Dogs: loglinear model for
   binary data

Lindley (19??) analyses data from Kalbfleisch (1985) on the Solomon-Wynne experiment on dogs, whereby they learn to avoid an electric shock. A dog is put in a compartment, the lights are turned out and a barrier is raised, and 10 seconds later an electric shock is applied. The results are recorded as success (Y = 1 ) if the dog jumps the barrier before the shock occurs, or failure (Y = 0) otherwise.

Thirty dogs were each subjected to 25 such trials. A plausible model is to suppose that a dog learns from previous trials, with the probability of success depending on the number of previous shocks and the number of previous avoidances. Lindley thus uses the following model

    p _j    =   A ^{x j} B ^{j-x j}

for the probability of a shock (failure) at trial j, where x _j   = number of success (avoidances) before trial j and j - x _j   = number of previous failures (shocks).   This is equivalent to the following log linear model

   log p _j    =    a x _j + b ( j-x _j )

Hence we have a generalised linear model for binary data, but with a log-link function rather than the canonical logit link. This is trivial to implement in BUGS:

    model
   {
      for (i in 1 : Dogs) {
         xa[i, 1] <- 0; xs[i, 1] <- 0 p[i, 1] <- 0
         for (j in 2 : Trials) {
            xa[i, j] <- sum(Y[i, 1 : j - 1])
            xs[i, j] <- j - 1 - xa[i, j]
            log(p[i, j]) <- alpha * xa[i, j] + beta * xs[i, j]
            y[i, j] <- 1 - Y[i, j]
            y[i, j] ~ dbern(p[i, j])
         }
      }
       alpha ~ dflat()T(, -0.00001)
       beta ~ dflat()T(, -0.00001)
       A <- exp(alpha)
       B <- exp(beta)
   }


Data ( click to open )

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


Results