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I believed I knew what multiple comparison or repeated testing meant but upon reading more on the subject and listening to others I have become more confused.

My confusion started with A/B testing and hence I will use that as context.

$$H_0: \quad \text{percentage conversion of } A \leq \text{percentage conversion of } B\\ H_1: \quad \text{percentage conversion of } A > \text{percentage conversion of } B$$

  1. This is my understanding of multiple testing from an A/B standpoint. Running the test for a sample of say 1000 each (Test and Control), then performing a chi-square test to check for significance. If the results are not significant then continuing the test for another 1000 and repeating the process until significance is reached. This is a classic case oh multiple testing and is wrong without some form of alpha correction like Bonferroni

  2. But if i run the test on a sample of 100K each and measure if the difference in % conversion between the 2 groups are significantly different once then this is not considered multiple comparison

  3. In (2) if i perform chi-sq test on the same 100K x 2 population 10 times does that also constitute as a repeated comparison, even though the data and hypothesis are the same?

  4. Similar to (2) but instead of just measuring the difference for only %conversion, if I also measure if the difference in height, weight etc are also significant between test and control without alpha correction is that also considered as multiple testing? These are separate hypotheses run on the same data.

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(1) is usually referred to as a stopping rule and will indeed inflate the false positive rate.

(2) only consists of one test, one hypothesis, so I don't see how that could be a multiple testing issue.

I don't understand (3), are you saying you conduct the exact same test on the same data $10$ times? That could never yield different results so does not constitute multiple testing.

(4) is the perfect example of a multiple testing issue, namely, you test multiple hypotheses on the same data, so this would decidedly warrant an $\alpha$ correction or some false discovery rate procedure.

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