Meta-analysis with inference on measurement error variability

Modeling
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8

The model is below. It is a hierarchical model with a few hierarchies and meta-regression, so you can safely ignore almost all of it (if interested in toxicology application just scihub the paper), the part that you’re possibly interested in will be at the end, with lv[i] ~ . Once we have the sampling distribution , the true \sigma can then be modelled.

I am working on putting this “generic” approach (lv ~) in a meta-analysis with Stan package we’re developing (here), but was also thinking about these models for a couple of application papers (more tox/risk assessment, but also economics). Would love to talk applications/ideas around treating variance as unknown if you’re also thinking about this!

data {
  int  n;
  int  n_drugs;
  int  n_studies;
  int  n_groups;  
  int  drug[n];
  int  study[n];
  int  group[n];
  
  //observed data on means & variation:
  real ss[n];
  real lgm[n];
  real<lower=0> lv[n];
}

parameters {
  real mu_drug[n_drugs];     
  real mu_study[n_studies];  
  real<lower=0> sigma_study;
  real gamma_group[n_groups];
  real gamma_drug[n_drugs];
  
  // random effect of group:
  real logratio[n];
  real mu_group[n_groups - 1];
  real<lower=0> sigma_group;
  
  // fixed effect of group:
  // real logratio[n_groups - 1];
}

model {
  // Both true mu and sigma are simply introduced to clean 
  // up notation, rather than as extra parameters to output.
  // Hence they are defined here, in the model block.
  real sigma[n]; 
  real mu[n];
  for(i in 1:n){
    if(group[i] != 1)
      // random effect
      mu[i] = mu_drug[drug[i]] + mu_study[study[i]] + logratio[n];
      // fixed effect
      // mu[i] = mu_drug[drug[i]] + mu_study[study[i]] + logratio[group[i] - 1];
    else
      mu[i] = mu_drug[drug[i]] + mu_study[study[i]];
    sigma[i] = exp(gamma_drug[drug[i]] + gamma_group[group[i]]);
  }
  
  //priors dealing with means (mu):
  mu_drug ~ normal(0, 100);
  for(i in 1:n_studies)
    mu_study[i] ~ normal(0, sigma_study);
  sigma_study ~ normal(0, 5);
  gamma_drug ~ normal(0, 10);
  gamma_group ~ normal(0, 10);
  // random effect of group:
  mu_group ~ normal(0, 100);
  sigma_group ~ normal(0, 5);
  // fixed effect of group:
  // logratio ~ normal(0, 5);
  
  for(i in 1:n){
    // random effect of group:
    if(group[i] == 1)
      logratio[i] ~ normal(0, .0001);
    else
      logratio[i] ~ normal(mu_group[group[i] - 1], sigma_group);
    //observed quantities:
    lgm[i] ~ normal(mu[i], sigma[i]/sqrt(ss[i]));
    //chi-sq is Gamma(njk - 1 / 2, tau_j*(n_jk-1)/2)
    lv[i]  ~ gamma((ss[i] - 1) / 2, (ss[i]-1)/(2*(sigma[i]^2))); 
  }
}
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