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 .

Graphical model for rats example:
BUGS language for rats example:
Note the use of a very flat but conjugate prior for the population effects: a locally uniform prior could also have been used.
Data click on one of the arrows to open the data
(Note: the response data (Y) for the rats example can also be found in the file ratsy.odc in rectangular format. The covariate data (X) can be found in S-Plus format in file ratsx.odc. To load data from each of these files, focus the window containing the open data file before clicking on "Data" from the "Model" menu.)
Inits list( alpha.c = 150, beta.c = 10,
tau.c = 1, alpha.tau = 1, beta.tau = 1)
Gelfand et al 1990 also consider the problem of missing data, and delete the last observation of cases 6-10, the last two from 11-20, the last 3 from 21-25 and the last 4 from 26-30. The appropriate data file is obtained by simply replacing data values by NA (see below). The model specification is unchanged, since the distinction between observed and unobserved quantities is made in the data file and not the model specification.
click on one of the arrows to open the data for the missing value analysis
Missing at random gives beta.c estimate of 6.54 (6.26 to 6.84)
Extra data list for sensitivity analysis in informative dropout model
list(k=0.02)
Initial values for informative dropout model, probability of missing depending on Y[i,j] with a as intercept
Inits list( alpha.c = 250, beta.c =6, tau.c = 1, alpha.tau = 1, beta.tau = 1, a = 0)
Second set of initial values
list( alpha.c = 100, beta.c =4, tau.c = .1, alpha.tau = .1, beta.tau = .1, a = 1)
With k=.02, get estimate of beta.c
node mean sd MC error 2.5% median 97.5% start sample
beta.c 6.273 0.15 0.009028 5.976 6.276 6.561 1001 3000