Meta-regression

Explore whether prespecified study-level characteristics explain variation in study effects.

Method lesson · Updated October 6, 2026

Concepts and reasoning

Explain variation across studies

Meta-regression relates study effect estimates to study characteristics called moderators. A moderator might be mean age, intervention dose, follow-up duration, or a design feature. The goal is to describe whether effect estimates change systematically with those characteristics while accounting for their different precisions.

A study, rather than a participant, is the unit of analysis. A dataset containing 20 studies supplies roughly 20 independent units for a simple moderator model, even if those studies include thousands of participants. Adding many moderators to a small collection can overfit the observed study differences.

Study-level association is not patient-level interaction

Suppose studies with higher mean age show larger treatment effects. That does not establish that older people benefit more within each study. The older-population studies may also differ in disease severity, treatment dose, follow-up, or bias. This mismatch between aggregate and individual interpretation is often called ecological bias.

A moderator coefficient is observational even when every included study is randomized: studies are generally not randomized to their average ages or research methods. Prespecifying plausible moderators and grounding them in a causal or clinical rationale is more defensible than searching many variables for a small p-value. Cochrane’s meta-regression discussion explains these limitations.

Mathematical foundations

On a small screen, scroll wide equations and tables horizontally.

Extend the random-effects model

A simple model is

The observed effect is , the moderator is , represents residual between-study variation, and is the reported standard error. The intercept refers to , which may not be meaningful unless the moderator is centered.

For log risk ratios, a one-unit moderator increase changes the predicted log ratio by . Exponentiating gives a ratio of risk ratios, not a new absolute risk ratio for an unspecified patient. The predicted risk ratio at moderator value is .

Residual heterogeneity describes variation not explained by the included moderators. A reduction in can be suggestive, but it does not establish a causal explanation. The metafor coefficient guide gives examples of transformations and contrasts.

Worked example

Hypothetical data for learning; independent of your analysis inputs.

Interpret a centered age moderator

Suppose a hypothetical fitted model is . The intercept describes a study whose mean age is 50. A ten-year increase in mean study age adds 0.10 to the log ratio and multiplies its predicted risk ratio by .

Mean study age	Centered age	Predicted log risk ratio	Predicted risk ratio
40	−10	−0.400	0.670
50	0	−0.300	0.741
60	10	−0.200	0.819

On an adverse-event endpoint, these predictions describe effects becoming closer to no effect as mean study age increases. They do not show an individual patient’s risk. They also do not establish that age caused the differences.

Now suppose all studies with mean age above 55 used shorter follow-up. Age and follow-up are correlated at the study level. Including both may make coefficients unstable if the data contain little independent variation. A plot of moderator combinations can reveal this problem before interpreting a regression table.

If the actual study ages range from 40 to 60, predicting at 85 extrapolates the study-level relationship. A precise-looking fitted line beyond the data should not be treated as evidence for a geriatric population.

Interpreting the results

Interpret associations between studies

A moderator coefficient describes how the underlying study effect changes with a one-unit difference in the study-level moderator, conditional on other included moderators. The intercept describes the predicted effect when every moderator is zero; centering can make it more meaningful.

For an effect measured on the log-ratio scale, exponentiating a slope gives a ratio of effect ratios per moderator unit. It is not the original treatment-effect ratio. A study-level association between average age and effect does not establish the effect modification that would be seen by comparing older and younger individuals within a study.

Residual heterogeneity describes unexplained variation after the fitted moderators. A decrease does not prove that a moderator is causally responsible for the difference. Interpret coefficient uncertainty, the observed moderator range, and remaining heterogeneity together, particularly when only a few studies contribute.

Assumptions and limitations

Study-level predictors have limited information

Independent, comparable study effects and valid standard errors are required as in ordinary meta-analysis. The regression adds assumptions about a linear moderator relationship and the residual effect distribution. An apparent straight line can be driven by one extreme study or a cluster of related designs.

Moderators are observational characteristics of studies. Confounding by design, outcome definitions, location, or risk of bias can explain an association. Aggregate data can conceal within-study relationships, so ecological conclusions should not be translated directly into patient-level recommendations.

The number of studies, not participants across all studies, determines information for moderator coefficients. Many correlated moderators can exhaust residual degrees of freedom. Categories need deliberate indicator coding and a reference group; a numeric code such as 1, 2, 3 otherwise imposes a linear trend.

Diagnostics and sensitivity analysis

Check support for each fitted contrast

Plot effect estimates against moderators and inspect which studies support comparisons. Avoid interpreting predicted effects far outside the observed range or combinations of moderators that barely occur. Examine influential studies and whether the fitted association changes under defensible sensitivity analyses.

Inspect collinearity and residual heterogeneity. A nonsignificant moderator can reflect imprecision, while a small p-value after exploring many candidate moderators can reflect selection. Prespecify a limited set of scientifically motivated relationships and distinguish exploratory findings.

Center continuous moderators where a zero value is uninformative, and retain the transformation in reporting. For categorical moderators, interpret contrasts rather than a numeric trend unless that trend was intended. The metafor guide to factors and linear combinations illustrates how coding changes the meaning of coefficients.

References and other material

Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.

metafor: testing factors and linear combinations — Author examples of moderator models and their interpretation.
Cochrane Handbook, chapter 10: analysing data and undertaking meta-analyses — Free guidance on synthesis, heterogeneity and interpretation.