Precisely because of the evidential weakness of case studies, the scientific community has long since recognized the value of randomized controlled trials (RCTs). Because of the way RCTs are structured, if the patients who received the target treatment fare significantly better than those in the control group, then this cannot be explained by regression towards the mean, biased reporting or memory, or the placebo effect. This is not to say that RCTs provide proof of the effectiveness of a given intervention. There might be design flaws in a particular study, leading to a false positive result; there could be reporting errors, or even outright fraud. Such factors could explain a positive result. Finally, it still might be true that a given positive result was just a chance fluke: the people who were poised to improve regardless of treatment just happened to be overrepresented in the treatment group. Citing chance in this way is tantamount to leaving the positive result as an unexplained mystery, for citing chance is really just another way of saying that we don't have an explanation. However, sometimes this can be exactly the right thing to say. Even if one were testing an utterly worthless medication against a placebo, in one out of 20 trials one would expect a result that would be expected to occur again only if the same trial was repeated 20 times, that is, one with a P < 0.05. Moreover, because of publication bias or the filedrawer effect, a large number of studies with negative results might remain unpublished, leaving only those studies which attained a positive result purely by chance. So, while citing chance is to leave a particular positive result as an unexplained mystery, it may not be leaving an incredibly large or puzzling mystery.
There have been a number of RCTs testing homeopathic remedies. Some have yielded apparently positive results, while many others have failed to show that homeopathic remedies do better than placebos. In a meta-analysis published in Lancet in 2005, the authors concluded that when analyses were restricted to large trials of higher quality there was no convincing evidence that homeopathy was superior to placebo.