I occasionally read reference to a cartridge's expansion ratio. The gist of it is usually that the higher the ratio, the more suitable slow burning powder are. So within the same family of cartridges, I would expect a .338 Federal to prefer faster powders than a .308 Win or .260 Rem. So how exactly do they measure this ratio?

Actually, you got that backwards. The higher the expansion ratio, the more efficient a cartridge is. Or in other words, the 338 federal would have a higher expansion ratio than a 243, 260, 708, 308 etc given equal barrel length. You are right that the 338 fed would need faster burning powders for optimal performance. The expansion ratio is the ratio of the volume of the bore from the muzzle to the flashhole of the case to the volume of the powder space of the cartridge. To figger it out, divide the volume of the powder space into the volume of the powder space and bore added together. When one thinks about this, one might see that combustion happens inside the brass and the gas expansion is what propels the bullet. Therefor, a longer bullet allows the exansion to be behind the bullet for longer time and would increase the expansion ratio.