Evidence supporting the current World Health Organization recommendations of early antiretroviral therapy (ART) initiation for adolescents is inconclusive. We leverage a large observational data and compare, in terms of mortality and CD4 cell count, the dynamic treatment initiation rules for human immunodeficiency virus‐infected adolescents. Our approaches extend the marginal structural model for estimating outcome distributions under dynamic treatment regimes, developed in Robins et al. (2008), to allow the causal comparisons of both specific regimes and regimes along a continuum. Furthermore, we propose strategies to address three challenges posed by the complex data set: continuous‐time measurement of the treatment initiation process; sparse measurement of longitudinal outcomes of interest, leading to incomplete data; and censoring due to dropout and death. We derive a weighting strategy for continuous‐time treatment initiation, use imputation to deal with missingness caused by sparse measurements and dropout, and define a composite outcome that incorporates both death and CD4 count as a basis for comparing treatment regimes. Our analysis suggests that immediate ART initiation leads to lower mortality and higher median values of the composite outcome, relative to other initiation rules.