How to use Bonferroni correction for multiple hypothesis testing

Navigating the Maze of Multiple Comparisons: How to Avoid False Positives

Ever run an experiment and found yourself overwhelmed by the number of metrics screaming for attention? You're not alone. In the world of data analysis, especially when dealing with A/B tests, the more hypotheses we test, the higher the chance we'll stumble upon a false positive—a result that seems significant but is actually just a fluke.

This phenomenon, known as the multiple comparisons problem, isn't just a statistical quirk. It can lead us down the wrong path, causing us to make changes based on results that aren't truly meaningful. But don't worry—there are ways to navigate this maze, and tools like Statsig are here to help.

Understanding the multiple comparisons problem

When we test multiple hypotheses, we inadvertently increase the risk of false positives. This issue, called the multiple comparisons problem, can lead to incorrect conclusions and misguided decisions. For instance, if you're running 20 tests with a significance level of 0.05, there's a whopping 64% chance of seeing at least one false positive without any correction.

False discoveries aren't just statistical nuisances—they can seriously impact how we interpret data and make decisions. In an A/B testing scenario, a variation might seem like a winner purely by chance. Acting on this could lead to unnecessary product updates that don't really improve the user experience. That's why it's crucial to control the family-wise error rate (FWER) in our experiments.

So, what's the FWER? It's the probability of making at least one Type I error (false positive) across all our comparisons. By adjusting our significance level using methods like the Bonferroni correction, we can keep the FWER at the desired level. This ensures we control the chance of encountering false positives, even when juggling numerous tests at once.

Ignoring the multiple comparisons problem can make us overconfident in our results and lead to costly mistakes. By understanding this problem and using appropriate correction methods, we can make more reliable, data-driven decisions. This is especially important in experimentation, where too many false positives can quickly derail our efforts to improve products or services.

Introducing the Bonferroni correction

Enter the Bonferroni correction—a simple yet powerful tool to adjust significance levels and reduce the risk of false positives in multiple comparisons. The idea is straightforward: divide your desired significance level (α) by the number of tests (m). This gives you the Bonferroni-corrected p-value threshold (α/m), ensuring the family-wise error rate (FWER) stays controlled at your desired level.

Let's make this concrete. If you're running an experiment with 20 metrics and aiming for an α of 0.05, your Bonferroni-corrected p-value for each metric becomes 0.05/20 = 0.0025. Any p-value below this threshold is considered significant after the correction.

Why divide α by the number of tests? It's based on the union bound. Essentially, the probability of at least one false positive is less than or equal to the sum of the individual false positive probabilities. By adjusting α this way, we control the FWER at our desired level.

However, the Bonferroni correction can be quite conservative, especially when dealing with lots of tests. It assumes independence between tests, which may not always be true. This can lead to overcorrection and a loss of statistical power. In these cases, alternative methods like the Benjamini-Hochberg procedure might be better for controlling the false discovery rate (FDR).

Applying the Bonferroni correction in practice

Putting the Bonferroni correction to work in real-world A/B testing is straightforward. Just divide your desired significance level (α) by the number of comparisons to get the Bonferroni-corrected p-value threshold. Then, evaluate each hypothesis against this adjusted threshold.

But here's the catch: after applying the Bonferroni correction, balancing Type I and Type II errors becomes crucial. While the correction reduces false positives, it can increase the risk of false negatives, especially with many tests. So, carefully consider the trade-offs between controlling for false positives and maintaining statistical power.

When you're documenting and interpreting results with Bonferroni-corrected p-values, be transparent. Clearly state the original significance level, the number of comparisons, and the adjusted threshold. This helps stakeholders understand the stringent criteria used to determine statistical significance.

A few tips to keep in mind:

• Prioritize your primary metrics by allocating a larger portion of the family-wise error rate (FWER) to them.

• Consider alternative methods like the Holm-Bonferroni or Benjamini-Hochberg procedures for less conservatism when dealing with numerous tests.

Remember, the Bonferroni correction is a valuable tool in your experimentation toolkit. By adjusting p-values for multiple comparisons, you can make more informed decisions based on reliable statistical evidence—avoiding the pitfalls of false positives. And with platforms like Statsig, applying these corrections becomes even more seamless.

Exploring alternatives to the Bonferroni correction

While the Bonferroni correction is helpful, it can be overly conservative, leading to increased Type II errors (false negatives). This is especially problematic when dealing with a large number of tests, as the adjusted significance level becomes more stringent.

Enter alternative methods like the Benjamini-Hochberg procedure. This approach aims to strike a balance between controlling false positives and maintaining statistical power. Instead of controlling the family-wise error rate, it controls the false discovery rate (FDR)—the expected proportion of false positives among all significant results. By allowing a small proportion of false positives, the Benjamini-Hochberg procedure is less conservative than the Bonferroni correction and more powerful in detecting true positives.

Choosing the right correction method depends on your experimental context and how you weigh Type I and Type II errors. If false positives are more costly than false negatives, the Bonferroni correction might be your go-to. However, when dealing with many tests or when missing true effects is a bigger concern, methods like the Benjamini-Hochberg procedure might be more suitable.

It's all about balancing the trade-offs between controlling false positives and maintaining statistical power. Carefully evaluate your experiment's specific needs and choose a method that aligns with your goals. Consulting with a statistician—or leveraging platforms like Statsig—can help navigate these decisions and ensure you're using the most appropriate approach.

When reporting results, transparency is key. Clearly state the correction method used and why you chose it. This allows others to interpret your findings accurately and assess the validity of your conclusions. By openly communicating the statistical methods employed, you promote reproducibility and facilitate meaningful discussions within the community.

Closing thoughts

Dealing with multiple comparisons is tricky, but understanding how to navigate this challenge is essential for making sound, data-driven decisions. Whether you choose the Bonferroni correction or an alternative like the Benjamini-Hochberg procedure, what's important is selecting the method that best fits your experimental needs and being transparent about your approach. Platforms like Statsig can make this process smoother, offering tools to help control error rates and interpret results accurately. Hope you found this helpful! For more insights on statistical testing and experimentation, feel free to explore additional resources or reach out to our team.