Tests can be run on numerous kinds of prayer. In fact, they have been run on the efficacy of praying on illness. But here's a simple one.
You may have come into the conversation late, but I've already admitted that some beliefs in prayer are irrational, particularly those that can and have been proven wrong. However, many Christian beliefs in prayer, particularly those in conjunction with the belief that God might not want us to know for certain that he exists, cannot be "shown to be extremely unlikely."
In my experience, praying has no significant influence on the outcome of events. I know of no non-religious studies that show that praying has a significant influence on the outcome of events, but if there are, you can show them to me and I'll retract the statement.
I completely agree that there is no evidence of prayer working, but absence of evidence is only evidence if a particular testable outcome would be expected. Since many Christians don't believe that God grants all prayers, there is no reason to assume that a particular outcome will necessarily occur even if prayer works.
There are some academic studies that show that prayer has minimal to insignificant influence on healing. Most researchers consider it to be so unlikely as to not be worth study.
The research that was done recently on the efficacy of praying on illness doesn't prove that prayer doesn't work. Most researchers consider most beliefs in prayer untestable, and therefore ridiculous to study. I tend to agree.
Now of course, this is only testable for intercessory prayers.
It is only testable for certain beliefs in intercessory prayers, specifically those beliefs that involve God granting prayers under testable conditions. If a Christian believes that God must grant prayers under testable conditions, then I agree that research would indicate otherwise.
For prayers where you are simply talking to God, the protocol would require that you have a way of determining whether or not God heard you. That's a bit of a sticky point, considering that it would first require a comprehensive definition of "God" and some evidence that He exists. In short, you cannot show communication with an entity that you cannot show exists.
Yes, which is why we specifically eliminated this sort of belief from consideration earlier in the thread. We were talking only about belief that prayer can influence events. Unfortunately, these beliefs cannot necessarily be "shown to be extremely unlikely."
-Bri