Meta-analysis

A meta-analysis combines statistical evidence from several studies addressing a common estimand. It may assume a shared effect or use a random-effects meta-analysis to represent between-study variation. Inverse-variance weighted means combine independent effect estimates when their variances are known or estimated; differences in study validity and effect modifiers require substantive assessment, not just weighting.

Divide studies by a specified characteristic and estimate an effect in each subgroup, then test the difference using an independent subgroup contrast in meta-analysis or an appropriate interaction model. The test of a between-group difference is distinct from separate tests against no effect within each group. Exploratory splits can be affected by multiple testing and by confounding between study characteristics.

A forest plot shows each study's effect estimate and confidence interval, together with a pooled estimate when appropriate. A vertical null-effect line allows comparison on a common scale. For an odds ratio the null is one and a logarithmic axis treats reciprocal ratios symmetrically; marker areas commonly encode study weights.

A funnel plot displays study effect estimates against a measure of their precision, often standard error increasing down the vertical axis. Under a common-effect model, imprecise estimates spread more widely around the effect. Asymmetry is evidence of a small-study effect and can be consistent with publication bias, but it has other possible explanations.

A small-study effect is a systematic relationship between study precision or size and estimated effect. It may result from selective publication, design differences, effect modification or other heterogeneity. A regression of effect on its standard error tests one such relationship; it does not by itself determine which mechanism generated it.

Publication bias arises when the availability of studies depends on the results, such as their significance, magnitude or direction. A meta-analysis of the available studies can then differ systematically from one including all eligible evidence. Funnel plot asymmetry can motivate investigation but does not establish this mechanism; heterogeneity and other small-study effects can also cause asymmetry.

A selection model combines a distribution for all study results with a probability $\pi (y)$ that a result becomes available. The observed distribution is reweighted by this probability and renormalized. Assumed or estimated selection mechanisms permit sensitivity analysis of a meta-analysis, but unavailable results generally prevent a fully assumption-free correction.

For $n$ studies conditionally independent given the truth of one relationship, the event that at least one is positive has probability $1-{\beta }^n$ under the alternative and $1-(1-\alpha {)}^n$ under the null. Its Bayes factor therefore gives the displayed posterior odds. For interior error probabilities these tend back to the prior odds $R$. If power exceeds size, the Bayes factor decreases to one: the quotient of geometric sums is a weighted average of decreasing powers $[\beta /(1-\alpha ){]}^k$.

This result concerns the coarsened selection event. Full observed results instead multiply their individual Bayes factors, and concordant evidence can accumulate without this bound. It is a useful example of information discarded by publication bias.

Trim and fill estimates missing studies from funnel plot asymmetry by trimming extreme studies to locate a centre and filling in counterparts on the less represented side. The adjusted meta-analysis includes these imputed estimates. Its interpretation depends on a symmetry model for missing evidence; it is not a guarantee that genuine publication bias has been removed.

For independent subgroup estimates ${\hat{\mu }}_1,{\hat{\mu }}_2$ with variances $v_1,v_2$, the null contrast has Wald statistic $Z=({\hat{\mu }}_1-{\hat{\mu }}_2)/\sqrt{v_1+v_2}$. If evidence overlaps, include minus twice the covariance in the contrast variance. A nonsignificant comparison is not proof of equivalence.

Systematic displacement of an individual study estimate caused by design, conduct, analysis or selective reporting. Combining estimates can reduce random error without reducing this bias. Bias-adjusted meta-analysis and analyses restricted or stratified by relevant bias domains can assess its impact.

Model study estimates as noisy versions of the target effects plus study-specific biases. External information or sensitivity parameters describe those biases and their uncertainty. Removing an assumed bias without propagating its uncertainty overstates precision; observed study data alone generally do not identify the adjustment.

A common-effect synthesis treats study estimates as estimates of one shared effect. For independent approximately unbiased estimates $y_i$ with variances $v_i$, use $\hat{\mu }={\sum }_i{w}_i{y}_i/{\sum }_i{w}_i$ with $w_i=v_i^{-1}$. This model does not accommodate additional between-study heterogeneity.

For independent two-by-two tables with counts $a_i,b_i,c_i,d_i$ and total $n_i$, the Mantel–Haenszel pooled odds ratio is the displayed ratio of sums. When all $b_i{c}_i>0$, it is also the weighted arithmetic mean of study odds ratios with weights $b_i{c}_i/n_i$. These differ from inverse estimated variance weights for log odds ratios. Both constructions can represent a fixed-effect meta-analysis, but they should not be silently interchanged when interpreting a supplied plot.

Refit a meta-analysis after deleting each study, comparing fitted effects, uncertainty and heterogeneity to the full fit. For fixed weights $w_i>0$, total weight $W$ and weighted effect $\hat{\mu }$, direct subtraction of the two weighted means gives $\hat{\mu }-{\hat{\mu }}_{(-i)}=w_i(y_i-\hat{\mu })/(W-w_i)$. Re-estimating heterogeneity additionally measures influence through the weights. This is a sensitivity diagnostic, not an automatic rule for removing discordant studies.

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.

A network meta-analysis jointly compares several treatments using a graph of direct randomized controlled trials and indirect treatment comparisons. If effects are measured relative to a reference treatment, consistency means ${\delta }_{ij}={\delta }_{i0}-{\delta }_{j0}$. A connected graph identifies all relative effects under that model. Transitivity in network meta-analysis is the substantive comparability assumption supporting these relations; inconsistency can be assessed when the network has loops.

For independent studies comparing treatments $A$ and $B$ to common control $C$, a consistent log odds ratio comparison uses ${\hat{\delta }}_{AB}={\hat{\delta }}_{AC}-{\hat{\delta }}_{BC}$, with variance $v_{AC}+v_{BC}$. This follows from subtracting two estimates of additive effects relative to the same reference. With overlapping evidence, subtract twice their covariance. Causal comparability across the studies is supplied by transitivity in network meta-analysis, not by the variance calculation.

Transitivity requires the studies of different treatment comparisons to be sufficiently comparable in distributions of effect modifiers, outcome definitions and other design features to support an indirect treatment comparison in one target population. For example, a treatment that works differently by disease severity cannot safely be compared indirectly across trials with systematically different severity distributions. Statistical consistency is an implication of suitable transitivity and modeling assumptions, not a substitute for assessing them.