Independence of total species richness and richness difference are not the same thing

Leprieur and Oikonomou (2014) criticized that a replacement index β− 3, a partitioned component of Jaccard index βjac, was not richness independent and should not be used in biogeographic and ecological studies. However, Leprieur and Oikonomou failed to recognize the difference between richness and richness difference. Independence of total species richness is not equal to independence of richness difference. Theoretically and ideally, it is true that β− 3 (and βjac as well) is not independent of richness difference while Simpson index (βsim) is fully independent of richness difference. However, all these indices actually are independent of total species richness. At last, it is worth mentioning that the ideal independence studied here is easily violated in computational simulation and real-world settings.