One of the big studies on the impact of dyes on hyperactivity had two things different in the test drinks as compared to the control (placebo) but the general public and the press ignore the possible impact of the preservative only focusing upon the absence or presence of the dyes.
I'm not saying that the dyes were not responsible; only that the study does not allow one to distinguish between the dyes, the preservative, or dye & preservative as the source of the impact.
This is a small example of the short comings with correlations; typically not just one factor is changed between the test case and the control to definitively identify the likely cause.