What is difficult to understand about this? You can't perform reliable experiments with something when the following conditions are inevitably involved:
1. It is not about an ability that works or doesn't work but an instance of asking a Being to do something. If I say I can read minds then experiments could show I'm either doing it at the time of the experiments or not doing it, but if I ask someone to do something then whether they don't do it doesn't automatically indicate by any measure whether or not they exist or whether or not my asking has really influenced their behavior if they do.
2. This and #3 go along the same lines. If the thing prayed for happens, there is no scientifically verifiable way of knowing whether it has happened because the Being accepted the supplication.
3. If the thing doesn't happen, there is no scientifically verifiable way of knowing whether it has happened because the Being did not accept the supplication.
4. The very existence of the Being in question has not and cannot be verified by science in the first place since the Being is supernatural and science is the study of nature, not supernature.
5. Only a miniscule slice of the total number of petitionary prayers that have been answered or not answered throughout history could be viewed anyway. Not that this particularly matters in light of 1-4, but I suppose it at least bears mention.
Whether a cause is capricious or simply variable does not preclude its study.
Supernatural causes can be studied by science, since science asks only for an effect. It doesn't make any distinction a priori as to the nature of that effect.
Linda