Bayesian Nonparametric Survival Analysis

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Abstract
This article considers a Bayesian nonparametric approach to a (right) censored data problem. Although the results are applicable to a wide variety of such problems, including reliability analysis, the discussion centers on medical survival studies. We extend the posterior distribution of percentiles given by Hill (1968) to obtain predictive posterior probabilities for the survival of one or more new patients, using data from other individuals having the same disease and given the same treatment. The analysis hinges on three assumptions: (a) The new patients and the previous sample patients are all deemed to be exchangeable with regard to survival time. (b) The posterior prediction rule, in the case of no censoring or ties among (say n) observed survival times, assigns equal probability of 1/(n + 1) to each of the n + 1 open intervals determined by these values. (c) The censoring mechanisms are “noninformative.” Detailed discussion of these assumptions is presented from a Bayesian point of view. In addition to obtaining a general representation for the predictive probabilities, intuition-building algorithms for the calculations are discussed. The derivation of these probabilities involves the use of an approximation with regard to the available censoring information. We present upper and lower bounds for the predictive probabilities under the exact censoring information. Substantial discussion is devoted to the practical implementation of the results. Relationships with other Bayesian approaches are discussed, as well as comparisons with the product limit rule of Kaplan and Meier (1958). To enhance the value of these comparisons, we analyze some examples of Kaplan and Meier (1958) and Gehan (1965).

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