Does anyone work for or know a company that is looking for free analysis from a doctoral candidate as it relates to their human capital?

This is a decent forum, but have you also asked your school's alumni office or career center for any leads?

In '87, I did what morphed into a market factors paper on Costco of all outfits. The reason I was able to do so was their long term CEO, Jim Sinegal, apparently was thinking about the same kind of questions that I was interviewing him on the phone about. I think it was more a matter of dumb luck, but his interest was piqued when a non MBA student that was working on a MS in Systems was looking into metrics that are actually worthwhile and rather than respond to the marketplace, shape it, AKA create your own future. I will say, even back then, he'd be way beyond my peak many years later. His brilliance was at an incredible level. Never talked or ran into him after, but he was one that showed me there were possibilities I'd never see on my own.

You can chase down various candidate companies, but I seriously recommend a specific question set for their CEO. Don't drill immediately. I can remember some I asked Jim.
- What gives you pride in your company?
- What eats at you most frequently?
- Where do you want to go?
- What changes are you interested in, AND, what difference are you looking to make with them?

That and a number of other questions I can't remember now helped put a fence around the world Jim wanted to move forward in. It also honed me in on areas where metrics that make a difference might be useful.