I'm making an intermediate claim. I find that most studies that claim "A does not cause B" to be extremely unconvincing exactly because of the statistical methodologies used and the fact that they don't actually do any statistics that would imply that A does not cause B.

The upshot is that I (and I'm more scientifically minded than most) am less than fully trusting of the interpretations given to a lot of claims resulting from scientific trials. Furthermore, I think a large part of this issue originates in both awful language used in actual papers and in the fact that people aren't doing the types of analysis that they should be doing.

The fact that I so frequently see news exclaiming some clinical result and that the statistical analysis (even if done rigorously and completely honestly) does not imply what the news claims means that something is very wrong in the way that statistics is used.

At least in Physics (my field), people tend to be a bit more careful about it. Physicists may do somewhat silly p-value tests, but they mostly insist on p values very very close to zero, and papers tend to at least answer the right question (such-and-such effect is upper-bounded by some tiny number to five sigmas).

I will backtrack a bit, though. I dug up "Vaccines are not associated with autism: An evidence-based meta-analysis of case-control and cohort studies". They do establish bounds on the odds ratio. Good for them.