Interval censoring might happen in survival analysis when it is only known that the survival response belongs to a time interval. Usually in survival analysis, a single cause for the occurrence of the event is considered, however, some studies may be interested in the occurrence of more than one event, called competing risks. The Cox proportional hazards model is model might be used to the cause-specific failure rate function. We propose a cause-specific regression model for correlated data and interval censoring. The cause-specific regression model is an extension of the traditional Cox regression model for each event type, in which failures of competing events are treated as censored observations. The present study was motivated by a longitudinal study conducted at the Dental Trauma Clinic at the Federal University of Minas Gerais, Brazil aiming to evaluate luxation of permantent teeth for their pulp prognosis as well as its prognostic factors. In this paper, the proposed methodology considers a GEE-type model using an independent work matrix to accommodate the presence of clusters. A Taylor series is used to approximate the log baseline hazard function in Cox proportional hazards regression. With this formulation, the likelihood ratio test can be used to select an appropriate order for the Taylor series approximation and maximum likelihood is used to estimate model parameters and provide statistical inference. The model performance was evaluated using a simulated data set and showing that the proposed methodology has good small sample properties. The proposed methodology is applied to a real dental trauma dataset.