We propose to model the cause-specific cumulative incidence function of multivariate competing risks data using a random effects model that allows for within-cluster dependence of both risk and timing. The model contains parameters that makes it possible to assess how the two are connected, e.g. if high-risk is related to early onset. Under the proposed model, the cumulative incidences of all failure causes are modeled and all cause-specific and cross-cause associations specified. Consequently, left-truncation and right-censoring are easily dealt with. The proposed model is assessed using simulation studies and applied in analysis of Danish register-based family data on breast cancer.