I dont think that setting by the moon is supposed to increase hatch rate. I think that it is supposed make chicks healthier and grow faster. I don't know if any of it is true, but it's a fun experiment.
Old Farmers Almanac:
Those born under a waxing Moon, in the fruitful signs of Cancer, Scorpio, and Pisces, are healthier and mature faster. Another saying from them also says: Set eggs to hatch on the Moon's increase, but not if a south wind blows.