I would guess the advanced stat at b-r.com that correlates best to Running rating is XBT% (Extra Bases Taken). The percent of times the runner takes more than one base on a single or more than two bases on a double per opportunities to do so.
XBT% does not appear to take into account the outs a runner makes attempting to take an extra base and consequently cannot be looked at in isolation. I'm sure running into outs on the bases would (and should) have a very negative effect on your running rating in DMB.