Coseismic and rainfall-triggered landslides are a common hazard in many terrains, and the risk associated with them can be quantified, usually by probabilistic modelling. These events are well-documented as a special case of a cascading hazard chain, and the assessment is commonly done via spatial modelling of susceptibility (suppressing temporal dependence) or tailoring models to specific areas and events. The interaction between Earthquakes and rainfall is not usually implemented in a model, as it is considered coincidental. However, because landslides have multiple triggering factors, there is a need for a statistical model that incorporates both features, in a manner such that the separate and joint effects can be estimated. This helps with understanding the interactions between primary events in the triggering of a single secondary hazard type that is crucial for generally applicable multi-hazard methodologies. The presented work aims at the apportioning of the relative and combined effect on landslide triggering by earthquakes and rainfall using a discrete approximation to a multivariate hierarchical point process. Doing so provides a building block in a general framework with the potential to be extended to other chains of events. A case study on the Italian region of Emilia-Romagna is included, using one of the longest and most complete landslide data sets known. Multiple models for the rainfall-earthquake interaction in landslide triggering are trialed, with the best explanation being additive effects from rainfall intensity, rainfall duration and coseismic triggering.