We consider weighted logrank tests for interval censored data when assessment times may depend on treatment, and for each individual, we only use the two assessment times that bracket the event of interest. It is known that treating finite right endpoints as observed events can substantially inflate the type I error rate under assessment–treatment dependence (ATD), but the validity of several other implementations of weighted logrank tests (score tests, permutation tests, multiple imputation tests) has not been studied in this situation. With a bounded number of unique assessment times, the score test under the grouped continuous model retains the type I error rate asymptotically under ATD; however, although the approximate permutation test based on the permutation central limit theorem is not asymptotically valid under every ATD scenario, we show through simulation that in many ATD scenarios, it retains the type I error rate better than the score test. We show a case where the approximate permutation test retains the type I error rate when the exact permutation test does not. We study and modify the multiple imputation logrank tests of Huang, Lee, and Yu (2008, Statistics in Medicine, 27: 3217–3226), showing that the distribution of the rank-like scores asymptotically does not depend on the assessment times. We show through simulations that our modifications of the multiple imputation logrank tests retain the type I error rate in all cases studied, even with ATD and a small number of individuals in each treatment group. Simulations were performed using the interval R package. Published 2012. This article is a US Government work and is in the public domain in the USA.