Survival Rate Calculator & Statistics Guide
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Statistical Survival Rate Calculator
Calculate survival probabilities, failure rates, and confidence intervals using the Actuarial Life Table method.
Survival Rate:
–%
Failure / Mortality Rate:
–%
Effective Sample Size ($N_{eff}$):
—
Standard Error (SE):
—
Confidence Interval (Lower):
–%
Confidence Interval (Upper):
–%
Understanding Survival Rate Statistics
Survival analysis is a branch of statistics for analyzing the expected duration of time until one or more events happen, such as death in biological organisms or failure in mechanical systems. This calculator uses the Actuarial Method (Life Table) approach for a single time interval.
Key Concepts Defined
- Initial Population ($N_{start}$): The number of subjects being observed at the beginning of the study or time period.
- Events ($N_{events}$): The specific outcome being measured (e.g., patient death, part failure, subscription cancellation).
- Censored Data ($N_{lost}$): Subjects who leave the study for reasons other than the event of interest. For example, a patient moving to a different city or a machine being removed from service before it fails.
- Effective Sample Size ($N_{eff}$): An adjustment made to account for censored data. It assumes that censored individuals were at risk for, on average, half of the observation period.
Calculation Formulas
The statistics provided by this calculator are derived using the following mathematical logic:
1. Effective Sample Size ($N_{eff}$):
$N_{eff} = N_{start} – (0.5 \times N_{lost})$
2. Probability of Event ($q$):
$q = N_{events} / N_{eff}$
3. Survival Probability ($p$):
$p = 1 – q$
Survival Rate % = $p \times 100$
4. Standard Error (SE):
$SE = p \times \sqrt{ \frac{q}{N_{eff} \times p} }$
5. Confidence Interval (CI):
$CI = p \pm (Z \times SE)$
Where Z is 1.96 for 95% confidence.
Applications of Survival Statistics
While often associated with medical research (e.g., 5-year cancer survival rates), these calculations are vital in various industries:
- Engineering: Reliability analysis to predict when mechanical parts will fail.
- Business: Customer churn analysis (survival of a subscription).
- Sociology: Event history analysis for life events like marriage or employment duration.
Why is Censoring Important?
If you simply ignore subjects who dropped out (censored data), you might underestimate the survival rate because you are treating them as if they didn't survive or didn't exist. Conversely, treating them as full survivors for the whole period overestimates survival. The "Effective Sample Size" calculation provides a statistical compromise by assuming they contributed to the data for half the interval.
function calculateStats() {
// 1. Get input values
var startPop = document.getElementById("initialPop").value;
var eventCount = document.getElementById("eventCount").value;
var censoredCount = document.getElementById("censoredCount").value;
var zScore = parseFloat(document.getElementById("confidenceLevel").value);
var errorDiv = document.getElementById("errorDisplay");
var resultContainer = document.getElementById("resultContainer");
// 2. Validate inputs
if (startPop === "" || eventCount === "" || censoredCount === "") {
errorDiv.style.display = "block";
errorDiv.innerHTML = "Please fill in all fields.";
return;
}
var N = parseFloat(startPop);
var E = parseFloat(eventCount);
var C = parseFloat(censoredCount);
// Check for non-numeric or negative inputs
if (isNaN(N) || isNaN(E) || isNaN(C) || N < 0 || E < 0 || C N) {
errorDiv.style.display = "block";
errorDiv.innerHTML = "Error: The sum of Events and Censored subjects cannot exceed the Initial Population.";
return;
}
// If inputs are valid, hide error and show full opacity results
errorDiv.style.display = "none";
resultContainer.style.opacity = "1";
// 3. Perform Calculations (Actuarial Method for Single Interval)
// Calculate Effective Sample Size (Neff)
// Assumption: Censored individuals are at risk for half the interval
var Neff = N – (0.5 * C);
// Prevent division by zero if Neff is 0
if (Neff 1)
if (p 1) p = 1;
// Standard Error (SE) using Greenwood's approximation for a single interval
// Formula: SE = p * sqrt( q / (Neff * p) )
// If p is 0 or 1, SE calculation might return NaN or Infinity, handle that.
var SE = 0;
if (p > 0 && p < 1) {
SE = p * Math.sqrt(q / (Neff * p));
} else {
SE = 0; // If survival is 100% or 0%, SE is effectively 0 for this calculation
}
// Confidence Intervals
var marginOfError = zScore * SE;
var lowerCI = p – marginOfError;
var upperCI = p + marginOfError;
// Clamp CI between 0 and 1
if (lowerCI 1) upperCI = 1;
// 4. Update HTML
document.getElementById("displaySurvival").innerHTML = (p * 100).toFixed(2) + "%";
document.getElementById("displayMortality").innerHTML = (q * 100).toFixed(2) + "%";
document.getElementById("displayNeff").innerHTML = Neff.toFixed(1);
document.getElementById("displaySE").innerHTML = SE.toFixed(4);
document.getElementById("displayLowerCI").innerHTML = (lowerCI * 100).toFixed(2) + "%";
document.getElementById("displayUpperCI").innerHTML = (upperCI * 100).toFixed(2) + "%";
}
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document.getElementById("confidenceLevel").value = "1.96";
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document.getElementById("displaySurvival").innerHTML = "–%";
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document.getElementById("displaySE").innerHTML = "–";
document.getElementById("displayLowerCI").innerHTML = "–%";
document.getElementById("displayUpperCI").innerHTML = "–%";
}