||* Two alternative frameworks have been proposed to partition compositional dissimilarity into replacement and nestedness-resultant component or into replacement and richness-difference components. These are, respectively, the BAS (Baselga 2010, Global Ecology and Biogeography, 19, 134–143) and POD (Podani & Schmera . Oikos, 120, 1625–1638) frameworks. * We conduct a systematic comparison of parallel components in alternative approaches. We test whether the replacement components derived from the BAS and POD frameworks are independent of richness difference. We also evaluate whether previously reported tests of monotonicity between indices and ecological processes are informative to assess the performance of indices. Finally, we illustrate the consequences of differences between the BAS and POD frameworks using the North American freshwater fish fauna as an empirical example. * In the BAS framework, the nestedness-resultant component (βjne or βsne) accounts only for richness differences derived from nested patterns while, in the POD framework, richness-difference dissimilarity (βrich or βrich.s) accounts for all kind of richness differences. Likewise, the replacement components of both alternative methods account for different concepts. Only the replacement component of the BAS framework (βjtu or βsim) is independent of richness difference, while the parallel component in the POD framework (β−3 or β−3.s) is not (i.e. it is mathematically constrained by richness difference). * Therefore, only the BAS framework allows separating (i) the variation in species composition derived from species replacement which is independent of richness difference (i.e. not mathematically constrained by it) and (ii) the variation in species composition derived from nested patterns.