Type I Error Rate Calculator for Multiple Tests

Type I Error Rate Calculator

Estimate the chance of at least one false positive across multiple independent tests, compare Bonferroni and Sidak thresholds, plan sequential looks, and keep Type I error separate from Type II error and power.

🎯Type I Error & Multiple Testing Presets

📝False-Positive Risk Inputs

Testing scenario

Sets a realistic starting family size; you can edit every number.

Multiple-comparison method

Bonferroni and Sidak convert a family target into a per-test alpha.

Entered per-test alpha (%)

Common single-test choice: 5%, written alpha = 0.05.

Number of hypotheses or tests (m)

Count tests in the same family before reading results.

Target family-wise error rate (%)

Used for Bonferroni and Sidak thresholds.

Sequential looks per test

Repeated looks inflate false-positive risk unless alpha is spent.

Sequential alpha plan

Sequential results here assume independent looks; formal trials need a prespecified monitoring rule.

Estimated true null tests (%)

100% shows the required all-true-null false-positive expectation.

Observed significant findings

Optional interpretation aid; it does not calculate Type II error.

Per-test alpha used 0.0500 threshold for each hypothesis

Family-wise error rate 0.0% 1 – (1 – alpha)^m

Expected false positives 0.00 m × alpha when all nulls are true

Per-look alpha 0.0500 sequential boundary approximation

🔢Current Error-Rate Snapshot

10Tests in family

10Decision chances

0.0050Bonferroni alpha

0.0051Sidak alpha

⚙Formula Breakdown

Per-test alphaAlpha is the probability of rejecting a true null hypothesis for one planned test. Example: alpha = 0.05 means a 5% Type I error rate for that single test.

Independent FWERFor m independent tests, family-wise error rate = 1 – (1 – alpha)^m. This is the chance of at least one false positive when all tested nulls are true.

Bonferroni alphaPer-test alpha = target FWER / m. This controls FWER by the union bound and is valid even when tests are dependent, though it can be conservative.

Sidak alphaPer-test alpha = 1 – (1 – target FWER)^1/m. This is exact for independent tests and slightly less conservative than Bonferroni.

Expected false positivesWhen every null hypothesis is true, expected false positives = m × alpha. With repeated looks, this calculator uses the per-test ever-reject probability in that same expectation.

Sequential looksIf the same alpha is reused at k independent looks, per-test ever-reject probability = 1 – (1 – alpha)^k. Bonferroni and Sidak look plans spend alpha across looks.

Interpretation boundaryType I error is a false positive rate under true nulls. It is not Type II error, false negative rate, statistical power, effect size, or the probability that one observed significant result is false.

📐Correction Method Reference

Method	Per-Test Formula	Controls	Best When	Watch Point
Unadjusted alpha	entered alpha	Single-test Type I error	One planned test or exploratory scan	FWER rises quickly with m
Bonferroni	target / m	FWER at or below target	Any dependence structure	Often conservative for many tests
Sidak	1 – (1 – target)^1/m	FWER at target if independent	Independent planned comparisons	Assumes independent tests
Same sequential alpha	alpha at each look	No spending control	Explaining inflation from peeking	Raises false positives
Bonferroni looks	alpha / k	Per-test ever-error by bound	Simple interim-look planning	Conservative boundaries
Sidak looks	1 – (1 – alpha)^1/k	Per-test ever-error if independent	Independent look approximation	Not a full group-sequential design

🔍Unadjusted 5% Alpha Lookup

Tests (m)	Per-Test Alpha	Independent FWER	Expected False Positives	Plain Meaning
1	0.05	5.0%	0.05	Single-test false-positive risk
2	0.05	9.8%	0.10	About 1 in 10 families
5	0.05	22.6%	0.25	False positive becomes common
10	0.05	40.1%	0.50	Nearly 2 in 5 families
20	0.05	64.2%	1.00	About one expected false hit
50	0.05	92.3%	2.50	At least one is very likely
100	0.05	99.4%	5.00	Many false positives expected

⏱Sequential Look Reference at 5% Per-Test Alpha

Looks (k)	Same 0.05 Each Look	Bonferroni Look Alpha	Sidak Look Alpha	Planning Note
1	5.0%	0.0500	0.0500	No interim inflation
2	9.8%	0.0250	0.0253	One interim plus final
3	14.3%	0.0167	0.0170	Use a prespecified plan
4	18.5%	0.0125	0.0127	Repeated peeking matters
5	22.6%	0.0100	0.0102	Simple approximation only
10	40.1%	0.0050	0.0051	Formal monitoring preferred

