Infants with hypoplastic left heart syndrome require an initial Norwood operation, followed some months later by a stage 2 palliation (S2P). The timing of S2P is critical for the operation’s success and the infant’s survival, but the optimal timing, if one exists, is unknown. We attempt to identify the optimal timing of S2P by applying an extension of propensity score analysis to data from the Single Ventricle Reconstruction Trial (SVRT). In the SVRT, the procedure used in the initial Norwood operation was randomized, but the timing of the S2P was not chosen randomly. Because there was systematic collection of surgery times and patient follow-up data, the trial constitutes a thoroughly documented observational study. To achieve the extension of propensity score analysis, we model the time to surgery as a function of confounders using a discrete competing-risk model. We then apply inverse probability weighting to estimate a spline hazard model for predicting survival from the time of S2P. Our analysis suggests that S2P conducted at 6 months after the Norwood gives the patient the best chance of survival.