Longitudinal meta-analysis • bayesma

Introduction

Many clinical and psychological interventions produce effects that evolve over time. When studies report outcomes at multiple follow-up points, a longitudinal meta-analysis (LMA) models the trajectory of the treatment effect across time, rather than selecting a single time-point for pooling.

LMA answers questions that univariate meta-analysis cannot:

Does the treatment effect grow, decay, or plateau over time?
At which time-point is the effect largest?
Is there evidence that long-term and short-term effects differ?

Model specification

Let $y_{it}$ be the effect estimate from study $i$ at time $t \in \{t_1, \ldots, t_{T_i}\}$ , with known standard error $s_{it}$ . The longitudinal model has two levels.

Level 1 (within-study):

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

Level 2 (between-study trajectory):

$\theta_{it} = f(t;\, \boldsymbol{\mu}) + u_i + v_{it}$

where:

$f(t;\, \boldsymbol{\mu})$ is the population-level trajectory (parametric or non-parametric)
$u_i \sim \mathcal{N}(0, \tau_u^2)$ is a study-level random intercept
$v_{it} \sim \mathcal{N}(0, \tau_v^2)$ is a study-by-time random deviation

Trajectory specifications

Linear trajectory

$f(t;\, \boldsymbol{\mu}) = \mu_0 + \mu_1 t$

Appropriate when the effect changes at a roughly constant rate. $\mu_0$ is the effect at $t = 0$ (baseline) and $\mu_1$ is the rate of change per unit time.

Exponential decay

$f(t;\, \boldsymbol{\mu}) = \mu_\infty + (\mu_0 - \mu_\infty) e^{-\lambda t}$

Appropriate for effects that peak near treatment initiation and decay toward a long-run asymptote $\mu_\infty$ . $\lambda > 0$ controls the decay rate.

Piecewise linear (spline)

$f(t;\, \boldsymbol{\mu}) = \mu_0 + \mu_1 t + \sum_j \mu_{2j} (t - \kappa_j)_+$

where $\kappa_j$ are knot positions and $(x)_+ = \max(0, x)$ . Flexible enough to capture non-monotone trajectories.

Priors

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

Trajectory-specific parameters (e.g., $\lambda$ , $\mu_\infty$ ) require domain-informed priors. Exponential decay requires $\lambda > 0$ ; a $\text{Half-Normal}(0, 1)$ prior is a reasonable weakly informative choice.

Fitting longitudinal models

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

The time_col argument names the column containing the time variable. trajectory accepts "linear", "exponential", or "spline".

Estimands

bayesma() returns the posterior trajectory $f(t)$ evaluated on a user-specified grid via bayesma_marginal(). This can be used to:

plot the trajectory with pointwise credible bands
identify the time of maximum effect
compare trajectories across subgroups (via meta_reg())

Alignment across studies

Studies rarely measure outcomes at identical time-points. The model handles this naturally — each study contributes $(y_{it}, s_{it}, t)$ triplets, and the trajectory is estimated across all studies simultaneously. No imputation for missing time-points is needed.

Limitations

Longitudinal models require that time is measured comparably across studies (same origin, same units).
The trajectory form must be specified. Model comparison between trajectory specifications is supported via compare_models().
With $k < 10$ and heterogeneous time-points, both $\tau_u$ and $\tau_v$ are poorly identified.