Hearts: a mixture model for count data

The table below presents data given by Berry (1987) on the effect of a drug used to treat patients with frequent premature ventricular contractions (PVCs) of the heart.
Farewell and Sprott (1988) model these data as a mixture distribution of Poisson counts in which some patients are "cured" by the drug, whilst others experience varying levels of response but remain abnormal. A zero count for the post-drug PVC may indicate a "cure", or may represent a sampling zero from a patient with a mildly abnormal PVC count. The following model thus is assumed:

   x_i   ~   Poisson(λ_i)   for all patients
   y_i   ~   Poisson(βλ_i)   for all uncured patients
   P(cure)   =   θ
   
To eliminate nuisance parameters li, Farewell and Sprott use the conditional distribution of yi given ti = xi + yi. This is equivalent to a binomial likelihood for yi with denominator ti and probability p = b /(1+b) (see Cox and Hinkley, 1974 pp. 136-137 for further details of the conditional distribution for Poisson variables). Hence the final mixture model may be expressed as follows:

   P(y_i = 0 | t_i )   =   θ + (1 - θ) (1 - p) t_i
   P(y_i | t_i )   =   (1 - θ) (t_i! / (y_i! (t_i-y_i)!)) (p^yi (1 - p)^{(ti - yi)}   y_i = 1,2,...,t_i   

The BUGS code for this model is given below:

   model
   {
      for (i in 1 : N) {
         y[i] ~ dbin(P[state1[i]], t[i])
         state[i] ~ dbern(theta)
         state1[i] <- state[i] + 1
         t[i] <- x[i] + y[i]
         prop[i] <- P[state1[i]]
      }
      P[1] <- p
      P[2] <- 0
      logit(p) <- alpha
      alpha ~ dnorm(0,1.0E-4)
      beta <- exp(alpha)
      logit(theta) <- delta
      delta ~ dnorm(0, 1.0E-4)
   }

   Results