Rats-drop: the rats example, illustrating the
effect of different dropout assumptions
This example is taken from section 6 of Gelfand
et al
(1990), and concerns 30 young rats whose weights were measured weekly for five weeks. Part of the data is shown below, where Y
ij
is the weight of the ith rat measured at age x
j
.
Weights Y
ij
of rat i on day x
j
x
j
= 8 15 22 29 36
__________________________________
Rat 1 151 199 246 283 320
Rat 2 145 199 249 293 354
.......
Rat 30 153 200 244 286 324
A plot of the 30 growth curves suggests some evidence of downward curvature.
The model is essentially a random effects linear growth curve
Y
ij
~ Normal(
a
i
+
b
i
(x
j
- x
bar
),
t
c
)
a
i
~ Normal(
a
c
,
t
a
)
b
i
~ Normal(
b
c
,
t
b
)
where x
bar
= 22, and
t
represents the
precision
(1/variance) of a normal distribution. We note the absence of a parameter representing correlation between
a
i
and
b
i
unlike in Gelfand
et al
1990. However, see the
Birats
example in Volume 2 which does explicitly model the covariance between
a
i
and
b
i
. For now, we standardise the x
j
's around their mean to reduce dependence between
a
i
and
b
i
in their likelihood: in fact for the full balanced data, complete independence is achieved. (Note that, in general, prior independence does not force the posterior distributions to be independent).
a
c
,
t
a
,
b
c
,
t
b
,
t
c
are given independent ``noninformative'' priors. Interest particularly focuses on the intercept at zero time (birth), denoted
a
0
=
a
c
-
b
c
x
bar
.