Bonferroni correction in A/B testing: Controlling false positives across multiple tests

Ever run an A/B test where you tested 10 different things and got super excited when one showed a "significant" result? Yeah, that's probably a false positive. The dirty secret of A/B testing is that the more things you test, the more likely you are to find something that looks significant but isn't.

It's like flipping a coin 20 times and claiming it's rigged when you get heads 7 times in a row. Do it enough times, and weird patterns will show up just by chance. In the world of experimentation, we call this the multiple comparisons problem - and if you're not careful, it'll torpedo your test results faster than you can say "p-value."

The challenge of multiple hypothesis testing in A/B experiments

Here's the thing about running multiple tests: each one you add is another roll of the dice. Let's say you're testing 20 different metrics in your experiment. With a standard 5% significance level, you'd expect to see one false positive just by random chance. That's the family-wise error rate (FWER) in action - the probability that at least one of your "significant" results is actually bogus.

The statistics community has been wrestling with this for years. I came across a fascinating thread where someone asked if they still needed the Bonferroni correction when all 20 of their metrics showed significance. The answer? Absolutely yes. In fact, that's when you should be most suspicious. Real effects rarely impact everything equally.

Georgi Georgiev from Analytics Toolkit makes a great point about another common mistake: using the Mann-Whitney U Test for the wrong reasons. People throw it at non-normal data thinking it tests for differences in means, but it actually tests for stochastic difference. It's like using a hammer when you need a screwdriver - you might get the job done, but you'll probably break something in the process.

The good news? We've got tools to handle this. Methods like the Bonferroni correction, Holm-Bonferroni method, and Benjamini-Hochberg procedure can help keep your error rates in check. Each has its sweet spot, and picking the right one can mean the difference between finding real insights and chasing ghosts.

The Bonferroni correction: A method to control false positives

The Bonferroni correction is the sledgehammer of multiple testing corrections. Dead simple to use: take your significance level (usually 0.05) and divide it by the number of tests you're running. Testing 10 things? Now each test needs a p-value under 0.005 to count as significant.

It works by being incredibly conservative. The math guarantees that your chance of any false positive stays below your target level. If you want a 5% family-wise error rate, Bonferroni delivers - no questions asked.

But here's the catch: it's often too conservative. As you test more things, that significance threshold gets tighter and tighter. Pretty soon, you need massive effect sizes to detect anything at all. It's like turning down your microscope's sensitivity because you're worried about seeing dust particles - sure, you won't mistake dust for bacteria, but you might miss the bacteria too.

Some researchers on Reddit suggest a compromise: report both your regular p-values and the Bonferroni-corrected ones. This gives readers the full picture - they can see what might be significant and judge for themselves whether the correction is too harsh. It's especially useful when you're testing just a handful of hypotheses where Bonferroni won't completely kill your statistical power.

The bottom line? Bonferroni is perfect when false positives would be catastrophic. Testing whether a drug has dangerous side effects? Bonferroni all the way. But for everyday A/B tests where you're trying to squeeze out incremental improvements? You might want something with a bit more finesse.

Implementing the Bonferroni correction in A/B testing

Let me walk you through how this actually works. Say you're testing a new checkout flow and measuring 5 metrics:

• Conversion rate

• Average order value

• Cart abandonment

• Page load time

• Customer satisfaction score

With a standard 0.05 significance level, Bonferroni says each metric needs a p-value under 0.01 (that's 0.05 ÷ 5) to be considered significant. Simple math, big impact.

When your results come in, you just compare each p-value to that adjusted threshold. Conversion rate has p = 0.008? That's a win. Cart abandonment has p = 0.02? Not significant under Bonferroni, even though it would be in a single test.

The tricky part is knowing when to ease up. If you're running dozens of tests, Bonferroni might be overkill. That's where alternatives like the Holm-Bonferroni correction come in handy - they give you better power while still protecting against false positives.

Tools like Statsig can handle these calculations automatically, which is a lifesaver when you're juggling multiple experiments. The key is picking your correction method before you see the results. Shopping around for the method that makes your results look best? That's just p-hacking with extra steps.

Alternatives to the Bonferroni correction for multiple testing

The Benjamini-Hochberg procedure takes a different approach. Instead of controlling the family-wise error rate, it controls the false discovery rate (FDR). Translation: it's okay with a few false positives slipping through if it means catching more true effects.

Here's how they stack up:

• Bonferroni: "I will protect you from any false positive, no matter the cost"

• Benjamini-Hochberg: "I'll keep false positives to a reasonable percentage of your discoveries"

• Holm-Bonferroni: "I'm like Bonferroni, but a bit more chill"

The Holm-Bonferroni method is particularly clever. It ranks your p-values from smallest to largest, then applies increasingly lenient thresholds as it goes. The smallest p-value faces the full Bonferroni correction, but later ones get more breathing room. It's strictly better than regular Bonferroni - more power, same protection.

Choosing between them depends on your situation. Running exploratory analyses where you're hunting for patterns? Benjamini-Hochberg lets you cast a wider net. Testing critical features where false positives could derail your product? Stick with Holm-Bonferroni or regular Bonferroni.

One Redditor pointed out that sometimes you don't need any correction at all - like when you're doing a single multivariate analysis instead of multiple univariate tests. The key is understanding what question you're actually asking and picking the right statistical approach from the start.

Closing thoughts

Multiple testing corrections aren't just statistical housekeeping - they're what separates real insights from random noise. The method you choose shapes what you'll find, so pick wisely.

Start with these questions: How many tests are you running? What's the cost of a false positive versus a false negative? Are you exploring or confirming? Your answers will point you toward the right correction.

Want to dive deeper? Check out:

• Statsig's guide on implementing these corrections in practice

• The original papers by Benjamini & Hochberg (1995) and Holm (1979)

• Your platform's documentation on how they handle multiple comparisons

Remember, the goal isn't to make your results look good - it's to find effects you can actually trust. A properly corrected "no significant difference" beats a false positive every time.

Hope you find this useful!