In absence of a uniformly most powerful test, are hypothesis tests subjective? We know that uniformly most powerful tests don’t always exist for composite alternative hypotheses. When they don’t, how can we justify the choice of test, e.g., the choice of test-statistic in a hypothesis test?
I suppose it could still be based on power considerations, i.e., we could choose tests that maximise the power for particular choices of alternative hypothesis within the composite model? Is this a subjective choice?
Are there connections to subjectivity in Bayesian statistical approaches? Should we perform sensitivity checks to check the dependence of our result on our choice of test?
 A: In practice, we rarely justify selection of a test due to its being uniformly most powerful (against alternatives). In fact, in most standard software and published literature, analyses of GLMs and OLS models most often defer to using the Wald test rather than the LRT. Admittedly those methods are often very close, but nominally, it is not the UMP test that is used, but the Wald. The Wald is formally preferred because of its alignment with confidence intervals. You could go so far as to argue that, if a significant LR test is rendered insignificant when performed with Wald, then to be conservative one should report the Wald results. Theory and practice diverge.
When planning analyses we often justify a specific analysis because it achieves a target power for a given sample size and alpha-level at a pre-specified effect size. It goes without saying that the test may have decent power in a neighborhood of possible effect sizes. There are however, certain unknowns that are hard to quantify and harder to anticipate, such as contamination (the tendency of study participants to exchange knowledge of the study and treatments among each other), imbalanced clusters (family studies with differing levels of attrition over time), accidental unblinding (administration of rescue therapy, emergency unblinding issues) where in retrospect one might have preferred another testing modality to the one that was proposed.
Perhaps more precisely to your question: you might wonder if a test isn't UMP, what other optimality criteria might we try to identify among a family of tests. Another characteristic to identify in a test is that of being minimax. That is the maximum risk (formally, the maximum expected loss among all possible true values of $\theta$) is less than among the other tests. https://link.springer.com/content/pdf/10.1007%2F0-387-27605-X_8.pdf Minimax tests guarantee power like UMP tests, and robustify analyses by incorporating a Bayesian approach to quantifying uncertainty with a loss function and a prior.
