Beetles: a generalized logit model

Mortality of adult Beetle after five hours Exposure 
to Gaseous Carbon Disulphide
Example taken from Bliss (1935) and studied in
Prentice, R. L., (1976), A generalization of the Probit and 
Logit Methods for Dose Response Curves (Biometrics)

We list three model alternatives:
after the noninformative prior specification model, various 
attempts for lowering parameter autocorrelation and 
crosscorrelation follow
The best fit is performed fixing mu at its empirical value. 
Fixing sigma does not improve significantly the MCMC. Besides, 
inferential results show to be very sensitive to the value plugged in.   
It is noteworthy that mu is the node updated by Metropolis in WinBUGS, 
while m1 and tau are updated by Slice sampling. 
Finally, the overrelaxed version of Slice sampling, available in WinBUGS,
proves ro be helpful.

1) Non informative priors for mu and tau

model  {
                for (i in 1:8) {
                 
                       y[i] ~ dbin(pi[i],n[i])                    
                       x[i] <- (w[i]-mu)/sigma
                       logist[i] <- exp(x[i])/(1+exp(x[i]))
                       pi[i] <- pow(logist[i],m1)}

m1~ dgamma(1,1)
mu ~ dnorm(0,0.001)
tau ~ dgamma(0.001,0.001)

sigma <-1 /sqrt(tau)}


Data

list(y=c(6,13,18,28,52,53,61,60),
      n=c(59,60,62,56,63,59,62,60),
      w=c(1.6907,1.7242,1.7552,1.7842,1.8113,1.8369,1.861,1.8839))

Inits for 3 parallel chains

list(m1=0.5,mu=1.8,tau=1)
list(m1=1,mu=2.0,tau=1000)
list(m1=2,mu=1.0,tau=0.5)


2) Informative priors (from Carlin and Louis)

model  {
                for (i in 1:8) {
                 
                       y[i] ~ dbin(pi[i],n[i])                       
                       x[i] <- (w[i]-mu)/sigma
                       logist[i] <- exp(x[i])/(1+exp(x[i]))                        
                       pi[i] <- pow(logist[i],m1)}

m1~ dgamma(.25,.25)
mu ~ dnorm(2,10)
tau ~ dgamma(2.000004,0.001)
sigma<-1 /sqrt(tau)}


Data

list(y=c(6,13,18,28,52,53,61,60),
      n=c(59,60,62,56,63,59,62,60),
      w=c(1.6907,1.7242,1.7552,1.7842,1.8113,1.8369,1.861,1.8839))

Inits

list(m1=0.5,mu=1,tau=2)
list(m1=1,mu=2.0,tau=1000)


 node	 mean	 sd	 MC error	2.5%	median	97.5%	start	sample
	m1	0.3828	0.133	0.007278	0.2019	0.358	0.7226	4001	10000
	mu	1.811	0.01122	6.354E-4	1.785	1.811	1.83	4001	10000
	sigma	0.01888	0.003523	1.552E-4	0.01299	0.01855	0.02667	4001	10000

autocor for mu
0.95 (lag1), Exponential decay 

crosscor for (mu,sigma), (mu,m1), (m1,sigma)
              0.75-0.9  -0.1-0.25  -0.25-5


3) Fixing mu at the empirical value 

model  {
                for (i in 1:8) {
                 
                       y[i] ~ dbin(pi[i],n[i])                    
                       x[i] <- (w[i]-mu)/sigma                       
                       logist[i] <- exp(x[i])/(1+exp(x[i]))                        
                       pi[i] <- pow(logist[i],m1)}

m1 ~ dgamma(1,1)
tau ~ dgamma(0.001,0.001)
sigma <-1 /sqrt(tau)}


Data

list(mu=1.8,
      y=c(6,13,18,28,52,53,61,60),
      n=c(59,60,62,56,63,59,62,60),
      w=c(1.6907,1.7242,1.7552,1.7842,1.8113,1.8369,1.861,1.8839))

Inits

list(m1=0.5,tau=1)