"I need some way to know how hard the engineers are working."

Why? And what does "hard" even mean? (working more hours?) The goal is good results, not hard workers.

"...clear correlation between hours worked and results produced."

Working zero hours will produce zero results, so there is obviously a correlation, but this correlation cannot be linear, and may not even be positive in all cases. I would expect working 100 hours in a week would would make for worse results than 40 hours.

Hours are not a measure of output, or efficiency, or results; they are a measure of time.