You asked about scientific studies of prayer, so I provided some examples
It appears that you are now saying that those aren't scientific studies of prayer, but are studies of positive thinking.
The folks at DukeMed don't seem squeamish about using the word "prayer" in the titles of their press releases about their studies of prayer.
DukeMedNews: Prayer, Noetic Studies Feasible; Results Indicate Benefit to Heart Patient
And
I'm perfectly willing to use the term prayer instead of 'positive thinking' but the end result is the same.
I wonder if these studies mention to whom the subjects are praying. If one of them prayed to their cat, does that support the theory that their cat is God?