Maybe I'm reading this wrong, but it sounds like you're catching coyotes in snares set for fox. If I'm correct with that, then you need to do two things if you're going to keep using snares. First, make a smaller snare loop size (I prefer 6 - 7 inches max for fox) and set them so the bottom loop of the snare is 6 - 8 inches off the ground. You can use a small stick set directly below the loop to force the head up and into the snare.
Also, to keep coyotes and deer out of the snares you can brush in the area making a simple tunnel about 10 - 12 inches before and after the snare that the fox will go through but larger animals will go around/over. Red fox are caught each year in 5 x 5 body gripping traps so don't think a 6 inch loop is too small.
Lastly, as Alan suggested put some sort of monitoring device on the snare. This can be as simple as using the cellular cable Covert or HCO game trail cameras (you can get Covert cameras at www.shopWCS.com
and HCO cameras are available from Rob Erickson) or one of the electronic monitoring devices like Trap Alert or TrapSmart. The reason I'm recommending the game camera is that you can get additional data by setting it to movie or multiple picture mode and may help you see how the fox approaches and reacts to the set.
What you don't want to have happen during a research project is the following I've experienced with fox while targeting coyotes.
My coyote snares are usually 10 inches off the ground with a 10 inch loop, but if I'm concerned with fox I'll go 12 and 12. The reason for 12 inches off the ground is to allow the fox to walk under the snare but what I've noticed is that they tend to jump through the loop more times than they walk under it. This causes two issues if you're using cam locks with breakaways (or possibly any lock that doesn't have a deer stop).
First, the fox can fire the snare as it jumps through and end up getting caught by the tail. For fur trapping, this isn't as much of an issue as most of our sets are out of sight from the general public, but for research in a metropolitan area you don't want anyone seeing a fox or any animal for that matter caught that way. Second, the fox can get hip snared and die which is what has mainly happened to me. From what I've been told by researchers, it is a build up of uric acid that killed them. Personally, I don't know if that is true or not. All I know is that the fox were dead and I didn't want them to be.