The placebo effect is pretty well documented, and it is fascinating. Sugar pills can produce measurable benefits. I'd suggest that it's power may go some way to explaining the perceived efficacy of prayer.
There's also the matter of regression to the mean. I'd suggest looking at the Daniel Kahneman quote here -
Regression toward the mean - Wikipedia - and seeing if you can discern how I think his thoughts are applicable to prayer.
There are also a number of other powerful psychological biases which I would suggest are in play: - confirmation, selection and survivorship bias, which mean that people tend to notice instances of "successful" prayer, and discount or otherwise downplay unsuccessful prayer attempts. [edit: in the case of survivorship bias, they aren't even *there* to document the failure of their prayers to save them]
There's also the matter that even if prayer is effective (and not for reasons of psychology or statistics, but some mysterious force, perhaps of a spiritual or supernatural kind) - then it isn't at all clear what fountainhead we have to thank for these helpful intercessions. I'd be surprised if the power of prayer wasn't considered to be similarly efficacious for all faith groups. Assuming that's true, this would suggest that it isn't the specific dogma but something to do with the general emanation of psychic wobbles. But just to be clear, I don't think there's any particular evidence to suggest psychic wobbles have any part to play - I think psychology and statistics probably cover most bases.
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