What test to use after a PCA? Need some help answering a prac exam question
"An ecologist is interested in the morphological adaptations of goannas to their microhabitat. She wishes to determine if goanna shape differs for goannas that are terrestrial, arboreal or scansorial. She takes six morphometric measurements (body length, leg length, tail length, head length, body width and toe length) for each goanna, and also scores each individual for which of the three habitats it occupies. She uses a principal component analysis to combine her morphometric analysis into more general principle components that represent shape."
From the data given, my PCA showed that there are 2 principal components that should be retained. I'm just stuck on the follow up question - "How would you recommend the ecologist tests her null hypothesis for her overall experiment? Why have you selected this particular approach?"
In other words, what type of test is required from here to determine whether goanna shape differs across habitats?

Welcome! I will try to give you some hints on how you could proceed from here. This is a study question so I guess my answer shouldn't be complete.
With PCA, you just captured the main sources of variability in your data (as a linear combination of the starting features). Now you might see some groups or structure in the principal components plot.
As you said, those components capture the shape of your entities. What is the null hypothesis here? It's not clear to me, but I would guess that goannas from different environment have different shapes. Try to state your Null hypothesis, it's very important to do that before testing.
If that is the case, remember that your PCs are features (combinations of them) and it makes sense to use them as a characteristic of each goanna.
So what would you do to test if two features are different across discrete groups? Does this recall some previous problem you faced?
