Chapter 7 Exercises
Asthma with treatment effect
Solutions

In Example 7.2.3 (asthma with treatment effect), repeat the analysis with alternative prior distributions. For the baseline (treatment 0) log odds, use standard logistic priors, which correspond to uniform priors on the scale of probabilities. Also suppose we are 95\% certain that the odds ratio is less than 5, and derive the corresponding normal prior distribution, centred around zero, for the log odds ratio. How are the posterior distributions of the odds ratio and the treatment 0 transition probabilities, affected?


For the log odds ratio, obtain the prior precision tau = 1/sigma to solve P(standard normal < log(5) / sigma) = 0.5, giving sigma = log(5) / invphi(0.95) = 0.9784688, and tau = 1.044494.

model {
   for(i in 1:2){
count[i,1:5] ~ dmulti(q[i,1:5], M[i])
for (r in 1:5) {
q[i,r] <- phi[i, r] / sum(phi[i, ])
      log(phi[i, r]) <- a[r] + b.treat[r] * treat[i]
}
   }
#   for (r in 2:5){ a[r] ~ dnorm(0, 0.00001) } # old prior
   for (r in 2:5){ a[r] ~ dlogis(0, 1) } # new prior
a[1] <- 0
   b.treat[1] <- 0
#   b.treat[2] ~ dnorm(0, 0.00001)   # old prior
   b.treat[2] ~ dnorm(0, 1.044494) # new prior
   or.treat <- exp(b.treat[2])   
   for (r in 3:5) { b.treat[r] <- 0 }
   treat[1] <- 0
   treat[2] <- 1
}

data
list(count=structure(.Data=c(210,60,0,1,1,
                                       66,32,0,0,2),.Dim=c(2,5)),
   M=c(272, 100),
)

inits
list(a = c(NA, 0, 0, 0, 0), b.treat = c(NA, 0, NA, NA, NA))

The posterior distribution of the log odds ratio is shrunk slightly towards zero to reflect the stronger prior information, but not substantially, since the prior is only weakly informative compared to the data. The "baseline" transition probabilities (under treatment 0) are hardly changed when the prior is changed.

Under new priors:
   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   or.treat   1.636   0.4215   0.004297   0.9657   1.582   2.61   4001   96000
   q[1,1]   0.7562   0.02533   2.581E-4   0.7049   0.7568   0.8047   4001   96000
   q[1,2]   0.2245   0.02498   2.56E-4   0.1769   0.2241   0.275   4001   96000
   q[1,3]   0.002746   0.002739   1.898E-5   6.988E-5   0.001896   0.01015   4001   96000
   q[1,4]   0.005488   0.003875   2.645E-5   6.419E-4   0.004619   0.01523   4001   96000
   q[1,5]   0.01103   0.005455   3.728E-5   0.003106   0.01014   0.02395   4001   96000
   q[2,1]   0.668   0.0446   3.28E-4   0.5777   0.6689   0.7517   4001   96000
   q[2,2]   0.315   0.04523   3.33E-4   0.2302   0.314   0.4068   4001   96000
   q[2,3]   0.002426   0.002426   1.675E-5   6.134E-5   0.001672   0.009007   4001   96000
   q[2,4]   0.004846   0.003436   2.357E-5   5.699E-4   0.004066   0.01351   4001   96000
   q[2,5]   0.00974   0.004851   3.319E-5   0.002726   0.008932   0.02132   4001   96000
   
   
Under original priors:
   node   mean   sd   MC error   2.5%   median   97.5%   start   sample
   or.treat   1.71   0.4517   0.004383   0.9911   1.655   2.75   4001   96000
   q[1,1]   0.768   0.02522   2.599E-4   0.7175   0.7686   0.8159   4001   96000
   q[1,2]   0.2208   0.02503   2.565E-4   0.1737   0.2201   0.2711   4001   96000
   q[1,3]   5.483E-6   1.414E-4   8.109E-7   0.0   0.0   2.166E-7   4001   96000
   q[1,4]   0.002771   0.002742   2.013E-5   6.661E-5   0.001954   0.01014   4001   96000
   q[1,5]   0.008381   0.004769   3.21E-5   0.001788   0.007486   0.02   4001   96000
   q[2,1]   0.6707   0.04562   3.196E-4   0.579   0.6719   0.7569   4001   96000
   q[2,2]   0.3195   0.04604   3.212E-4   0.2329   0.3183   0.4124   4001   96000
   q[2,3]   4.722E-6   1.226E-4   7.001E-7   0.0   0.0   1.861E-7   4001   96000
   q[2,4]   0.002421   0.002405   1.78E-5   5.784E-5   0.001704   0.008948   4001   96000
   q[2,5]   0.007317   0.004192   2.808E-5   0.001544   0.006514   0.01763   4001   96000