When and how often is an associate editor not starting review process?
I'll give you my perspective as an co-Editor-in-Chief and previously an Associate Editor of the ACM Transactions on Mathematical Software (ACM TOMS):


As Editor-in-Chief, I take a brief look at each paper and decide whether it even makes sense to move forward with it. I would say that between 10 and 20% of the papers are already filtered out at this stage: They are not within scope of the journal, or are obviously not at the level we expect -- we get our fair share of submissions from second and third world countries (and some from first world countries) that are at the level of semester projects for advanced graduates. 
As Editor-in-Chief, many submissions I get are not within my research area. The ones already filtered out above are obviously not acceptable, but then there are many that I suspect are not good enough, but where I don't know the area well enough to make the call. These get handed on to the Associate Editors, often with a note that they should take a look whether I'm right or wrong in suspecting that the paper does not meet our criteria.
As Associate Editor (before I became EiC), I then reject another ~15% or so based on my subject area knowledge.


So I suspect that for this journal -- a fairly reputable one with one of the highest impact factors as far as math journals go --, about 1/4 of the papers get rejected without being sent out to review. Of course, then another 1/3 to 1/2 or so of the rest get rejected after review.