An introduction to the theory of competing risks is given, then the formulation of competing risks models with the mathematical relationships between Dagum type-1 and Beta-P models are derived. The formulation of the Dagum type-I model within the context of competing risks is given. Hence Dagum type-I probability distribution is used to fit competing risks data. The fitted model is compared to the Weibull probability/ distribution. Finally the model derived is used to fit the survival times of cancer patients and the maximum likelihood method is used to estimate the parameters of the model and their covariance structure.