WinBUGS Weibull Network Meta-Analysis

Question

I am currently working on a meta-analysis of survival data across several clinical trials.

To do this, I have code from a published analysis using the same methodology. However, when running this code using the data from the published analysis, I am unable to replicate their results. In fact, the results fail to converge to any kind of reasonable estimate.

The code itself (not including the data) should be correct as it comes directly from the authors. I assume the problem has to do w/ initial values or parameters of how the sampling is run, but after playing w/ many initial values, the length of burn in, thinning, etc... I haven't gotten meaningful results.

I would appreciate anyone suggestions about how to run this (initial values, etc...) to get it to run properly. Alternately, if there are problems in the code or if the data is set up in a way that doesn't match the code, that would be useful to know.

As a side note, I am doing the analyses using R2WinBUGs, though I have gotten the same kind of problems using WinBUGs alone.

A bit of extra background on the method:

The way this works is by estimating the difference in shape and scale parameters of a reparameterized Weibull distribution between treatments across multiple studies using random effects.

The Weibull distribution is reparameterized such that log of the hazard rate is a+b*log(t) where a is a scale parameter and b is a shape parameter. From this, you can calculate the the likelihood function of a given number of failures out of a given number of patients over an interval.

Unfortunately, the article is public, but if you can get access here is the link: http://onlinelibrary.wiley.com/doi/10.1002/jrsm.25/abstract;jsessionid=2BA8F0D9BEF9A33F84975618D33F8DD9.f03t03?userIsAuthenticated=false&deniedAccessCustomisedMessage=

A quick summary of the variables entered into the model:

NT: Number of separate treatments included.

N: Number of rows in the main dataset. NS: Number of studies

s: Study that the line of data corresponds to (this is numbered 1:6)

r: number of patients failing in interval for this treatment/study

n: number of patients at risk at start of interval for this treatment/study

t: Treatment that this line of data corresponds to (numbered 1:3)

b: Indicates which treatment is the baseline one to which others are compared (set to 1 for each line).

bs: Treatment that is the control arm of this study

bt: Treatment that is the research arm of this study

WinBUGS code (including data):

#Winbugs code for random effects networks meta-analysis model
Model
{
  for (i in 1:N)
  { # N=number of data points in dataset
    #likelihood
    r[i]~ dbin(p[i],n[i])
    p[i]<-1-exp(-h[i]*dt[i]) # hazard h over interval [t,t+dt] # expressed as deaths per unit person-time (e.g. months)
    #random effects model
    log(h[i])<-nu[i]+log(time[i])*theta[i]
    nu[i]<-mu[s[i],1]+delta[s[i],1]*(1-equals(t[i],b[i]))
    theta[i]<-mu[s[i],2]+ delta[s[i],2]*(1-equals(t[i],b[i]))
  }
  for(k in 1 :NS)
  { # NS=number of studies in dataset
    delta[k,1:2]~dmnorm(md[k,1:2],omega[1:2,1:2])
    md[k,1]<-d[ts[k],1]-d[bs[k],1]
    md[k,2]<-d[ts[k],2]-d[bs[k],2]
  }
  # priors
  d[1,1]<-0
  d[1,2]<-0
  for(j in 2 :NT)
  { # NT=number of treatments
    d[j,1:2] ~ dmnorm(mean[1:2],prec2[,])
  }
  for(k in 1 :NS)
  {
    mu[k,1:2] ~ dmnorm(mean[1:2],prec2[,])
  }
  omega[1:2, 1:2] ~ dwish(R[1:2,1:2],2)
}
# Winbugs data set
list(N=242, NS=6, NT=3,
mean=c(0,0),
prec2 = structure(.Data = c(
0.0001,0,
0,0.0001), .Dim = c(2,2)),
R = structure(.Data = c(
0.01,0,
0,0.01), .Dim = c(2,2))
)

