In randomized trials to prevent breast milk transmission of human immunodeficiency virus (HIV) from mother to infant, investigators are often interested in assessing the effect of a treatment or intervention on the cumulative risk of HIV infection by time (age) t in infants who are alive and uninfected at a certain time point τ0 τ0 within the principal stratum of infants who would be alive and uninfected by τ0 regardless of randomization assignment. Large sample non-parametric bounds and a semi-parametric sensitivity analysis model are developed for drawing inference about this causal effect. A simulation study is presented demonstrating that the proposed methods perform well in finite samples. The proposed methods are applied to a large, recent MTCT trial
