Web-based Supplementary Materials for Weighted Likelihood Method for Grouped Survival Data in Case-Cohort Studies with Application to HIV Vaccine Trials

Web Appendix A: Derivation of P (∆i = δi, Ri = j|Xi) Clearly the pair of random variables (∆i, Ri), or equivalently (∆i,Ri+1, Ri), is completely determined by (Ti, Ci). In particular, the set {∆i,Ri+1 = 0, Ri = j} is equivalent to observing the event in (tj−1, tj], which in turn is equivalent to the set {Ti ∈ (tj−1, tj], Ci ≥ tj}; and the set {∆i,Ri+1 = 1, Ri = j} is equivalent to censoring the event at time tj, which in turn is equivalent to the set {Ti ≥ tj, Ci ∈ (tj−1, tj]}. Let δi denote the realized vector values of ∆i. Then by the conditional independence of Ti and Ci given Xi, the conditional probability mass function of (∆i, Ri) given Xi can be written as