In a recent paper we defined a new filtration of the mapping class group – the Lagrangian filtration. We here determine the successive quotients of this filtration, up to finite index. As an application we show that, for any additive invariant of finite-type (e.g. the Casson invariant) and any level of the Lagrangian filtration, there is a homology 3-sphere which has a Heegaard decomposition whose gluing diffeomorphism lies at that level, on which this invariant is non-zero. In a final section we examine the relationship between the Johnson and Lagrangian filtrations.