There is the anecdotal evidence that phones with Marshmallow use up excessive battery when hunting for weak data signal compared to under Lollipop. But I never saw any solid evidence.
I would think the only way to prove/disprove that would be to have two identical phones, set up identically on the same carrier network towers... one with LP and the other with MM. And base line them one versus the other at the same times.
Joey... for the heck of it I ran my phone last night www surfing, amazon surfing, YT videos as I normally do in a dim lit room and screen brightness turned all the way down. I was also downloading Android Wear files for my moto-360 watch. I also turned on the weather widget, to sync that with my watch. Thats something I don't normally do. After a full working day plus ~5 hours of evening dim screen on time I dropped down to 68%.
I then cranked the screen all the way up to 100% for ~50 minutes where the battery dropped from 68% to 44%. I NEVER do this btw, 100% brightness causes eye fatigue, as my eyes are sensitive to bright LED lighting. How does this compare to your usage numbers?