I have had a paper accepted to a good journal (Animal Ecology) and have just received the proofs for checking. However my supervisor has suddenly decided that we need to redo some of the statistics(!) and present the statistics in a different way. I know that you can't make lots of changes, but this won't be too many words, just some statistics. Is is possible at this stage or is it too late?
For all the journals I am familiar with, this is a big no-no.
At the galley proof stage, your paper has already gone through peer review, and any substantial changes to the manuscript, especially in terms of its scientific content (such as the statistics you mentioned in your post), can be seen as an attempt to contravene the peer review process.
Galley proofs, taking the literal limited sense, should mostly be concerned with fixing differences that arise in the production process, where the publisher mistakenly changed the content/intent of your article through misprints, omissions, etc. Most journal would also not mind when you want to make stylistic/grammatical improvements (basically things that copy-editors should be able to sign-off on). But the science should stay fixed once the peer-review stage is over.
However, you can try to contact the editor to see what your options are. Depending on the publisher and the editor, it could be anything from the editorial board approving your proposed change (because they have the expertise to assess that the change does not significantly change the scientific content of the article), them rejecting your proposal, or perhaps even a decision to send the paper out for a second round of peer review.