s[] r[] n[] t[] b[] time[] dt[]
1 15 152 3 1 3 3
1 11 140 3 1 6 3
1 8 129 3 1 9 3
1 9 121 3 1 12 3
1 9 112 3 1 15 3
1 3 83 3 1 18 3
1 4 80 3 1 21 3
1 5 76 3 1 24 3
1 2 71 3 1 27 3
1 2 41 3 1 30 3
1 1 39 3 1 33 3
1 3 38 3 1 36 3
1 2 35 3 1 39 3
1 1 33 3 1 42 3
1 3 32 3 1 45 3
1 3 29 3 1 48 3
1 2 26 3 1 51 3
1 1 24 3 1 54 3
1 1 23 3 1 57 3
1 1 22 3 1 60 3
1 10 149 1 1 3 3
1 11 140 1 1 6 3
1 9 128 1 1 9 3
1 5 119 1 1 12 3
1 6 114 1 1 15 3
1 3 72 1 1 18 3
1 5 70 1 1 21 3
1 4 65 1 1 24 3
1 7 61 1 1 27 3
1 2 34 1 1 30 3
1 2 32 1 1 33 3
1 3 30 1 1 36 3
1 2 27 1 1 39 3
1 2 25 1 1 42 3
1 1 23 1 1 45 3
1 2 22 1 1 48 3
1 1 19 1 1 51 3
1 2 19 1 1 54 3
1 1 17 1 1 57 3
1 0 16 1 1 60 3
2 4 125 2 1 3 3
2 4 121 2 1 6 3
2 2 117 2 1 9 3
2 5 114 2 1 12 3
2 2 109 2 1 15 3
2 3 107 2 1 18 3
2 2 104 2 1 21 3
2 4 94 2 1 24 3
2 4 90 2 1 27 3
2 3 81 2 1 30 3
2 4 78 2 1 33 3
2 3 61 2 1 36 3
2 5 58 2 1 39 3
2 1 48 2 1 42 3
2 2 47 2 1 45 3
2 3 41 2 1 48 3
2 0 38 2 1 51 3
2 3 29 2 1 54 3
2 3 26 2 1 57 3
2 2 18 2 1 60 3
2 0 16 2 1 63 3
2 1 10 2 1 66 3
2 0 9 2 1 69 3
2 0 3 2 1 72 3
2 0 3 2 1 75 3
2 0 3 2 1 78 3
2 15 196 1 1 3 3
2 9 179 1 1 6 3
2 10 170 1 1 9 3
2 9 162 1 1 12 3
2 9 153 1 1 15 3
2 5 141 1 1 18 3
2 5 136 1 1 21 3
2 10 121 1 1 24 3
2 5 111 1 1 27 3
2 7 92 1 1 30 3
2 7 85 1 1 33 3
2 4 71 1 1 36 3
2 6 67 1 1 39 3
2 4 53 1 1 42 3
2 5 49 1 1 45 3
2 6 36 1 1 48 3
2 3 30 1 1 51 3
2 2 26 1 1 54 3
2 2 24 1 1 57 3
2 0 13 1 1 60 3
2 1 13 1 1 63 3
2 1 11 1 1 66 3
2 1 10 1 1 69 3
2 0 6 1 1 72 3
2 0 6 1 1 75 3
2 0 6 1 1 78 3
3 6 113 2 1 3 3
3 4 105 2 1 6 3
3 3 101 2 1 9 3
3 1 97 2 1 12 3
3 9 96 2 1 15 3
3 4 84 2 1 18 3
3 2 80 2 1 21 3
3 4 74 2 1 24 3
3 3 70 2 1 27 3
3 2 59 2 1 30 3
3 0 57 2 1 33 3
3 6 51 2 1 36 3
3 2 45 2 1 39 3
3 1 37 2 1 42 3
3 3 36 2 1 45 3
3 1 27 2 1 48 3
3 1 26 2 1 51 3
3 2 25 2 1 54 3
3 7 116 1 1 3 3
3 6 111 1 1 6 3
3 4 105 1 1 9 3
3 3 99 1 1 12 3
3 9 96 1 1 15 3
3 5 85 1 1 18 3
3 5 80 1 1 21 3
3 3 68 1 1 24 3
3 7 65 1 1 27 3
3 8 48 1 1 30 3
3 4 40 1 1 33 3
