Explain multiple testing correction with an example

Question

Explain multiple testing correction in simple terms and provide a concrete example.

Codemia · Accepted Answer

In this example, we are considering a scenario where Notion is testing the effectiveness of three different features on user engagement. Each feature is tested against a null hypothesis (H0) that the feature has no effect on user engagement. Let's say we conduct three separate A/B tests for each feature. Each test yields a p-value representing the probability of observing the data given that the null hypothesis is true. If we test multiple hypotheses, the chance of obtaining at least one statistically significant result due to chance increases, leading to a higher Type I error rate. We need to correct for this phenomenon to ensure our conclusions are valid.

To correct for multiple testing, we can use the Bonferroni correction method. The Bonferroni correction adjusts the significance level (alpha) by dividing it by the number of tests conducted. If we set our original alpha level at 0.05 and conduct three tests, the adjusted alpha level becomes $ \alpha_{adjusted} = \frac{0.05}{3} \approx 0.0167 $. We will evaluate the results of our tests against this new alpha level to determine statistical significance.

Assume we conduct tests for three features: Feature A, Feature B, and Feature C, yielding the following p-values:
- Feature A: p = 0.02
- Feature B: p = 0.10
- Feature C: p = 0.03

Applying the Bonferroni correction, we compare each p-value to 0.0167:
- For Feature A: 0.02 > 0.0167 (not significant)
- For Feature B: 0.10 > 0.0167 (not significant)
- For Feature C: 0.03 > 0.0167 (not significant)

None of the features are statistically significant after correction, suggesting that we do not have enough evidence to conclude that any of the features have an effect on user engagement.

The result indicates that after applying the Bonferroni correction, we cannot claim that any of the features significantly enhance user engagement. This highlights the importance of multiple testing corrections, as failing to apply them could lead to false positives. In practical terms, Notion should be cautious about implementing these features based on the current test results, as they may not lead to the desired improvements in user engagement.

Beyond the Bonferroni correction, it's also useful to understand the False Discovery Rate (FDR) method, which controls the expected proportion of false discoveries among the rejected hypotheses. This is particularly relevant for high-dimensional data or when conducting a large number of tests. Additionally, discussing potential confounders such as user demographics or seasonal effects on engagement could provide further insight into the results. Addressing these confounders through stratification or regression analysis can enhance the validity of the conclusions drawn from the tests.

Explain multiple testing correction with an example

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