My company's CEO says he looks at the survey answers to "Would you recommend Company X as a good place to work" as a health indicator of how the company's doing. Which makes sense to me, since if the employees overall would recommend it as a place to work, it's probably reasonably stable and rewarding, has reasonably trusted managers, etc.
I've never delved deep into actual statistics on this, though, so consider this just an anecdote.
This is basically called Net Promoter Score or NPS. Often people use it for products and it's not a bad method for understanding how a company is doing from employees perspective.
I've never delved deep into actual statistics on this, though, so consider this just an anecdote.