Is it still necessary to correct for multiple comparisons/testing if the sample sizes are large? I was having a discussion with a colleague today regarding corrections for multiple testing. We're planning on running a large number of tests (probably in the hundreds as a rough estimate) and thus I brought up multiple testing and how we should go about it. Specifically the fact that various outcomes are related (not independent) and thus this doesn't meet the assumptions of a Bonferroni, Benjamini-Hochberg or whatever method of adjustment. Their suggestion was that we don't have to worry about it, because the sample size of each test is going to be big enough (we're looking at minimum of n=100 but frequently n=1000+). This is the first time I've ever heard of such an approach and thus was a bit surprised by this. From my research and trying to think it through myself, perhaps their logic was that as n gets larger, sampling error becomes less of an issue, therefore it counteracts any increase in false positives?
My question ultimately is, is my colleague correct in suggesting the large sample sizes are sufficient for handling multiple testing and if so, is my reasoning of it being related to the sampling error accurate?


Their suggestion was that we don't have to worry about [multiple testing], because the sample size of each test is going to be big enough (we're looking at minimum of n=100 but frequently n=1000+).

Here's a scenario where your colleagues would be right in practice even if wrong in theory.
Your many tests fall neatly into two categories: $H_0$ is correct (or very close to be) or $H_0$ is extremely inapprioriate. Also, you are not too interested in which tests reject $H_0$ but rather in the overall proportion of accept/reject or you are not too worried about the occasional false positive. In such a case adjusting or not the p-values would make little difference (I still don't see why shouldn't do it though...).
For example, you have a bag of coins, some are legal and some with both sides the same. If you flip each coin a hundred times or so, you are nearly certain to spot the faulty coins by looking at the pvalue, whether you adjust it or not. If the "cost" of a false negative is much higher than a false positive you may just work with nominal p-values.
(I'm not saying your colleagues are right of course, I'm just imaging a scenario that would give them some grounds).
