The impact of risk factors on the amount of time taken to reach an endpoint is a common parameter of interest. Hazard ratios are often estimated using a discrete-time approximation, which works well when the by-interval event rate is low. However, if the intervals are made more frequent than the observation times, missing values will arise. We investigated common analytical approaches, including available-case (AC) analysis, last observation carried forward (LOCF), and multiple imputation (MI), in a setting where time-dependent covariates also act as mediators. We generated complete data to obtain monthly information for all individuals, and from the complete data, we selected “observed” data by assuming that follow-up visits occurred every 6 months. MI proved superior to LOCF and AC analyses when only data on confounding variables were missing; AC analysis also performed well when data for additional variables were missing completely at random. We applied the 3 approaches to data from the Canadian HIV–Hepatitis C Co-infection Cohort Study (2003–2014) to estimate the association of alcohol abuse with liver fibrosis. The AC and LOCF estimates were larger but less precise than those obtained from the analysis that employed MI.