SUMMARY Cox & Oakes (1984, p. 66) introduced the 'strong version' of the accelerated failure time model with time-dependent exposures. We provide conditions under which this model could be used to estimate, from observational data, the causal effect of a time- varying exposure or treatment on time to an event of interest in the presence of time- dependent confounding variables. We propose a class of semiparametric tests and estimators for the model parameters. This class contains an estimator that is semipara- metric efficient in the sense of Begun et al. (1983). The purpose of this paper is first to provide conditions under which it would be possible to estimate, from observational data, the causal effect of a time-dependent treatment or exposure on time to an event of interest in the presence of time-dependent confounding covariates and, then, to construct estimators for the treatment effect under these condi- tions. Our approach will be based on the semiparametric estimation of the parameters of the 'strong version' of the accelerated failure time model with time-dependent exposures or treatments introduced by Cox & Oakes (1984, p. 66). The usual approach to the estimation of the effect of a time-varying treatment on survival is to model the hazard of failure at t as a function of past treatment history using a time-dependent proportional hazards model. We show that the usual approach may be biased, whether or not one further adjusts for past confounder history in the analysis, when (a) there exists a time-dependent risk factor for, or predictor of, the event of interest that also predicts subsequent treatment, and (b) past treatment history predicts subsequent risk factor level. The following four examples demonstrate conditions (a) and (b) will often be true in an observational study in which there is 'treatment by indication'. The drug zidovudine, formerly AZT, used in the treatment of the acquired immunodeficiency syndrome, is a direct red-blood cell toxin that is contra-indicated in anaemic subjects, i.e. subjects with depressed red-cell counts, since the toxic effects of zidovudine can worsen the anaemia. Further, anaemic patients are at increased risk of death. Thus in an observational study of the effect of zidovudine on survival of patients with the acquired immunodeficiency syndrome, anaemia is both a risk factor for death