Stan code — longitudinal model

Model description

The longitudinal meta-analysis model estimates how the treatment effect evolves over time across studies. Studies contribute multiple time-point observations; a parametric trajectory describes the population-level effect at each time, with study-level random intercepts capturing between-study heterogeneity.

This vignette documents the linear trajectory implementation. See Longitudinal meta-analysis for the statistical rationale and alternative trajectory forms.

Mathematical specification

Likelihood:

$$y_{it} \mid \theta_{it} \sim \mathcal{N}(\theta_{it},\, s_{it}^2)$$

Trajectory:

$$\theta_{it} = (\mu_0 + u_i) + \mu_1 \cdot t_{it} + v_{it}$$

Priors:

$$\mu_0 \sim \mathcal{N}(0,\, 1), \qquad \mu_1 \sim \mathcal{N}(0,\, 0.5), \qquad \tau_u \sim \text{Half-Cauchy}(0,\, 0.5), \qquad \tau_v \sim \text{Half-Cauchy}(0,\, 0.5)$$

Stan code (linear trajectory)

data {
  int<lower=1> N;
  int<lower=1> K;
  vector[N] y;
  vector<lower=0>[N] se;
  vector[N] time;
  array[N] int<lower=1> study;
}

parameters {
  real mu0;
  real mu1;
  real<lower=0> tau_u;
  real<lower=0> tau_v;
  vector[K] z_u;
  vector[N] z_v;
}

transformed parameters {
  vector[K] u = tau_u * z_u;
  vector[N] v = tau_v * z_v;
}

model {
  target += normal_lpdf(mu0   | 0, 1);
  target += normal_lpdf(mu1   | 0, 0.5);
  target += cauchy_lpdf(tau_u | 0, 0.5);
  target += cauchy_lpdf(tau_v | 0, 0.5);
  target += std_normal_lpdf(z_u);
  target += std_normal_lpdf(z_v);

  target += normal_lpdf(y | (mu0 + u[study]) + mu1 * time + v, se);
}

generated quantities {
  real b_Intercept = mu0;
  real b_time      = mu1;
}

How bayesma calls this model

bayesma(
  data,
  model_type = "random_effect",
  time_col   = "weeks",
  trajectory = "linear"
)

The time vector in the Stan data is constructed from data[[time_col]], centred at the median observation time for interpretability.

Parameterisation notes

• $u_i$ is a study-level random intercept (deviation from $\mu_0$ at $t = 0$).
• $v_{it}$ is a study-by-time residual capturing deviations from the linear trajectory within studies.
• For the exponential decay trajectory, an additional parameter $\lambda > 0$ is required with a $\text{Half-Normal}(0, 1)$ prior.
• b_time reports the posterior for the rate of change $\mu_1$ per unit of the centred time variable.

Known sampling difficulties

The residual term $v_{it}$ can be weakly identified when each study contributes only one or two time-points. In this case, fix $\tau_v = 0$ (drop the $v_{it}$ term) to improve mixing. The model then reduces to a random-intercept longitudinal meta-analysis.