When I was extremely ill, I suffered horrible insomnia and often went to sleep at 7am. During that time, I was not able to be genuinely productive. A lot of what I did amounted to churn and time-filling. When I began working, I took an evening shift job because I knew that although I was healthier, I still wasn't healthy enough to cope with getting up early. I now work dayshift, and that correlates to an ability to get up earlier because of improved health.

So if there is any correlation generally (not just for me) between health and schedule, then I would say that being an early bird may be a proxy for health but real productivity is rooted in sound health and the ability it provides to think clearly and do sustained work, which isn't specifically related to the hours you keep.