I will reply to this - if JPM wants to move the discussion, fine.
TA, I'm glad you demonstrated my point so well - that Christian and Naturalist worldviews agree only to the extent that the former adapts to naturalism for convenience or out of inconsistency. Let me expand, using my original example. In my career I did something in the order of a quarter of a million face-to-face consultations. Of that, a significant proportion involved coronary artery disease. My uniform experience was that, given certain angiographic criteria, operative intervention was required.
With one notable exception: my friend P, slated for surgery, attended a prayer meeting where he was convinced God had healed him. Angiography confirmed he was certainly healed (as do his annual Christmas cards). That is the phenomenon to be explained.
The Christian physician (me), once appraised of the facts, gave thanks to God. If we may take your reponse as a proxy for my "naturalist consultant", what you have actually done is to lose the phenomenon, and the human individual, altogether in the word "anecdote", and to replace it with a statistical fog of what are invariably poorly designed studies, to show (in the end) that there is no explanation. Not least in their poor design is that they are all unconsciously seeking to explain God as a phenomenon within Nature by naturalism - whereas my point on worldviews was that one is as justified in seeking to explain Nature as a phenomenon within God. It is a straight choice about where you ground the reality behind phenomena.
Let's examine those worldviews. The Christian takes God and his will as the source of all reality. Often God acts, or causes entities in the world, to act, regularly. Sometimes, he chooses to act contingently. Lets label that worldview [God].
The Naturalist believes there is a thing called "Nature" that is the the source of reality. This is usually ill-defined, but seems usually to be considered as a system of universal laws of unknown origin that govern all regular events. Contingent events are attributed to "chance", but those Naturalists who think a bit more conclude that Laws acting in concert are actually the source of those contingencies - only in ways that are not actually amenable to investigation because too complex, too small-scale, unknown etc. We'll call that worldview [Nature].
Your first set of studies (I won't quibble on the details, as it doesn't affect the point) actually assume that the reality behind the psychology of support, or of trial blinds, is some kind of law based on [Nature], which being still unclear retain the character of chance. But under [God] any regularities are equally explicable by his will, and unexplained contingencies are also governed by the same will. Beneficial human relationships are from God as much as anything else.
The very act of designing a blind-trial of that kind is a giveaway of naturalistic assumptions: it is only necessary because of the universal observation that telling a volitional subject the aims of sociological or medical research will affect the outcomes. Yet there's no attempt in the studies to involve God, as the volitional agent under test, in the blind. Nor could there be, of course, for he knows all things. Yet he is treated as a blind hypothesis within [Nature], whereas (under theism) Nature is a hypothesis under [God], and God is always the volitional Subject, and never an object. maybe that's what JPM meant in his post - if so, I agree.
Your second set of studies uses statistical outcomes to negate a "God hypothesis". Not only does that do nothing to account for the actual phenomenon of my friend's cure (it's the equivalent of suggesting that there is no Usain Bolt because statistics show that on average people can't run fast), but also it can be used in exactly the same way against Nature, seen as the inviolable laws that govern reality, should one choose to make that the hypothesis instead of God.
For whatever it was that cured my friend, it was something, and under [Nature] it was the action of laws. So maybe the prayer had some psychological or other effect, or maybe it was an unrelated coincidence of blood chemistry, or something else. So your trials seek to control for as much of that as possible, and find that some people get better, some get worse. In other words the fixed laws sometimes produce one outcome, and sometimes another, for hidden reasons. If the laws were being treated treated as the hypothesis under test, there is no evidence for them in these trials. All you have is a set of contingent results, which can be ignored altogether if you treat them as a mass with a statistical distribution.
Remember how Sir Robert Maxwell, the father of statistical science, viewed the events making up the statistics:
Would it not be more profound and feasible to determine the general constraints within which the deity must act than to track each event the divine will enacts?
Statistics generalise events to reveal any regularities: all sociology is based on the fact that many of those individual events are free choices.
But there, still sitting having his dinner amidst the inconclusive studies of [Nature], is my friend who once had CAD, but no longer has, who has a perfectly good explanation under [God]. He's not trying to prove, or disprove anything. He's just living in a different reality - that works well.