What is meant by the jurists' saying, "This is contrary to analogy (qiyas)"?

The jurists' statement "contrary to analogy" (على خلاف القياس) means that the ruling on an issue was established in deviation from jurisprudential rules. This is a point of contention between some Hanbalis, Shaykh al-Islam Ibn Taymiyyah, and his student Ibn al-Qayyim, on one side, and the majority of scholars, on the other. For example, they differed on the permissibility of ijarah (leasing/hiring) and qard (loan):

The majority of jurists hold that ijarah was legislated contrary to analogy because it is "selling what does not exist" (an exchange of benefit for money), and selling what does not exist is invalid based on the hadith "Do not sell what is not with you." However, it was exceptionally permitted due to necessity.

In contrast, some Hanbalis, Ibn Taymiyyah, and Ibn al-Qayyim maintained that it is permissible in accordance with analogy, and that it is not a sale of tangible assets but rather of benefits, which are difficult to possess at the time of contract.

Similarly, they differed on qard (loan):

The majority of scholars hold that it is permissible contrary to analogy, out of compassion for those in need; because it is the ownership of something with the return of its like, thus making it a usurious sale of its kind with a delayed قبض (possession), so its permissibility is an exception.

As for Ibn Taymiyyah and Ibn al-Qayyim, they held that it is permissible in accordance with analogy, and that it is of the same type as donating benefits, like 'ariyah (gratuitous loan of an item for use).

This difference of opinion extends to other issues such as the removal of impurities, cupping (hijama), tayammum (dry ablution), hawalah (transfer of debt), rahn (pledge/mortgage), and the liability of the 'aqilah (male relatives on the father's side) for blood money (diyah).