I agree what Bill said. In general I would go for fit seperately, as I probably get more useful models for each response by removing highly non-significant effects for one response (aka starting to fit the noise), which might be important for other responses. You can still see the overall model in the profiler (having flat lines at responses for those effects beeing not in the model anymore)
You can take a look individually and see if the same factors pop up to be significant. Then you should see almost the same result as when you fit all together. If you just look for the overall model you might start fitting noise for the one or other response as you add non-significant factors.
A good approach is in general to try different models as well, so LS, Stepwise, Screening, Decision tree, just to name some. If all tell the same story you increase your confidence which is always good, especially (and most important) if the model turns out to be useful 😉
... View more