Suppose we have a standard regression model and want to identify whether we have violated the assumptions of the model. Traditionally, we might utilize a significance test to determine whether (for example) the residuals are normally distributed (where the null model states that the residuals belong to a normal distribution).

I think there's consensus that using hypothesis tests to evaluate model assumptions isn't the best idea. These tests are sensitive to sample size (failing to detect major departures when N is small, and detecting even minor discrepancies when N is large). Personally, I think the biggest problem with these tests is they evaluate whether violated assumptions can be *detected*, not whether they are *problematic*.

When I evaluate assumptions, I exclusively use visuals. Unfortunately, visuals are more subjective (and they don't really tell you whether the observed violation is problematic). So, here's my question:

**Are there any objective ways to evaluate statistical assumptions that do not rely on significance tests?**

And, do these objective means of evaluating assumption evaluate whether the assumption violation *matters*, rather than whether it can be detected?

I'd love to see references to articles that I can study, if you have it. Thanks!

*Edit*

Some of the comments/answers mention that it depends on what one's research goals are. Yes, I absolutely agree and appreciate you all pointing this out. I suppose what I'm asking for is some literature that says, "If your goal is A, index B is appropriate. If your goal is C, criteria D is appropriate."