Statistical Methods in Medical Research 4ed

dispersed reversed J-shape. With that assumption, the posterior distribution
becomes Gamma (x ‡ 1
2 , 1), and 2m has a x2 distribution on 2x ‡ 1 DF.
Example 6.3
Example 5.2 described a study in which x ˆ 33 workers died of lung cancer. The number
expected at national death rates was 20 0.
The Bayesian model described above, with a non-informative prior, gives the posterior
distribution Gamma (33 5, 1), and 2m has a x2 distribution on 67 DF. From computer
tabulations of this distribution we find that P (m < 20 0) is 0 0036, very close to the
frequentist one-sided mid-P significance level of 0 0037 quoted in Example 5.2.
If, on the other hand, it was believed that local death rates varied around the national
rate by relatively small increments, a prior might be chosen to have a mean count of 20
and a small standard deviation of, say, 2. Setting the mean of the prior to be ab ˆ 20 and
its variance to be ab2 ˆ 4, gives a ˆ 100 and b ˆ 0 2, and the posterior distribution is
Gamma (133, 0 1667). Thus, 12m has a x2 distribution on 266 DF, and computer tabulations
show that P(m < 20 0) is 0 128. There is now considerably less evidence that the local
death rate is excessive. The posterior estimate of the expected number of deaths is
(133) (0 1667) ˆ 22 17. Note, however, that the observed count is somewhat incompatible
with the prior assumptions. The difference between x and the prior mean is 33 20 ˆ 13.
Its variance might be estimated as 33 ‡ 4 ˆ 37 and its standard error as 3
p 7 ˆ 6 083. The
difference is thus over twice its standard error, and the investigator might be well advised
to reconsider prior assumptions.
Analyses involving the ratio of two counts can proceed from the approach
described in §5.2 and illustrated in Examples 5.2 and 5.3. If two counts, x1 and
x2, follow independent Poisson distributions with means m 1 and m 2, respectively,
then, given the total count x1 ‡ x2, the observed count x1 is binomially distributed
with mean …x1 ‡ x2†m 1=…m 2 ‡ m 2†. The methods described earlier in this
section for the Bayesian analysis of proportions may thus be applied also to
this problem.
6.4 Further comments on Bayesian methods
Shrinkage
6.4 Further comments on Bayesian methods 179
The phenomenon of shrinkage was introduced in §6.2 and illustrated in several of
the situations described in that section and in §6.3. It is a common feature of
parameter estimation in Bayesian analyses. The posterior distribution is determined
by the prior distribution and the likelihood based on the data, and its
measures of location will tend to lie between those of the prior distribution and
the central features of the likelihood function. The relative weights of these two
determinants will depend on the variability of the prior and the tightness of the
likelihood function, the latter being a function of the amount of data.