In clinical trials of long-term therapies, patients often terminate their treatments earlier than planned. When analysing time-to-failure data, one approach to account for early treatment termination censors failure at the time of termination of therapy. In general, however, this does not produce valid inferences about the distribution of time to failure that would have occurred had treatment not been terminated. In contrast, intent-to-treat analyses, which are based on time to failure regardless of whether and when treatment is terminated, always produce valid inferences about the unconditional distribution of time to failure. Early treatment termination does not distort the size (type I error rate) of intent-to-treat tests but can cause a loss in power. Modifications to ordinary logrank tests can be used to recover some of the lost power without affecting test size, and can be most useful when the proportion of at-risk patients still taking their treatment changes substantially during periods when failures are observed. Extensions of the modified test to include strata are straightforward, although important design questions require further research.