If you can determine the capacitance, you can determine capacitive reactance (Xc in Ω) where C=capacitance in farads, ƒ=frequency in Hertz, ∏=3.1416, as:
Xc=1/(2∏ƒC)
... but that will only help. I haven't run across a magic bullet engineering formula for determining efficiency of hat materials, arm lengths, radius, or shape. All I know for sure is that Xc is reduced by C.
It makes loading more effective, meaning you could reduce the number of turns, and therefore loss, in a loading coil. So make your antenna, a LC with half the number of turns for a given frequency, add your hat arms longer than you think they should be, and then start clipping the ends off a little at a time until it resonates.
Ya know ... I'd be willing to bet that particularly with AM BCB radio, a hat would be more effective at lower frequencies where many more turns are needed in a loading coil for a 3m antenna. No idea what the efficiency break point would be though, i.e., how big the hat should be as opposed to coil turns and antenna length -- remember hats are considered part of the antenna length under part 15 rules, whereas coil wire length is not.
The loading coil shouldn't count except where the antenna is a coil of wire wrapped around a supporting rod referred to as a continuously loaded antenna. I'm told the length of wire for the coil is counted as the radiator. This would be like a "rubber duck" antenna on a much larger scale.
In some past threads here, there has occassionally been mention of cases where the loading coil is considered in the length of the antenna subject to the Inspectors interpretation whereas the coil would need to be in a shielded box.
There is a copy of an old low power antenna design manual here:
Medium Frequency Antenna Construction Plans
Offering several antenna designs, tuning coil options and the mathematical formulas needed to approximate performance.
I'm going to set those pages up as a PDF.
Oops ... supposed to be on the air .... seeya ...
I mentioned it in another thread too, but I went ahead and created a multi-page PDF of the old antenna manual mentioned by scwis:
http://fhtinyradio.com/Library/DocLib/MediumFrequencyAntennas.pdf