Familial aggregation and the role of genetic and environmental factors can be investigated through family studies analysed using the liability-threshold model. The liability-threshold model ignores the timing of events including the age of disease onset and right censoring, which can lead to estimates that are difficult to interpret and are potentially biased. We incorporate the time aspect into the liability-threshold model for case-control-family data following the same approach that has been applied in the twin setting. Thus, the data are considered as arising from a competing risks setting and inverse probability of censoring weights are used to adjust for right censoring. In the case-control-family setting, recognising the existence of competing events is highly relevant to the sampling of control probands. Because of the presence of multiple family members who may be censored at different ages, the estimation of inverse probability of censoring weights is not as straightforward as in the twin setting but requires consideration. We propose to employ a composite likelihood conditioning on proband status that markedly simplifies adjustment for right censoring. We assess the proposed approach using simulation studies and apply it in the analysis of two Danish register-based case-control-family studies: one on cancer diagnosed in childhood and adolescence, and one on early-onset breast cancer.