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Development of a flexible Erlang mixture model for survival analysis is introduced. The model for the survival density is built from a structured mixture of Erlang densities, mixing on the integer shape parameter with a common scale parameter. The mixture weights are constructed through increments of a distribution function on the positive real line, which is assigned a Dirichlet process prior. The model has a relatively simple structure, balancing flexibility with efficient posterior computation. Moreover, it implies a mixture representation for the hazard function that involves time-dependent mixture weights, thus offering a general approach to hazard estimation. Extension of the model is made to accommodate survival responses corresponding to multiple experimental groups, using a dependent Dirichlet process prior for the group-specific distributions that define the mixture weights. Model properties, prior specification, and posterior simulation are discussed, and the methodology is illustrated with synthetic and real data examples. 

Summary A sequentially adaptive Bayesian design is presented for a clinical trial of cord-blood-derived natural killer cells to treat severe haematologic malignancies. Given six prognostic subgroups defined by disease type and severity, the goal is to optimize cell dose in each subgroup. The trial has five co-primary outcomes: the times to severe toxicity, cytokine release syndrome, disease progression or response and death. The design assumes a multivariate Weibull regression model, with marginals depending on dose, subgroup and patient frailties that induce association between the event times. Utilities of all possible combinations of the non-fatal outcomes over the first 100 days following cell infusion are elicited, with posterior mean utility used as a criterion to optimize the dose. For each subgroup, the design stops accrual to doses having an unacceptably high death rate and at the end of the trial selects the optimal safe dose. A simulation study is presented to validate the design's safety, ability to identify optimal doses and robustness, and to compare it with a simplified design that ignores patient heterogeneity.