3 2 33 1 1 36 3
3 0 31 1 1 39 3
3 1 28 1 1 42 3
3 2 27 1 1 45 3
3 3 20 1 1 48 3
3 1 17 1 1 51 3
3 0 16 1 1 54 3
4 10 167 2 1 3 3
4 5 149 2 1 6 3
4 6 145 2 1 9 3
4 3 138 2 1 12 3
4 4 135 2 1 15 3
4 5 128 2 1 18 3
4 2 122 2 1 21 3
4 2 120 2 1 24 3
4 7 104 2 1 27 3
4 9 98 2 1 30 3
4 5 89 2 1 33 3
4 2 57 2 1 36 3
4 2 55 2 1 39 3
4 4 53 2 1 42 3
4 2 49 2 1 45 3
4 2 26 2 1 48 3
4 1 24 2 1 51 3
4 1 23 2 1 54 3
4 1 11 2 1 57 3
4 0 10 2 1 60 3
4 0 10 2 1 63 3
4 2 164 1 1 3 3
4 5 153 1 1 6 3
4 7 148 1 1 9 3
4 6 141 1 1 12 3
4 12 135 1 1 15 3
4 6 119 1 1 18 3
4 4 113 1 1 21 3
4 3 109 1 1 24 3
4 5 98 1 1 27 3
4 2 94 1 1 30 3
4 2 92 1 1 33 3
4 4 55 1 1 36 3
4 3 50 1 1 39 3
4 1 48 1 1 42 3
4 2 47 1 1 45 3
4 1 22 1 1 48 3
4 1 21 1 1 51 3
4 0 20 1 1 54 3
4 1 7 1 1 57 3
4 0 6 1 1 60 3
4 0 6 1 1 63 3
5 12 152 2 1 3 3
5 7 135 2 1 6 3
5 9 128 2 1 9 3
5 8 120 2 1 12 3
5 7 112 2 1 15 3
5 1 77 2 1 18 3
5 3 76 2 1 21 3
5 2 73 2 1 24 3
5 4 71 2 1 27 3
5 5 45 2 1 30 3
5 3 40 2 1 33 3
5 2 37 2 1 36 3
5 3 35 2 1 39 3
5 3 32 2 1 42 3
5 3 32 2 1 45 3
5 1 32 2 1 48 3
5 9 149 1 1 3 3
5 4 131 1 1 6 3
5 5 127 1 1 9 3
5 8 122 1 1 12 3
5 11 114 1 1 15 3
5 5 76 1 1 18 3
5 5 71 1 1 21 3
5 5 66 1 1 24 3
5 6 61 1 1 27 3
5 3 35 1 1 30 3
5 4 32 1 1 33 3
5 1 28 1 1 36 3
5 1 27 1 1 39 3
5 6 26 1 1 42 3
5 5 26 1 1 45 3
5 0 26 1 1 48 3
6 22 179 2 1 3 3
6 13 151 2 1 6 3
6 3 138 2 1 9 3
6 5 135 2 1 12 3
6 1 130 2 1 15 3
6 5 104 2 1 18 3
6 7 99 2 1 21 3
6 6 92 2 1 24 3
6 6 66 2 1 27 3
6 7 60 2 1 30 3
6 4 53 2 1 33 3
6 0 30 2 1 36 3
6 2 29 2 1 39 3
6 3 27 2 1 42 3
6 1 24 2 1 45 3
6 0 16 2 1 48 3
6 1 15 2 1 51 3
6 0 14 2 1 54 3
6 1 14 2 1 57 3
6 0 14 2 1 60 3
6 13 178 1 1 3 3
6 7 160 1 1 6 3
6 7 153 1 1 9 3
6 10 146 1 1 12 3
6 10 136 1 1 15 3
6 2 97 1 1 18 3
6 5 95 1 1 21 3
6 3 90 1 1 24 3
6 5 57 1 1 27 3
6 2 52 1 1 30 3
6 6 50 1 1 33 3
6 3 37 1 1 36 3
6 1 34 1 1 39 3
6 2 33 1 1 42 3
6 4 31 1 1 45 3
6 0 17 1 1 48 3
6 0 17 1 1 51 3
6 1 17 1 1 54 3
6 0 16 1 1 57 3
6 0 16 1 1 60 3
END


