Multiple comparisons in testing: how to control type I errors

Ever run multiple tests and wonder why you're getting so many positive results? You're not alone. In the world of experiments and data analysis, it's easy to stumble upon findings that seem significant but might just be due to chance.

But here's the thing: the more tests you run, the higher the chance of encountering false positives. This is known as the multiple comparisons problem, and it's something we need to talk about.

The multiple comparisons problem in testing

When you run several tests at the same time, the odds of getting false positives just by chance shoot up. This is what's called the multiple comparisons problem, and if we don't tackle it right, we might end up drawing the wrong conclusions. Basically, the more comparisons you make, the higher the family-wise error rate gets—meaning there's a bigger chance you'll make one or more Type I errors in your tests.

There are real-world stories that show what happens when we ignore this problem. For example, there was a famous study claiming that certain parts of the brain lit up when people listened to music. But here's the kicker: the researchers didn't adjust for multiple comparisons, so their findings were, well, pretty dubious. Similarly, a company once ran tons of A/B tests without any corrections and ended up making changes based on false positives—ouch!

Luckily, we've got some tricks up our sleeves to deal with this. One is the Bonferroni correction, a conservative approach where you adjust the significance level for each test by dividing your overall alpha (α) by the number of tests. Another option is the Benjamini-Hochberg procedure, which focuses on controlling the false discovery rate. This method lets you find more true positives while keeping the false ones in check.

So when you're setting up experiments, it's key to think about the multiple comparisons problem right from the start. You might want to limit how many metrics and variants you're testing, focus on your primary metrics, and tweak your significance levels based on how many comparisons you're making. By using the right correction techniques, like the ones available through Statsig, you can make sure your results are solid and worth acting on.

Understanding Type I errors and α inflation

Let's talk about —that's when we mistakenly reject a true null hypothesis and end up with a false positive. The more hypotheses we test, the more likely we are to run into Type I errors. This increase is called α inflation, and it's at the core of the .

Think about it this way: if you run 100 independent tests with a significance level (α) of 0.05, even if all the null hypotheses are true, you'd still expect about five false positives just by chance. So as you increase the number of comparisons, the chance of making at least one Type I error—the family-wise error rate (FWER)—goes up pretty fast.

Controlling the FWER is super important to keep your statistical inferences solid when dealing with multiple comparisons. If you don't adjust properly, the risk of false discoveries can mess up your results. This is especially true in fields like genomics, where you might be testing thousands of hypotheses at once.

To tackle α inflation and keep the FWER or the false discovery rate (FDR) in check, researchers have come up with several methods. These include the and the , both of which are explained in depth on the Statsig blog.

Finding the right balance between controlling Type I and Type II errors is crucial when applying multiple comparison corrections. Sure, stricter thresholds reduce false positives, but they might also bump up the chance of false negatives (Type II errors), meaning you could miss out on real discoveries. So, it's important to , the number of comparisons you're making, and what the consequences of errors might be to strike that balance.

Methods for controlling Type I errors in multiple comparisons

When we're dealing with multiple comparisons, we've got to juggle the risk of Type I errors (false positives) and the need to keep our statistical power strong. One method is the Bonferroni correction, a conservative approach where you adjust the significance level for each test by dividing your overall α by the number of tests. It's great for controlling the family-wise error rate (FWER), but it can make it harder to spot real effects, especially if you're running a lot of tests.

Another way to go is the Benjamini-Hochberg procedure, which focuses on controlling the false discovery rate (FDR)—the expected proportion of false positives among all your significant results. It works by ranking your p-values from smallest to largest and comparing each one to a threshold adjusted for its rank. This method lets you find more true effects while keeping false positives in check. The Benjamini-Hochberg procedure offers a nice balance between tight error control and keeping statistical power high, making it a favorite for online experiments and other scenarios with lots of comparisons.

When deciding which method to use, think about how Type I and Type II errors impact your specific situation. If false positives would be really bad news, the Bonferroni correction might be your best bet. But if you're trying to uncover as many real effects as possible and can live with a few false positives, then the Benjamini-Hochberg procedure could be the way to go. There are also other options out there, like the Holm-Bonferroni method or the Šidák correction, which offer middle-ground choices between Bonferroni and Benjamini-Hochberg.

At the end of the day, the best way to control Type I errors when facing multiple comparisons depends on your goals, how many tests you're running, and how important it is to avoid false positives versus keeping your statistical power. Tools like Statsig can help you apply these methods effectively in your experiments. By weighing these factors and choosing the right correction method, you can make sure your results are both statistically sound and scientifically valuable.

Best practices for applying multiple comparison corrections in testing

When you're running experiments that involve multiple comparisons, it's super important to pick correction methods that match your testing situation and how much risk you're willing to take. The Bonferroni correction is more conservative and works well if you're testing fewer hypotheses, while the Benjamini-Hochberg procedure is less strict and better suited for testing lots of hypotheses.

Be sure to implement corrections that effectively balance the risks of Type I and Type II errors. If your corrections are too strict, you'll cut down on false positives but might ramp up false negatives, which could mean missing out on some valuable insights. So think about your experimental goals and what might happen when you're choosing your thresholds.

Make sure your whole team knows about these methods to ensure you get reliable and valid results. Understanding the multiple comparisons problem and the right correction techniques is key to making good, data-driven decisions. Encourage everyone—data scientists, engineers, product managers—to collaborate and get on the same page with your testing strategies.

Optimize your testing by focusing on your primary metrics and keeping the number of variants in check. This way, you maintain your statistical power while controlling error rates. It's also a good idea to regularly review and tweak your experimentation practices to keep them effective and efficient.

By following these best practices, you'll be better equipped to handle the challenges of multiple comparisons in your testing. Embrace a culture of continuous learning and improvement, and stay up-to-date with the latest in experimentation methodologies. This way, you'll drive meaningful insights and help your business grow.

Closing thoughts

Dealing with the multiple comparisons problem is crucial for anyone running experiments with lots of tests. By understanding and applying the right correction methods—like the Bonferroni correction or the Benjamini-Hochberg procedure—you can control false positives and make sure your findings are reliable. Tools like Statsig can make this process easier, helping you get actionable insights from your data.

If you want to learn more, check out resources on statistical testing and experimental design. Keep exploring, keep testing, and happy experimenting!