In randomized trials comparing a new therapy to standard therapy, the sharp null hypothesis of equivalent therapeutic efficacy does not imply the intent-to-treat null hypothesis of equal outcome distributions in the two-treatment arm if non-compliance is present. As a consequence, the development of analytic methods that adjust for non-compliance is of particular importance in equivalence trials comparing a new therapy to standard therapy. This paper provides, in the context of equivalence trial, a unified overview of various analytic approaches to correct for non-compliance in randomized trials. The overview focuses on comparing and contrasting the plausibility, robustness, and strength of assumptions required by each method and their programming and computational burdens. In addition, several new structural (causal) models are introduced: the coarse structural nested models, the non-nested marginal structural models and the continuous-time structural nested models, and their properties are compared with those of previously proposed structural nested models. The fundamental assumption that allows us to correct for non-compliance is that the decision whether or not to continue to comply with assigned therapy at time t is random (that is, ignorable or explainable) conditional on the history up to t of measured pre- and time-dependent post-randomization prognostic factors. In the final sections of the paper, we consider how the consequences of violations of our assumption of conditionally ignorable non-compliance can be explored through a sensitivity analysis. Finally, the analytic methods described in this paper can also be used to estimate the causal effect of a time-varying treatment from observational data.