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We propose to estimate the exposure effects through the cause-specific proportional hazards frailty model where random effects are introduced into the model to account for the within-cluster correlation. We propose a weighted penalized partial likelihood method where the weights represent the probabilities of the occurrence of events, and the weights can be obtained by fitting a classification model for the event types on the training dataset. Alternatively, we propose an imputation approach where the missing event types are imputed based on the predictions from the classification model. We derive the analytical variances, and evaluate the finite sample properties of our methods in an extensive simulation study. 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