Menna Arafat

Introduction to meta-analysis

Meta-analysis combines results from multiple studies that address the same or closely related research question. Rather than treating each study separately, it summarizes the available evidence using a common effect-size measure.

A meta-analysis usually involves four main components:

  • estimating an effect size for each study;
  • weighting studies according to their statistical precision;
  • combining the study-specific effects into an overall estimate;
  • assessing how much the effects vary across studies.

1.1 Effect sizes

Studies must be expressed on a comparable scale before they can be combined. The appropriate effect measure depends on the type of outcome.

Common effect measures include:

  • mean difference (MD) for continuous outcomes measured on the same scale;
  • standardized mean difference (SMD) when continuous outcomes are measured using different scales;
  • risk ratio (RR) or odds ratio (OR) for binary outcomes;
  • hazard ratio (HR) for time-to-event outcomes;
  • correlation coefficients for associations between continuous variables.

For study \(i\), let the estimated effect be

\[ \hat{\theta}_i \]

with standard error

\[ SE_i. \]

A study with a smaller standard error provides a more precise estimate and therefore usually receives greater weight in the meta-analysis.

1.2 Inverse-variance weighting

A common approach is to weight each study by the inverse of its variance:

\[ w_i=\frac{1}{SE_i^2}. \]

The pooled effect is then calculated as

\[ \hat{\theta}_{pooled} = \frac{\sum_{i=1}^{k} w_i\hat{\theta}_i} {\sum_{i=1}^{k} w_i}, \]

where \(k\) is the number of studies.

This means that more precise studies contribute more strongly to the pooled estimate.

1.2.1 Simple inverse-variance example

Suppose four studies report effect estimates and their standard errors.

effect <- c(-2.1, -1.5, -2.8, -1.9)
se <- c(0.50, 0.40, 0.70, 0.45)

weight <- 1 / se^2
pooled <- sum(weight * effect) / sum(weight)

pooled
## [1] -1.907936

The pooled estimate is a weighted average rather than a simple arithmetic mean.

1.3 Fixed-effect and random-effects models

Two common approaches are the fixed-effect model and the random-effects model.

1.3.1 Fixed-effect model

A fixed-effect model assumes that all studies estimate the same underlying true effect:

\[ \theta_1=\theta_2=\cdots=\theta_k=\theta. \]

Differences between observed study estimates are therefore attributed to sampling variation.

This model is most appropriate when the studies are sufficiently similar and the scientific question concerns a common effect.

1.3.2 Random-effects model

A random-effects model allows the true effect to vary across studies:

\[ \theta_i \sim N(\mu,\tau^2), \]

where:

  • \(\mu\) is the average underlying effect;
  • \(\tau^2\) represents between-study heterogeneity.

The random-effects model therefore incorporates both within-study uncertainty and variation between studies.

1.4 Heterogeneity

Heterogeneity describes differences in effect sizes across studies beyond what would be expected from sampling variation alone.

Two commonly reported measures are \(Q\) and \(I^2\).

The \(I^2\) statistic is often written as

\[ I^2 = \max\left(0,\frac{Q-(k-1)}{Q}\right)\times100\%. \]

It describes the proportion of observed variation in study estimates that is attributed to heterogeneity rather than sampling error.

For example:

  • \(I^2=0\%\) indicates little evidence of heterogeneity;
  • larger values indicate increasing between-study variation.

These values should not be interpreted using rigid cut-offs alone. The magnitude and clinical importance of heterogeneity should also be considered.

1.5 Forest plots

A forest plot displays the effect estimate and confidence interval from each study together with the pooled estimate.

It allows the reader to examine:

  • the direction of effects;
  • the precision of individual studies;
  • consistency between studies;
  • the pooled effect and its confidence interval.

Studies with narrower confidence intervals are generally more precise and receive greater statistical weight.

1.6 Interpretation

A pooled estimate should not be interpreted without considering the studies that produced it. Important questions include:

  • Are the study populations comparable?
  • Are the interventions or exposures sufficiently similar?
  • Are the outcomes defined in comparable ways?
  • Is there substantial statistical or clinical heterogeneity?
  • Are some studies at high risk of bias?

Meta-analysis can increase precision by combining evidence, but it cannot correct systematic bias in the included studies. The validity of the pooled result therefore depends on both the statistical model and the quality and comparability of the underlying evidence.

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