🧪Scenario Comparison Grid

Scenario	Tests	Nominal Alpha	FWER if Unadjusted	Bonferroni Alpha	Sidak Alpha	Expected False Positives
Single primary endpoint	1	0.05	5.0%	0.0500	0.0500	0.05
Education study outcomes	8	0.05	33.7%	0.0063	0.0064	0.40
Clinical safety labs	12	0.05	46.0%	0.0042	0.0043	0.60
A/B test variant checks	24	0.05	70.8%	0.0021	0.0021	1.20
Manufacturing sensors	50	0.05	92.3%	0.0010	0.0010	2.50
Brain imaging regions	60	0.05	95.4%	0.0008	0.0009	3.00
Exploratory biomarkers	200	0.05	100.0%	0.0003	0.0003	10.00
Genome marker pilot	1000	0.05	100.0%	0.0001	0.0001	50.00

💡Actionable Type I Error Tips

Family scope tip: Define the family before analysis: primary endpoints, secondary endpoints, subgroup checks, repeated looks, or dashboard alerts. A 5% alpha on 20 independent true null tests has a 64.2% chance of at least one false positive.

Sequential tip: If you plan interim checks, write the look schedule and alpha spending rule first. Five independent looks at 0.05 each create a 22.6% per-test ever-reject chance, not a 5% test.

Somewhere along the line you’ll run some sort of analysis and get a significant result supporting your hypothesis, everything’s looking perfect with your data. It’s a breakthrough. And then you recall that you ran it through twenty other variables first, none of which worked out. You’re suddenly no longer excited because now you suspect you’ve fallen into the multiple testing trap.

This calculator does the math for you after you input how many tests was in your test family. Saving you from having to guess if that one shiny result was actualy important or simply statistical noise.

Why You Need This Calculator

This is where most researchers gets tripped up. They know about type I error (the probability of rejecting the null when it’s actually correct). That’s called a false positive. That means if you perform a test and your alpha is 0.05, then you’re willing to accept a five percent chance of declaring something as discovered when in fact nothing are there. That’s fine. Most folks can grasp that with just one test.

The issue arise when we perform five tests. Or ten. Or one thousand. At that point the risk doesn’t remain at five percent. Instead, it multiplies. At 0.05 alpha, if you perform 20 unconnected tests, the probability of having at least one false positive climbs past sixty-four percent.

This is why this piece comes so late for most researcher: someone questions their publication. To help visualize this inflation, the tool also allows you to change size of your family (i.e., how many hypotheses). Before viewing the results, however, you must first establish what you mean by “family.” Do you have one main endpoint and three secondary endpoints? Or do you have thousands of markers from exploring some part of genome? These situations is treated differently.

When you choose the Bonferroni correction method, the tool calculates the division of your targeted family-wise error rate by the number of tests. That maintains overall risk while making each individual threshold quite strict. You’d need a lot more evidence to call something significant. It is conservative, but that is why it works.

The Sidak adjustment is a little easier on the eyes for independent tests. Because Sidak assumes that the tests has no impact on one another, it lets you take a little more alpha per test and stay within the same level of family-wide risk. You’ll see the comparison in the calculator, which will let you pick whichever works better for your setup. Bonferroni is safer because presumably your tests are all related. But if they’re really independent, then Sidak provides a little extra power. It is a small exchange, sure, but it is important when you need to find a small effect in noisy data without being overwhelmed by false alarms.

Sequential monitoring also makes things harder. Looking repeatedly at the results of a clinical trial amounts to performing several separate test without correcting your alpha. To account for this, the calculator has fields to input number of sequential looks; it’ll display the resulting increase in error rates. At the default 0.05 level with five interim checks, the error probability rise above twenty-two percent. Unless you assign some alpha over each look, you’ve lost control of your false positive rate. This is spelled out concisely in the reference table on the page.

Many analysts commonly confuse Type I error with Type II error. Type I is a false positive. Type II is a false negative (missing a real effect). Lowering Type I error by adjusting for multiple comparisons makes the bar of significance higher, and will therefore increase Type II error. This is a tradeoff. There’s no free lunch in statistics.

What error are you willing to tolerate for your particular study? For drug safety, false positives can be dangerous. At earlier stages of exploration, missing out on what might become a lead could be more costly. This decision isn’t made by the calculator. It quantifies the risk; it doesn’t choose on your behalf. That’s up to you to bring.

Is it worth inflating the p-value slightly if you’re doing an A/B test on button colors for your website? Probably. Is it okay to do that if you just launched a new medication? No way. Use the tool to get the numbers so you can debate the strategy, not the arithmetic.

There are two keys to conducting honest research: 1) Understand the distinction between a test vs. A family of tests; and 2) Admit when you look at the data more than once that the game has changed. So how do we handle it? How do we control for error? The answer: We don’t get rid of it entirely; we just keep it under control.

Without losing all our firepower, there’s no way to reach zero false positives. But by drawing a line in the sand beforehand, by selecting an approach to correcting those mistakes, then defining that test family with well-defined boundaries, we find ourselves in the sweet spot. From here, the math handles itself.

You don’t need to worry anymore about the hidden traps in your data. You simply need to worry about what the data is trying to tell you. And even when the numbers lie (which they will, as numbers always do), at least you’ll know where the limits are.