Statistical Methods in Medical Research 4ed

180 Bayesian methods
The examples discussed in §6.2 and §6.3 involved the means of the prior and
posterior distributions and the mean of the sampling distribution giving rise to
the likelihood. For unimodal distributions shrinkage will normally also be
observed for other measures of location, such as the median or mode. However,
if the prior had two or more well-separated modes, as might be the case for some
genetic traits, the tendency would be to shrink towards the nearest major mode,
and that might be in the opposite direction to the overall prior mean. An
example, for normally distributed observations with a prior distribution concentrated
at just two points, in given by Carlin and Louis (2000, §4.1.1), who refer to
the phenomenon as stretching.
We discuss here two aspects of shrinkage that relate to concepts of linear
regression, a topic dealt with in more detail in Chapter 7. We shall anticipate
some results described in Chapter 7, and the reader unfamiliar with the principles
of linear regression may wish to postpone a reading of this subsection.
First, we take another approach to the normal model described at the start of
§6.2. We could imagine taking random observations, simultaneously, of the two
variables m and x. Here, m is chosen randomly from the distribution N…m0, s2 0 †.
Then, given this value of m, x is chosen randomly from the conditional distribution
N…m, s2 =n†. If this process is repeated, a series of random pairs …m, x† is
generated. These paired observations form a bivariate normal distribution (§7.4,
Fig. 7.6). In this distribution, var…m† ˆs2 0 ,var…x† ˆs2 0 ‡ s2 =n (incorporating
both the variation of m and that of x given m), and the correlation (§7.3) between
m and x is
s0
r0 ˆ
…s2 0 ‡ s2 p :
=n†
The regression equation of x on m is
E…x j m† ˆm,
so the regression coefficient b x : m is 1. From the relationship between the regression
coefficients and the correlation coefficient, shown below (7.11), the other
regression coefficient is
bm : x ˆ r20 ˆ r
bx : m
2 0 ˆ
s 2 0
s 2 0 ‡ s2 =n :
This result is confirmed by the mean of the distribution (6.1), which can be
written as
E…m j x† ˆm 0 ‡
s 2 0
s 2 0 ‡ s2 =n …x m 0†:
The fact that bm : x…ˆ r2 0 † is less than 1 reflects the shrinkage in the posterior
mean. The proportionate shrinkage is