Looking at the static characteristic (input/output graph), the ratio R is the reciprocal of the slope above the threshold. (Below the threshold the slope is 1.) In the doc example, above threshold the output increases 3 dB for a 15 dB increase in input, so the slope is 3/15 = 0.2. The ratio is the reciprocal of that: 15/3 = 5.
In the doc for compressor you can find the equation for the static characteristic above threshold, where you can see that the slope is 1/R: