Off pat & Off your own bat

Any ideas as to the origins of these?

The alphabetical meanings-and-origins list on this website explains "on your Pat" as "on your own" (Cockney or Irish rhyming slang: "on your Pat Malone"). Maybe our British friends can tell us whether "off pat" is related. The Oxford Engl. Dict. traces "off his own bat" to cricket: "in reference to the score made by a player's own hits" and, figuratively, "solely by his own exertions, by himself."