I've only done one comparison trial so its nothing conclusive but the results were in favour of planting by the moon when looking at germination rates. Here's how the experiment went
I made two beds side by side using the same technique and
compost from the same pile, built within a few days of eachother
Red kidney beans were planted in both beds
One bed was planted during the last quarter about a week or more before the new moon
The other bed was planted the day of the new moon.
RESULTS: The beans planted the day of the new moon sprouted first and took off better. Unfortunately I left that farm before I could weigh the yields.