Try this:

A is better than C

B is not better than C

Therefore A must be better than B.

Doesn't feel quite right, does it?

Now try this one:

A is significantly better than C

B is not significantly bettter than C

Therefore A must be significantly better than B

And finally:

Confidence interval on A excludes C

Confidence interval on B does not exclude C

Therefore A's confdience interval must exclude B

Hopefully you can now see clearly what's been wrong all along.

This failure of logic can be disguised statistical testing: A vs C, B vs C. But the only way you can be sure is to do the direct comparison: A vs B. The golden rule in statistical testing is to use the direct, appropriate test *for the thing you are interested in*, not some other indirect comparison. Insert the names of a few common drugs instead of A, B, C and you can see why this is hugely important.

This and more in a compendium of Ben Goldacre's writings:

I think you'll find it's a bit more complicated than that

An excellent way to fill those long days between Christmas and New Year!