Birats: a bivariate normal
hierarchical model
We return to the Rats example, and illustrate the use of a multivariate Normal (MVN) population distribution for the regression coefficients of the growth curve for each rat. This is the model adopted by Gelfand etal (1990) for these data, and assumes
a priori
that the intercept and slope parameters for each rat are correlated. For example, positive correlation would imply that initially heavy rats (high intercept) tend to gain weight more rapidly (steeper slope) than lighter rats. The model is as follows
Y
ij
~ Normal(
m
ij
,
t
c
)
m
ij
=
b
1i
+
b
2i
x
j
b
i
~ MVN(
m
b
,
W
)
where Y
ij
is the weight of the ith rat measured at age x
j
, and
b
i
denotes the vector
(b
1i
,
b
2i
). We assume 'non-informative' independent univariate Normal priors for the separate components
m
b
1
and
m
b
2
. A Wishart(R,
r
) prior was specified for
W
, the population precision matrix of the regression coefficients. To represent vague prior knowledge, we chose the the degrees of freedom
r
for this distribution to be as small as possible (i.e. 2, the rank of
W
). The scale matrix was specified as