[salm0]     Salm: extra - Poisson variation
         in dose - response study

Breslow (1984) analyses some mutagenicity assay data (shown below) on salmonella in which three plates have been processed at each dose i of quinoline and the number of revertant colonies of TA98 Salmonella measured. A certain dose-response curve is suggested by theory.

[salm1]
This is assumed to be a random effects Poisson model allowing for over-dispersion. Let x i be the dose on the plates i 1, i 2 and i 3. Then we assume

   y ij ~ Poisson( m ij )

   log( m ij ) = a + b log(x i + 10) + g x i + l ij

    l ij ~ Normal(0, t )

a , b , g , t are given independent ``noninformative'' priors. The appropriate graph is shown


Graphical model for salm example
[salm2]



BUGS language for salm example

   model
   {
      for( i in 1 : doses ) {
         for( j in 1 : plates ) {
            y[i , j] ~ dpois(mu[i , j])
            log(mu[i , j]) <- alpha + beta * log(x[i] + 10) +
               gamma * x[i] + lambda[i , j]
            lambda[i , j] ~ dnorm(0.0, tau)   
            cumulative.y[i , j] <- cumulative(y[i , j], y[i , j])
         }
      }
      alpha ~ dnorm(0.0,1.0E-6)
      beta ~ dnorm(0.0,1.0E-6)
      gamma ~ dnorm(0.0,1.0E-6)
      tau ~ dgamma(0.001, 0.001)
      sigma <- 1 / sqrt(tau)
   }   
Data ( click to open )

Inits for chain 1    Inits for chain 2    ( click to open )



Results

A 1000 update burn in followed by a further 10000 updates gave the parameter estimates

[salm3]

These estimates can be compared with the quasi-likelihood estimates of Breslow (1984) who reported a = 2.203 +/- 0.363, b = 0.311 +/- 0.099, g = -9.74E-4 +/- 4.37E-4, s = 0.268