I've never seen that kind of formula, probably too difficult to determine through impirical data. Most gearmotor manufacturers (Dodge, SEW-Eurodrive, Nord, Falk, etc) will list "generalized" efficiencies for a particular type of gearing configuration. For example, helical gears = 0.96, worm gears = .85, etc.
The efficiency can vary a lot with temperature with any gearing depending on the lube and range of temperature. A worm gear will be affected much more by temperature changes since it uses a sliding action instead of a rolling action for the reduction. You can minimize this with synthetic oil and/or oil coolers/heaters.
The rule of thumb I use for efficiency is 3%-5% per reduction for spur and helical and the ratio equals the loss on worm gears. A two stage helical reducer will be 94%-90% efficient and a 60:1 worm gear will be 40% efficient. Remember that worm gears have high starting friction for the first half turn of the output shaft until the oil gets in mesh.
Hypoid gear efficiency can vary greatly between manufactures. I think GTR/Brother has the only high efficiency high reduction hypoid. There are a few manufactures that can do a good hypoid, 5%, up to about 5:1, GTR/Brother and Sumitomo are two. SEW has a something that resembles a hypoid, Spiroplan, but is very inefficient and should be avoided.
Dudley's Gear Handbook gives formulas for gear mesh efficiency, which include terms for friction. Friction depends, of course, on surface roughness and lube viscosity, which in turn depends on temperature.
For overall gearbox efficiency, you'll also need to include beaaring friction, windage, lube churning, and seal friction.
In my opinion, There is 1% loss in every stage in the case of spur/Helical gear unit. This is applicable in the case of gear unit wih Mineral oil and proper cooling. In a 4 stage gear unit the efficiency is 96%. If you want to arrive a formula it is possible with KA factor. (Application Factor), Since this determines the amount of load that the tooth experience.
