I am testing a path model on a sample of 12.000 subjects. All the path coefficients have extremely low values (e.g. 0.003, 0.012) yet they prove significant.
How do I know I am detecting a biologically meaningful effect when the estimates themselves are so comically low? Am I not finding significant results simply because my sample is extremely large?
The hypothesis test is doing what it advertises itself as doing: it is detecting a small but real effect that you are able to detect because your large sample size gives you that much sensitivity. It’s like the Princess and the Pea: she is right to feel the pea, since it really is there. For most of us, the pea does not matter, but maybe it does in some situation. Either way, it really is there.
As far as if the effect is biologically meaningful, that’s up to the biology subject matter experts. You say the values are comically low. Perhaps they will agree; perhaps they will not.
I agree entirely with @Dave's answer. I will add that the circumstance where you have such a large sample is one where the level of significance of the estimates is not particularly helpful in communicating the system. The all-or-none significance/not significance of a hypothesis test is even worse. Instead, you could present confidence intervals (or the like) for the coefficients or of any other meaningful effect size estimates, like you did in the question.
There are many circumstances where statistical tests are less helpful than might be expected. They are just tools to help with inference, and they only guide you directly about statistical inference. Expertise in the subject matter is usually an important ingredient in scientific inferences about the world outside of statistical models.
There is no statistical cutoff for "biologically meaningful", it's entirely domain dependent. If you're looking at a pathway related to a common, highly variable process like glucose production, tiny changes are not likely going to have large effects on the organism. Being very sure that your treatment increases glucose production by 0.001%, for example, probably doesn't have much biological significance.
If you're looking at something like exposure to botulinum toxin, however, small effects can have large biological consequences, as just a few nanograms of the toxin can kill. If are very sure that your treatment exposes patients to a non-zero amount of botulinum toxin, it's likely meaningful even with very, very small effect sizes.
P-values just express your certainty of whether something changed or didn't change, it doesn't say whether those changes are actually meaningful.