ts[] bs[]
3 1
2 1
2 1
2 1
2 1
2 1
END

Answer 1

Ultimately, I was unable to get the model to run properly in WinBUGS. However, I was able to port the model over to STAN and get a very close match. See below for the STAN code:

data { 


 int<lower=1> N;
  int<lower=1> NS;
  int<lower=1> NT;

  cov_matrix[2] prec2;
  matrix[2,2] R;
  vector[2] means;

  int<lower=0> bs[NS];
  int<lower=0> ts[NS];

  int<lower=0> s[N];
  int<lower=0> t[N];
  int<lower=0> n[N];
  int<lower=0> r[N];
  real<lower=0> dt[N];
  real<lower=0> time[N];
}
parameters {
  vector[2] mu[NS];
  vector[2] delta[NS];
  vector[2] dj[NT-1];
  cov_matrix[2] omega;
} 
transformed parameters{
  real<lower=0,upper=1> p[N];
  real<lower=0> h[N];
  real nu[N];
  real theta[N];
  vector[2] md[NS];
  vector[2] d[NT];

  d[1][1] <- 0;
  d[1][2] <- 0;
  for(j in 2:NT){
    d[j] <- dj[j-1];
  }
  for(k in 1:NS){
    md[k] <- d[ts[k]] - d[bs[k]];
  }
  for(i in 1:N){
    if(t[i] == 1){
      nu[i] <- mu[s[i]][1];
      theta[i] <- mu[s[i]][2];
    }else{
      nu[i] <- mu[s[i]][1] + delta[s[i]][1];
      theta[i] <- mu[s[i]][2] + delta[s[i]][2];
    }
    h[i] <- exp(nu[i] + log(time[i]) * theta[i]);
    p[i] <- 1 - exp(- h[i] * dt[i]);
  }
}
model {
  omega ~ inv_wishart(2,R);
  for(j in 1:(NT-1)){
    dj[j] ~ multi_normal(means,prec2);
  }
  for(k in 1:NS){
    delta[k] ~ multi_normal(md[k],omega);
    mu[k] ~ multi_normal(means,prec2);
  }
  for(i in 1:N){
    r[i] ~ binomial(n[i],p[i]);
  }
}
generated quantities{
  vector[N] log_lik;
  for (l in 1:N) {
    log_lik[l] <- binomial_log(r[l], n[l], p[l]);
  }
}

Note that due to differences between STAN/BUGS, confusion over precision/variance can make it confusing what to enter for data. For R and prec2, I loaded:

dataList$R <- matrix(c(0.01,0,0,0.01),nrow=2,ncol=2,byrow=TRUE)
dataList$prec2 <- matrix(c(10^4,0,0,10^4),nrow=2,ncol=2,byrow=TRUE)

Answer 2

Some suggestions for you, that I hope will be helpful:

1. I think your question could be better answered in https://stats.stackexchange.com/; Although if you want to move the question there, you should rewrite it rather than posting a code dump.

2. WinBUGS is several years old, and it is 8 years since its last update! I think you should forget it, because there are several better alternatives.

3. You may try virtually the same code in Jags or Stan, in which both are usable in R via rJags and RStan. Stan is specially important, because it uses HCMC which converges in many situations that WinBUGS does not.

4. I think you should read the easy book: Doing Bayesian Data Analysis 2e from John K. Kruschke to be able to understand and do Bayesian data analysis by yourself.