I didn’t do the research into the literature, but this is the one paper a coworker found with discussion of measuring “engagement” as:
(Beta / (Alpha + Theta))
http://www.sciencedirect.com/science/article/pii/S0301051199000022
We didn’t end up including this in our software and stopped our pursuit of this measurement, but you’re free to look at the paper and see for yourself if it is convincing. Looks like it’s been cited 99 times since it came out in 1999, so that’s encouraging.