When "facts" are not facts: what does p value really mean, and how does it deceive us?

The recent paper in JAMA alleging that frozen embryo transfer causes twice the risk of childhood cancer in the offspring is an excellent example of the erroneous use of statistical tests (and the misinterpretation of p value) that is common in much of the medical literature, even in very high impact journals. These myths backed by misleading statements of "statistical significance" can cause far-reaching harm to patients and doctors who might not understand the pitfalls of specious statistical testing.