Between-study heterogeneity (${\tau }^2$)

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.

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.