**
****
**Seeds: Random effect logistic

regression

This example is taken from Table 3 of Crowder (1978), and concerns the proportion of seeds that germinated on each of 21 plates arranged according to a 2 by 2 factorial layout by seed and type of root extract. The data are shown below, where r
_{i
} and n
_{i
} are the number of germinated and the total number of seeds on the
* i
*th plate,
*i
* =1,...,N. These data are also analysed by, for example, Breslow: and Clayton (1993).

r

logit(p

b

where x

BUGS

model

{

for( i in 1 : N ) {

r[i] ~ dbin(p[i],n[i])

b[i] ~ dnorm(0.0,tau)

logit(p[i]) <- alpha0 + alpha1 * x1[i] + alpha2 * x2[i] +

alpha12 * x1[i] * x2[i] + b[i]

}

alpha0 ~ dnorm(0.0,1.0E-6)

alpha1 ~ dnorm(0.0,1.0E-6)

alpha2 ~ dnorm(0.0,1.0E-6)

alpha12 ~ dnorm(0.0,1.0E-6)

tau ~ dgamma(0.001,0.001)

sigma <- 1 / sqrt(tau)

}

Results

A burn in of 1000 updates followed by a further 10000 updates gave the following parameter estimates:

We may compare simple logistic, maximum likelihood (from EGRET), penalized quasi-likelihood (PQL) Breslow and Clayton (1993) with the

Heirarchical centering is an interesting reformulation of random effects models. Introduce the variables

m

r

logit(p

b

The graphical model is shown below