Risk Ratio Calculator

Exposure Variable Name (Rows)

Outcome Variable Name (Columns)

2 × 2 Contingency Table Frequencies Enter non-negative integer cell counts

Cell a (Exposed, Event Yes)

Cell b (Exposed, Event No)

Cell c (Unexposed, Event Yes)

Cell d (Unexposed, Event No)

All 4 cells must contain valid non-negative integers.

Significance Level (α) Default threshold is 0.05 (95% Confidence Interval)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Prospective & Clinical Association

What Is a Risk Ratio?

A Risk Ratio (RR), also known as Relative Risk, compares the probability of an event occurring in an exposed group to the probability of the event occurring in an unexposed group.

An $RR = 1.0$ indicates no difference in risk between groups. An $RR > 1.0$ suggests an increased risk of the outcome due to exposure (risk factor), while an $RR < 1.0$ implies a decreased risk (protective factor).

When to Use It

Cohort Studies & Clinical Trials: The standard effect size metric when prospective incidence or risk can be tracked directly.
Incidence Rate Comparisons: Evaluating cumulative risk across randomized experimental treatment arms.
Public Health Reporting: Communicating relative hazard and exposure risk straightforwardly to general audiences.
Prospective Study Designs: Where subjects are selected based on exposure status and followed forward.

The Mathematical Formulas

$$Risk_{\text{exposed}} = \frac{a}{a + b}, \quad Risk_{\text{unexposed}} = \frac{c}{c + d}$$ $$RR = \frac{Risk_{\text{exposed}}}{Risk_{\text{unexposed}}} = \frac{a / (a + b)}{c / (c + d)}$$ $$SE(\ln(RR)) = \sqrt{\frac{b}{a(a+b)} + \frac{d}{c(c+d)}}$$ $$95\% \text{ CI for } RR = \left[ \exp\left( \ln(RR) - z_{\alpha/2} \cdot SE \right), \; \exp\left( \ln(RR) + z_{\alpha/2} \cdot SE \right) \right]$$

Where $a, b, c, d$ represent cell frequencies from a $2 \times 2$ contingency table. If any cell is zero, an automatic Haldane-Anscombe correction ($+0.5$) is applied to allow finite estimation.

Worked Example

Scenario: 200 subjects: Exposed ($a=40$ event, $b=60$ no event); Unexposed ($c=20$ event, $d=80$ no event) at $\alpha=0.05$.

Step 1: Compute Risks & Risk Ratio ($RR$):
$Risk_{\text{exp}} = \frac{40}{100} = 0.4000, \quad Risk_{\text{unexp}} = \frac{20}{100} = 0.2000$
$RR = \frac{0.40}{0.20} = 2.0000$.

Step 2: Log Risk Ratio & Standard Error ($SE$):
$\ln(RR) = \ln(2.0) = 0.6931$
$SE = \sqrt{\frac{60}{40(100)} + \frac{80}{20(100)}} = \sqrt{0.015 + 0.040} = \sqrt{0.055} = 0.2345$.

Step 3: 95% Confidence Interval:
$\ln(\text{CI}) = 0.6931 \pm 1.96(0.2345) = [0.2334, 1.1528]$
$95\% \text{ CI for } RR = [e^{0.2334}, e^{1.1528}] = [1.2629, 3.1670]$.

Conclusion: Since the 95% CI does not include 1.0 ($p = 0.0031$), exposure is significantly associated with double the risk of the event.

Risk Analysis Complete

Risk Ratio Results

Computed Risk Ratio (Relative Risk)

RR = 2.0000

Statistically Significant Risk Factor (95% CI: 1.2878 to 3.1060, p = 0.0019)

Risk Ratio (RR) 2.0000

Exposed Risk 40.00%

Unexposed Risk 20.00%

95% CI Lower 1.2878

95% CI Upper 3.1060

p-value 0.0019

Risk Ratio Calculator

Risk Ratio Calculator

Risk Ratio Results

Step-by-Step Calculation

Interpretation

Log Risk Ratio & 95% Confidence Interval Forest Plot

Perfect for students, researchers, and professionals looking to build real statistical skills.