Compare incidence rates between two cohorts using person-time data.
Group 1 (Exposed/Intervention)
Group 2 (Control/Reference)
90%
95%
99%
Analysis Results
Incidence Rate 1
–
Incidence Rate 2
–
Rate Ratio (RR)
–
Confidence Interval
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Understanding the Rate Ratio
A Rate Ratio (RR), often used in epidemiology and medical research, compares the incidence rate of an event in one group to the incidence rate in another. Unlike a Risk Ratio, which uses the total number of people at risk, a Rate Ratio uses Person-Time (e.g., person-years or person-months) as the denominator, making it ideal for studies where participants are followed for different lengths of time.
How the Calculation Works
The calculator uses the standard logarithmic method to determine the confidence interval for the rate ratio:
Incidence Rate: Events divided by Person-Time.
Rate Ratio: Rate 1 divided by Rate 2.
Standard Error (SE) of ln(RR): Computed as √[(1 / Events₁) + (1 / Events₂)].
Confidence Interval: Calculated in log space and then exponentiated back to the linear scale using the Z-score for your chosen confidence level (e.g., 1.96 for 95%).
Example Calculation
Suppose you are studying the incidence of a side effect in two groups:
Group 1: 10 events in 500 person-years (Rate = 0.02)
Group 2: 25 events in 600 person-years (Rate = 0.0417)
Rate Ratio: 0.02 / 0.0417 = 0.48
In this case, the Rate Ratio of 0.48 suggests that Group 1 has a 52% lower incidence rate than Group 2. The Confidence Interval will tell you if this difference is statistically significant (i.e., if the interval does not include the value 1.0).
Interpreting Results
RR = 1.0: No difference between groups.
RR > 1.0: Higher incidence in Group 1.
RR < 1.0: Lower incidence in Group 1.
CI Includes 1.0: The result is generally considered not statistically significant at the chosen level.
function calculateRateRatio() {
var e1 = parseFloat(document.getElementById('e1').value);
var pt1 = parseFloat(document.getElementById('pt1').value);
var e2 = parseFloat(document.getElementById('e2').value);
var pt2 = parseFloat(document.getElementById('pt2').value);
var z = parseFloat(document.getElementById('confidenceLevel').value);
// Validation
if (isNaN(e1) || isNaN(pt1) || isNaN(e2) || isNaN(pt2) || pt1 <= 0 || pt2 <= 0) {
alert("Please enter valid positive numbers. Person-time must be greater than zero.");
return;
}
if (e1 <= 0 || e2 <= 0) {
alert("Standard error calculation requires at least one event in each group to provide a reliable confidence interval.");
return;
}
// Math
var rate1 = e1 / pt1;
var rate2 = e2 / pt2;
var rr = rate1 / rate2;
// SE of natural log of RR
var seLnRR = Math.sqrt((1 / e1) + (1 / e2));
// Confidence Interval bounds
var lnRR = Math.log(rr);
var lowerBound = Math.exp(lnRR – (z * seLnRR));
var upperBound = Math.exp(lnRR + (z * seLnRR));
// UI Updates
document.getElementById('res_rate1').innerText = rate1.toFixed(4);
document.getElementById('res_rate2').innerText = rate2.toFixed(4);
document.getElementById('res_rr').innerText = rr.toFixed(3);
document.getElementById('res_ci').innerText = "[" + lowerBound.toFixed(3) + " to " + upperBound.toFixed(3) + "]";
var interpretation = "";
if (lowerBound = 1) {
interpretation = "The confidence interval includes 1.0, suggesting the difference is not statistically significant.";
} else if (rr > 1) {
interpretation = "Incidence rate is significantly higher in Group 1.";
} else {
interpretation = "Incidence rate is significantly lower in Group 1.";
}
document.getElementById('res_interpretation').innerText = interpretation;
// Show results
document.getElementById('rr-results-box').style.display = 'block';
}