function calculateFWER() {
var alphaInput = document.getElementById('alphaLevel');
var testsInput = document.getElementById('numTests');
var resultDiv = document.getElementById('fwerResults');
var fwerDisplay = document.getElementById('resultFWER');
var probDisplay = document.getElementById('resultProb');
var bonfDisplay = document.getElementById('resultBonferroni');
var warningDisplay = document.getElementById('highRiskWarning');
var alpha = parseFloat(alphaInput.value);
var m = parseInt(testsInput.value);
// Validation
if (isNaN(alpha) || alpha 1) {
alert("Please enter a valid Significance Level between 0 and 1.");
return;
}
if (isNaN(m) || m 0.0001) {
bonfStr = bonferroniAlpha.toFixed(6);
}
// Update DOM
resultDiv.style.display = 'block';
fwerDisplay.innerHTML = fwerPercentage;
probDisplay.innerHTML = probDecimalStr;
bonfDisplay.innerHTML = bonfStr;
// Show warning if FWER > 10%
if (fwerDecimal > 0.10) {
warningDisplay.style.display = 'block';
warningDisplay.innerHTML = "Warning: performing " + m + " tests raises your False Positive risk to " + fwerPercentage + ".";
} else {
warningDisplay.style.display = 'none';
}
}
Understanding Family-Wise Error Rate (FWER)
When conducting statistical analysis, researchers often perform multiple hypothesis tests simultaneously. While a standard significance level (α) of 0.05 ensures a 95% confidence level for a single test, repeating this test multiple times drastically increases the probability of making at least one Type I error (a false positive). This accumulated risk is known as the Family-Wise Error Rate (FWER).
Our FWER Calculator helps statisticians, data scientists, and researchers quantify the risk of false discoveries when testing multiple hypotheses and determines the necessary adjustments, such as the Bonferroni correction, to maintain statistical integrity.
The Multiple Comparisons Problem
The "Multiple Comparisons Problem" occurs because the more inferences are made, the more likely erroneous inferences are to occur.
For example, if you test 20 independent hypotheses at an α of 0.05:
Probability of no error in one test: 0.95
Probability of no error in 20 tests: 0.9520 ≈ 0.36
FWER (Probability of at least one error): 1 – 0.36 = 0.64 (64%)
Without adjustment, you have a 64% chance of finding a "statistically significant" result purely by chance, rendering your conclusions unreliable.
How the FWER Formula Works
The calculator uses the Šidák formula for independent tests to determine the probability of making at least one Type I error:
FWER = 1 – (1 – α)m
Where:
α (Alpha): The significance level for an individual test (typically 0.05 or 0.01).
m: The total number of comparisons or hypothesis tests performed.
Bonferroni Correction
To control the FWER, researchers often adjust the alpha level for individual tests. The most common method is the Bonferroni Correction, calculated as:
αcorrected = αoriginal / m
If you aim for a family-wise error rate of 0.05 across 10 tests, each individual test must have a p-value below 0.005 (0.05 / 10) to be considered significant. The calculator above automatically provides this corrected threshold.
Frequently Asked Questions
What is a Type I Error?
A Type I error occurs when a null hypothesis that is actually true is rejected. It is essentially a "false positive"—detecting an effect that does not exist.
When should I use the Bonferroni correction?
You should use the Bonferroni correction or similar adjustments (like Holm-Bonferroni or FDR) whenever you perform multiple statistical tests on the same dataset to prevent the inflation of false positive rates.
Is Bonferroni too conservative?
Yes, often considered conservative, the Bonferroni correction strictly controls false positives but increases the risk of Type II errors (false negatives), meaning you might miss real effects. It assumes tests are independent.
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