A Bayesian Test of Some Classical Hypotheses—With Applications to Sequential Clinical Trials

原文摘要

Consider someone who sets out to collect data sequentially in such a way as to disprove a hypothesis, H 0, about the value of θ, the mean of a normal distribution. Observation is continued as long as the posterior probability of H 0 exceeds α1 and stopped when it falls below it. It is shown that if a non-zero prior probability, p, is assigned to H 0(0≤α1 < p) and the remaining prior probability is spread over alternate values of θ that the probability that H 0 will eventually be rejected when true is With non-zero assignment of p it is therefore not possible to sample to a foregone conclusion. The Wald 3-decision scheme is shown to be equivalent to setting and assigning prior probability of to the alternatives θ = θ0 + Δ and θ = θ0 − Δ. Anomalous features of this scheme, particularly for the case of unknown σ, are shown to be a consequence of the unrealistic nature of the alternatives assumed. It is suggested that meaningful assignments of p are possible in practice by considering the minimum number of cases that would lead one to accept H 0 under the most favorable possible evidence for it. Rejection of H 0 is shown to be insensitive to choice of p. Closed schemes are obtained by sampling as long as P(θ ≤ Δ) > α1 and P(θ > 0) < α2 and stopping when either inequality is infringed, where Δ is the smallest positive departure from H 0 of substantive interest and α1 < α2. Approximating the likelihood function for the logit of a binomial parameter with a normal likelihood function it is possible to extend these methods to the comparison of two binomial parameters, without the usual sequential assumption of pairing and equal number of cases.

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