Random-effects meta-analysis ($y_i\sim N(\mu ,v_i+{\tau }^2)$)

For independent study estimates $y_i$ with within-study variances $v_i$, a usual approximate model is $y_i\mid {\delta }_i\sim N({\delta }_i,v_i)$ and ${\delta }_i\sim N(\mu ,{\tau }^2)$. Its marginal distribution is $N(\mu ,v_i+{\tau }^2)$, so for fixed heterogeneity ${\tau }^2$ the inverse-variance weighted mean uses weights $(v_i+{\tau }^2{)}^{-1}$. Here $\mu$ describes the mean effect across comparable studies; ${\tau }^2$ represents between-study variation. Estimation and uncertainty for heterogeneity are essential, especially with few studies.

A Student t random-effect model assigns Student's t-distributions to exchangeable study effects, allowing heavier tails than a normal distribution hierarchy. An equivalent Gaussian scale mixture is ${\lambda }_j\sim {\chi }_{\nu }^2$, ${\beta }_j\mid {\lambda }_j,\mu ,\psi \sim N(\mu ,\nu {\psi }^2/{\lambda }_j)$, with independent latent draws. For $\nu >2$ the variance is $\nu {\psi }^2/(\nu -2)$, so $\psi$ is a scale rather than a standard deviation. Small latent precisions weaken shrinkage for atypical studies while retaining partial pooling for the rest.

Variation in the underlying study effects, additional to sampling error. The usual normal random-effects meta-analysis models it by a between-study variance ${\tau }^2$. It can reflect effect modification, design differences or within-study bias; it is not automatically a biological treatment difference.

Meta-regression relates study effect estimates to study-level explanatory variables. A common random-effects meta-analysis extension is $y_i=\alpha +\beta x_i+u_i+{\epsilon }_i$, where $\mathrm{Var}(u_i)={\tau }^2$ and $\mathrm{Var}({\epsilon }_i)=v_i$. For log odds ratios, $e^{\beta }$ is a ratio of underlying odds ratios per unit of $x$. Few studies, correlated design changes and exploratory variable selection limit interpretation; a study-level association does not identify an individual-level causal effect. The interpretation of study-level predictors is also discussed in the Cochrane methods handbook, section 10.11.

For $J\ge 2$ conditionally independent effects ${\beta }_j\sim N(\mu ,{\tau }^2)$, each pair difference has normal distribution $N(0,2{\tau }^2)$. With $m=\left(\genfrac{}{}{0ex}{}{J}{2}\right)$, an upper bound $A=R/[\sqrt{2}{\Phi }^{-1}(1-\epsilon /(2m))]$ on $\tau$ ensures, by the union bound, that $\mathbb{P}({\max }_j{\beta }_j-{\min }_j{\beta }_j>R)\le \epsilon$. Any proper scale prior distribution supported on $(0,A)$ preserves this bound after averaging. This is conservative simultaneous calibration, rather than the weaker statement about one selected pair. For log odds ratios, a bound $R$ corresponds to a ratio-of-odds-ratios bound $e^R$.

For $Q>0$ and $m$ studies, $I^2=\max \{0,[Q-(m-1)]/Q\}$. Usually reported as a percentage, it describes excess variation relative to the observed variation on this heterogeneity scale. It is not a proportion of studies with different effects or a proportion of an individual clinical outcome.

For $m$ studies let $w_i=v_i^{-1}$, $W={\sum }_i{w}_i$, $C=W-{\sum }_i{w}_i^2/W$, and let $Q$ be Cochran's Q statistic. The moment estimate is ${\hat{\tau }}^2=\max \{0,[Q-(m-1)]/C\}$ when $C>0$. It estimates the variance of underlying study effects, not their sampling variance.

With fixed inverse-variance weights and weighted mean $\hat{\mu }$, $Q={\sum }_i{w}_i(y_i-\hat{\mu }{)}^2$. Under a common-effect normal model with $m$ independent estimates and known variances it has a chi-squared distribution with $m-1$ degrees of freedom. Estimated variances make this calibration approximate.