Fixed and Random-effects models
Traditional meta-analyses, regardless of whether it is frequentist or Bayesian, aim to detect the significant difference in the effect sizes, $\delta$. Say there are $N$ studies. Then, fixed-effects meta-analyses assume the estimated effect sizes $Y_i$ for $i=1,\ldots,N$ reflect the underlying true effect $\delta$ across all $N$ studies. Random-effects meta-analyses rather assume that the underlying true effect $\delta_i$ for the $i$th study is a realization of the overall effect $\delta$. Under the random-effects models, $Y_i$ are estimates of $\delta_i$, and $s_i^2$ are the estimates for $\mathrm{Var}(Y_i)$ for $i=1,\ldots,N$. Such an approach permits the modeling of within-study and between-study variabilities.
Fixed-effects models
Fixed-effects meta-analysis models make two assumptions:
- the studies included are identical and thus all represent a homogeneous population
- the meta-analysis does not generalize to other populations
Assumption 1 inherently restricts the source of variation to be within studies, identified and included in the analysis, leaving no room for between-study variation. Assumption 2 implies the inference is nothing more than conditional.
The model is given by $Y_i = \delta + \epsilon_i$ where $\epsilon_i \sim N(0,\sigma_i^2)$. The classical estimation uses the inverse-variance method where the estimate is given by $$ \widehat{\delta} = \dfrac{\sum_{i=1}^N Y_i / s_i^2}{\sum_{i=1}^N 1/s_i^2}. $$
Random-effects models
The primary objective of (study-level) meta-analyses is to estimate the global effect $\delta$ based on the summary information from $N$ studies. Denote by $Y_i$ the summary value of $\delta$ from the $i$th study for $i=1,\ldots,N$. Let $\sigma_i^2$ be the associated dispersion parameter. The model widely adopted for such settings is the linear random-effects model by DerSimonian and Laird (1986) given by $$ \begin{align*} Y_i &= \delta_i + \epsilon_i,\quad \epsilon_i \sim N(0,\sigma_i^2)\\ \delta_i & = \delta + \nu_i,\quad \nu_i \sim N(0,\tau^2), \end{align*} $$ where $\epsilon_i$ and $\nu_i$ are assumed to be independent. $\tau^2$ under this formulation represents the heterogeneity across the $N$ studies whereas $\sigma_i^2$ stands for the within-study variation of each study.
DerSimonian and Laird (1986) suggests a two-step moment estimation.
- Get the estimate for $\tau^2$.
- Compute the weights $w_i(\hat{\tau}^2) = (\hat{\tau}^2 + s_i^2)^{-1}$ where $s_i^2$ is the variance of $Y_i$ reported from the $i$th study.
- Then, $$ \dfrac{\sum_{i=1}^N w_i(\widehat{\tau}^2)Y_i}{\sum_{i=1}^N w_i(\widehat{\tau}^2)} $$
Guolo and Varin (2012) further suggests a likelihood-based estimation method and extends the meta-analysis models to a regression framework to accommodate covariates.