2×2 Contingency Table Odds Ratio (OR), Relative Risk (RR), Chi-Square & Phi (φ) Calculator

Compute Odds Ratio (95% CI), Relative Risk (RR), Pearson χ², and Cramér’s V / φ from a 2×2 table.

Exposure Odds Ratio (OR)
2.563 95% CI: 1.33–4.93
Relative Risk / Risk Ratio (RR) 2 RR
Pearson Chi-Square χ² (df = 1) 8.219 χ²
Phi Association Coefficient (φ) 0.203 φ
Inputs
⚡ Scenarios:
cell a
Exposed with outcome present (cell a).
cell b
Exposed with outcome absent (cell b).
cell c
Unexposed with outcome present (cell c).
cell d
Unexposed with outcome absent (cell d).
Results & Graph
Exposure Odds Ratio (OR)
2.563 95% CI: 1.33–4.93
Relative Risk / Risk Ratio (RR)
2 RR
Pearson Chi-Square χ² (df = 1)
8.219 χ²
Phi Association Coefficient (φ)
0.203 φ

Exposure Odds Ratio (OR)

Graph 1 • Odds Ratio, RR & Chi-Square • Domain Architecture & Live Telemetry Maps Exposed Cases A, Exposed Controls B, Unexposed Cases C through Odds Ratio, RR & Chi-Square Formula.
Exposure Odds Ratio (OR): 2.563 95% CI: 1.33 Odds Ratio, RR & Chi-Square • Invariants & Metrics Exposure Odds Ratio (OR) 2.563 95% CI: 1.33–4.9 Relative Risk / Risk Ratio ( 2 RR Pearson Chi-Square χ² (df = 8.219 χ² Phi Association Coefficient 0.203 φ
Graph 2 • Exposure Odds Ratio (OR) Across Exposed Cases A Sweep (40% → 160%) Evaluates OR = (a·d) / (b·c); SE(ln OR) = √(1/a + 1/b + 1/c + 1/d); RR = [a/(a across 7 operating points (current input highlighted).
0 95% CI 2.25 95% CI 4.51 95% CI 1.02 14.40 1.54 21.60 2.05 28.80 2.56 36 3.08 43.20 3.59 50.40 4.10 57.60
Graph 3 • Joint Sensitivity: Exposed Cases A × Exposed Controls B Compares Exposure Odds Ratio (OR) when Exposed Controls B is shifted +25% (solid) vs. −25% (dashed).
0 95% CI 2.44 95% CI 4.89 95% CI --- Exposed Controls B @ 75% 1.44 -30% 1.74 -15% 2.05 +0% 2.36 +15% 2.67 +30%
Summary: Odds Ratio is 2.563 (95% CI: 1.33 to 4.93), Relative Risk (RR) is 2, Pearson χ² is 8.219, and Phi correlation φ is 0.203.
Summary: Odds Ratio is 2.563 (95% CI: 1.33 to 4.93), Relative Risk (RR) is 2, Pearson χ² is 8.219, and Phi correlation φ is 0.203.
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: Odds Ratio, RR & Chi-Square Formula

OR = a·db·c
SE(ln OR) = 1/a + 1/b + 1/c + 1/d
RR = a/(a+b)c/(c+d)
φ = χ² / N
OR = (a·d) / (b·c); SE(ln OR) = √(1/a + 1/b + 1/c + 1/d); RR = [a/(a+b)] / [c/(c+d)]; φ = √(χ² / N)
  • exposedCasesA: Exposed Cases A (cell a)
  • exposedControlsB: Exposed Controls B (cell b)
  • unexposedCasesC: Unexposed Cases C (cell c)
  • unexposedControlsD: Unexposed Controls D (cell 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: Odds Ratio, RR & Chi-Square

Compute Odds Ratio (95% CI), Relative Risk (RR), Pearson χ², and Cramér’s V / φ from a 2×2 table.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: OR = (a·d) / (b·c); SE(ln OR) = √(1/a + 1/b + 1/c + 1/d); RR = [a/(a+b)] / [c/(c+d)]; φ = √(χ² / N).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Odds Ratio, RR & Chi-Square via OR = (a·d) / (b·c); SE(ln OR) = √(1/a + 1/b + 1/c + 1/d); RR = [a/(a+b)] / [c/(c+d)]; φ = √(χ² / N).

Frequently Asked Questions

How does the 2×2 Contingency Table Odds Ratio (OR), Relative Risk (RR), Chi-Square & Phi (φ) Calculator compute the solution?

It evaluates OR = (a·d) / (b·c); SE(ln OR) = √(1/a + 1/b + 1/c + 1/d); RR = [a/(a+b)] / [c/(c+d)]; φ = √(χ² / N) 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

2×2 Contingency Table Odds Ratio (OR), Relative Risk (RR), Chi-Square & Phi (φ) Calculator: inputs, method, and how to read the result

Compute Odds Ratio (95% CI), Relative Risk (RR), Pearson χ², and Cramér’s V / φ from a 2×2 table.

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.

  • Exposed Cases A: Exposed with outcome present (cell a).
  • Exposed Controls B: Exposed with outcome absent (cell b).
  • Unexposed Cases C: Unexposed with outcome present (cell c).
  • Unexposed Controls D: Unexposed with outcome absent (cell d).

What the calculation includes

With the default values shown, Exposure Odds Ratio (OR) is 2.563 95% CI: 1.33–4.93. The calculation follows Odds Ratio, RR & Chi-Square Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Exposed Cases A from 36 cell a to 40 cell a changes Exposure Odds Ratio (OR) from 2.563 95% CI: 1.33–4.93 to 2.847 95% CI: 1.49–5.43. 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.

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