Parametric Competing Risks Models in Clinical Trials

Some fully general parametric competing risks models, which allow the underlying (possibly dependent) lifetimes to be modeled as a function of an arbitrary number of covariables, are formulated, and their resulting likelihood function is derived. Under carefully stated regularity conditions, large sample tests of hypotheses concerning the model parameters are obtained for the single point truncation (Type I censoring) and single point (Type II) censoring schemes, as well as for the case when all the observations are complete. Due to ethical considerations and possible loss of efficiency associated with single point truncation or censoring schemes, time-sequential procedures, which allow termination of the experiment at the earliest possible stage based on the accumulated statistical evidence, are often advocated for clinical trials. For parametric competing risks models a progressively truncated scheme (PTS) is used to develop a (large sample) time-sequential test procedure, and this procedure is modified to allow the presence of "nuisance" parameters and a form of stagger~d entry. Although it has limited applicability, a (restricted) sequential procedure is also obtained in the context of repeated significance testing, and a parallel procedure based on truncated observations is given. Under suitable regularity conditions, we show that the distributions of the proposed test statistics for both the time-sequential and sequential procedures converge weakly to those of a functional of certain Gaussian processes under the appropriate null hypotheses. iii Unfortunately, these null distributions are not available in the statistical literature, and theoretical derivation appears mathematically intractable, at least in general. However, the computer generation of general Gaussian processes is discussed and subsequently used to provide an algorithm which obtains the distributions of the appropriate functionals empirically. Finally, the regularity conditions for the procedures developed in this investigation are verified for a general independent risks exponential model and a dependent risks model based on Gumbel's bivariate exponential distribution, and numerical illustrations are provided for these procedures using simulated data from exponential populations. ACKNOWLEDGEMENTS Working with my adviser, Professor P. K. Sen, has been an honor and priviledge, and it is indeed a pleasure to express my sincere appreciation for his invaluable advice, constant guidance, and encouragement, as well as his patience, warm personality, and sense of humor. I also wish to thank the other committee members, Professors C. r wish to thank my wife, Mary Lou, for her continued support and encouragement during my entire graduate education. Appreciation is also expressed to my mother, Valeria, …