Multiplicity correction when using emtrends to see if slopes are different from 0

Question

I have a relatively straightforward question but I haven't been able to find the answer. Consider this example:

require(emmeans)
fiber.lm <- lm(strength ~ diameter*machine, data=fiber)

My hypothesis is that for each of the three machines, there is a relationship between strength and diameter, i.e. the slope for each machine is significantly different from 0. To test this, I use the emmeans package as follows:

summary(emtrends(fiber.lm, ~"machine", var = "diameter"), infer = TRUE)

My question is: should I use multiplicity correction on the pvalues and confidence intervals (e.g. by adding adjust="sidak") to conclude whether the slopes are significantly different from 0 or not? If yes, what may be the reason why this not done by default like when using pairwise comparisons in the emmeans package?

I understand there are no fixed rules about this, but I haven't been able to find discussions or examples in the literature about this specifically. In tutorials on the emmeans package, I have only found instances where no adjustment is performed (e.g. https://stats.idre.ucla.edu/r/seminars/interactions-r/#s4b and https://psu-psychology.github.io/r-bootcamp-2018/talks/correlation_regression.html#pairwise-differences-and-simple-slopes-in-regression).

Answer 1

First, a multiplicity correction is a multiplicity correction. That is, the importance of using one or not is due solely to the fact that you are doing more than one test, and the risks of making an error somewhere in that family of tests. Whether it is slopes or pairwise differences or whatever is not of key importance.

Second, there is also the possibility of doing one joint test, via test(emtrends(...), joint = TRUE). That avoids the multiplicity issue. Or rather, it incorporates it, as it considers all slopes at once.