Fixed-effects model: inverse-variance weighting
The fixed-effects model assumes all studies estimate the same true effect $\theta$. Each study $i$ reports an estimate $\hat{\theta}_i$ with known variance $\sigma_i^2$:
\[\hat{\theta}_i = \theta + \epsilon_i, \quad \epsilon_i \sim N(0, \sigma_i^2)\]The inverse-variance weighted estimator is the minimum-variance unbiased estimator:
\[\hat{\theta}_{\text{FE}} = \frac{\sum_i w_i \hat{\theta}_i}{\sum_i w_i}, \quad w_i = \frac{1}{\sigma_i^2}\]Variance and 95% CI:
\[\text{Var}(\hat{\theta}_{\text{FE}}) = \frac{1}{\sum_i w_i}, \quad \text{CI} = \hat{\theta}_{\text{FE}} \pm 1.96 / \sqrt{\sum_i w_i}\]The fixed-effects model is appropriate only when there is no reason to believe true effects differ across studies. In practice, this is rarely defensible.
Random-effects model: DerSimonian-Laird
The random-effects model treats each study’s true effect as a draw from a distribution:
\[\hat{\theta}_i = \mu + u_i + \epsilon_i, \quad u_i \sim N(0, \tau^2), \quad \epsilon_i \sim N(0, \sigma_i^2)\]where $\mu$ is the mean effect and $\tau^2$ is between-study variance (heterogeneity). The marginal model is:
\[\hat{\theta}_i \sim N(\mu, \tau^2 + \sigma_i^2)\]DerSimonian-Laird (DL) estimator of $\tau^2$: moments-based, using Cochran’s $Q$ statistic:
\[Q = \sum_i w_i(\hat{\theta}_i - \hat{\theta}_{\text{FE}})^2\]Under the null $\tau^2 = 0$, $Q \sim \chi^2_{k-1}$ where $k$ is the number of studies. The DL estimator:
\[\hat{\tau}^2_{\text{DL}} = \max\left(0,\; \frac{Q - (k-1)}{c}\right), \quad c = \sum_i w_i - \frac{\sum_i w_i^2}{\sum_i w_i}\]The random-effects pooled estimate uses updated weights $w_i^* = 1/(\sigma_i^2 + \hat{\tau}^2)$:
\[\hat{\mu}_{\text{RE}} = \frac{\sum_i w_i^* \hat{\theta}_i}{\sum_i w_i^*}\]Limitations of DL: $\hat{\tau}^2$ is biased downward; alternative estimators include REML, Paule-Mandel (iterative moments), and Hedges. With few studies ($k < 10$), all estimators of $\tau^2$ are imprecise.
Quantifying heterogeneity: $I^2$ and $H^2$
The $I^2$ statistic (Higgins & Thompson, 2002) quantifies the proportion of total variation attributable to heterogeneity:
\[I^2 = \max\left(0,\; \frac{Q - (k-1)}{Q}\right) \times 100\%\]Approximate benchmarks (not thresholds):
- $I^2 < 25\%$: low heterogeneity
- $25\% \le I^2 < 50\%$: moderate
- $50\% \le I^2 < 75\%$: substantial
- $I^2 \ge 75\%$: considerable
Important caveat: $I^2$ depends on the number of studies and their precision. With many large studies, even small $\tau^2$ yields high $I^2$. The prediction interval:
\[\hat{\mu} \pm t_{k-2, 0.975} \sqrt{\hat{\tau}^2 + \text{Var}(\hat{\mu})}\]captures the range of true effects in 95% of settings — often more informative than $I^2$ alone.
The $H^2 = Q/(k-1)$ statistic is the ratio of observed to expected $Q$; $H = 1$ means no heterogeneity.
Forest plot
A forest plot displays each study’s estimate and CI as a horizontal line with a central square (sized proportional to weight), plus a pooled estimate diamond at the bottom.
Standard elements:
- Study identifier and year
- Point estimate and 95% CI (horizontal line)
- Weight (square area)
- Pooled estimate and CI (diamond)
- Heterogeneity statistics ($I^2$, $Q$ test $p$-value, $\tau^2$)
The forest plot immediately reveals outlier studies (those whose CIs do not overlap the diamond) and influential studies (large squares).
Funnel plot and publication bias
A funnel plot plots each study’s effect estimate (x-axis) against a measure of precision — standard error (y-axis, inverted), sample size, or 1/SE. Under no publication bias and homogeneity, the plot should be symmetric around the pooled estimate, forming an inverted funnel.
Asymmetry can indicate:
- Publication bias (small studies with null/negative results unpublished)
- True small-study effects (e.g., smaller trials run in higher-risk populations)
- Reporting bias, clinical heterogeneity
Egger’s test: regress the standardized effect $\hat{\theta}_i / \text{SE}_i$ on precision $1/\text{SE}_i$:
\[\frac{\hat{\theta}_i}{\text{SE}_i} = a + b \cdot \frac{1}{\text{SE}_i} + \epsilon_i\]Under symmetry, the intercept $a = 0$. A t-test of $H_0: a = 0$ is the Egger test. Begg’s rank test is an alternative based on Kendall’s $\tau$ between standardized effect and precision.
Trim-and-fill method (Duval & Tweedie): iteratively trims asymmetric studies and imputes their mirror images around the adjusted pooled estimate. The imputed studies provide a bias-corrected estimate. This is exploratory — it assumes a specific publication bias mechanism and can adjust in the wrong direction.
Meta-regression
When heterogeneity can be explained by study-level characteristics (moderators), meta-regression extends the random-effects model:
\[\hat{\theta}_i = \beta_0 + \beta_1 z_{i1} + \cdots + \beta_p z_{ip} + u_i + \epsilon_i\]where $z_{ij}$ are study-level covariates (mean age, year of publication, risk of bias score, dose level). The residual heterogeneity $\tau^2_{\text{residual}}$ measures unexplained between-study variance.
R$^2$ analog: the proportion of between-study variance explained by the moderator:
\[R^2 = 1 - \frac{\hat{\tau}^2_{\text{with moderator}}}{\hat{\tau}^2_{\text{without moderator}}}\]Caution: meta-regression has low power with few studies and is susceptible to ecological fallacy (study-level associations may not reflect individual-level effects).
Network meta-analysis
When treatments A, B, C have not all been compared head-to-head, network meta-analysis (NMA) combines direct and indirect evidence. If trials A vs. B and B vs. C exist, the indirect estimate of A vs. C is:
\[\hat{\theta}_{AC}^{\text{indirect}} = \hat{\theta}_{AB} + \hat{\theta}_{BC}\]The consistency assumption requires that direct and indirect estimates agree. The node-splitting method tests consistency by comparing direct vs. indirect evidence for each comparison.
NMA simultaneously estimates all pairwise treatment effects within a network using a hierarchical model. Treatments can be ranked by their posterior probability of being best, producing SUCRA (surface under the cumulative ranking) scores.