Problems of practical interest in the analysis of data on contraceptive use, from Demographic and Health Surveys (DHS), include the estimation of the cause-specific probability of discontinuation by time t (the cumulative incidence function), in the presence of other competing causes and the evaluation of the effect of covariates on the cause-specific hazards of discontinuation. Methods of analysis of failure time data with competing risks are by now fairly well developed in the case of a simple random sample. However, the data from the DHS are clustered by geographical areas and include multiple episodes per woman. For a marginal (population average) approach, we propose using methods developed for simple random samples with standard errors calculated using a double bootstrap to take account of the clustered hierarchical nature of the data. In the conditional approach, the cause-specific hazards are modelled as log-linear functions of the covariates conditional on random effects of clusters and women, using a three-level multinomial discrete-time logit model. The methods are applied to data from Egypt 1992 DHS on the oral contraceptive pill use.