Statistical Methods in Medical Research 4ed

170 Bayesian methods
in this way. For example, in the present problem, the user might believe that the
mean lies in the neighbourhood of either of two values, u0 or u1. It might then be
appropriate to use a bimodal prior distribution with peaks at these two values. In
that case, the simplicity afforded by the conjugate family would be lost, and the
posterior distribution would no longer take the normal form (6.1).
Secondly, if either n is very large (when the evidence from the data overwhelms
the prior information) or if s2 0 is very large (when the prior evidence is
very weak and the prior distribution is non-informative), the posterior distribution
(6.1) tends towards the likelihood N(x, s2 =n).
In principle, once the formulations for the prior and likelihood have been
accepted as appropriate, the posterior distribution provides all we need for
inference about m. In practice, as in frequentist inference, it will be useful to
consider ways of answering specific questions about the possible value of m. In
particular, what are the Bayesian analogues of the two principal modes of
inference discussed in §4.1: significance tests and confidence intervals?
Bayesian significance tests
Suppose that, in the formulation leading up to (6.1), we wanted to ask whether
there was strong evidence that m < 0orm > 0. In frequentist inference we should
test the hypothesis that m ˆ 0, and see whether it was strongly contradicted by a
significant result in either direction. In the present Bayesian formulation there is
no point in considering the probability that m is exactly 0, since that probability
is zero (although m ˆ 0 has a non-zero density). However, we can state directly
the probability that, say m < 0 by calculating the tail area to the left of zero in the
normal distribution (6.1).
It is instructive to note what happens in the limiting case considered above,
when the sample size is large or the prior is non-informative and the posterior
distribution is N(x, s2 =n). The posterior probability that m < 0 is the probability
of a standardized normal deviate less than
0 x x n
p ˆ
s= n
p
s ,
and this is precisely the same as the one-sided P value obtained in a frequentist
test of the null hypothesis that m ˆ 0. The posterior tail area and the one-sided P
value are thus numerically the same, although of course their strict interpretations
are quite different.
Example 6.1
Example 4.1 described a frequentist significance test based on a sample of n ˆ 100 survival
times of patients with a form of cancer. The observed mean was x ˆ 46 9 months, and the