² Test

Test paired nominal shifts via McNemar’s χ², continuity-corrected χ², and matched Odds Ratio.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: χ²_McNemar = (b − c)² / (b + c); χ²_Edwards = (|b − c| − 1)² / (b + c); OR_matched = b / c.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes McNemar’s Paired χ² Test via χ²_McNemar = (b − c)² / (b + c); χ²_Edwards = (|b − c| − 1)² / (b + c); OR_matched = b / c.

Frequently Asked Questions

How does the McNemar’s Paired Chi-Square Test (χ² = (b − c)²/(b + c)) & Exact Discordant Odds Calculator compute the solution?

It evaluates χ²_McNemar = (b − c)² / (b + c); χ²_Edwards = (|b − c| − 1)² / (b + c); OR_matched = b / c directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Non-Parametric & Hypothesis Testing

McNemar’s Paired Chi-Square Test (χ² = (b − c)²/(b + c)) & Exact Discordant Odds Calculator: inputs, method, and how to read the result

Test paired nominal shifts via McNemar’s χ², continuity-corrected χ², and matched Odds Ratio.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Discordant Neg To Pos Cell B: Discordant pairs shifting Pre-Negative → Post-Positive (cell b).
  • Discordant Pos To Neg Cell C: Discordant pairs shifting Pre-Positive → Post-Negative (cell c).
  • Concordant Pairs A Plus D: Concordant pairs remaining unchanged (a + d).

What the calculation includes

With the default values shown, McNemar’s Chi-Square Statistic (df = 1) is 8.526 χ². The calculation follows McNemar’s Paired χ² Test Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Discordant Neg To Pos Cell B from 28 cell b to 31 cell b changes McNemar’s Chi-Square Statistic (df = 1) from 8.526 χ² to 10.756 χ². This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

More in Mathematics & Statistics: Scientific Calculator, Statistics & Standard Deviation Calculator, Permutations, Combinations, Probability, Sample Size & P-Value Calculator.