Understanding how past environmental changes have influenced the diversification of species is key for predicting the impact of current and future environmental changes on biodiversity, and the associated human, social and economic impact. Various environment-dependent phylogenetic comparative methods, that allow testing whether and how past. These methods build upon classical birth-death models of cladogenesis used to study speciation and extinction dynamics from phylogenies of extant species, where the evolutionary rates correspond to speciation and extinction rates. Even though several recent studies have fitted these models to comparative phylogenetic data, providing estimates of the association between evolutionary rates and environmental variables, the phylogenetic methods already developed have several limitations.