ADR 0003: Meta-regression prediction-interval choices

  • Status: Accepted
  • Date: 2026-07-23

Context

For a mixed-effects Meta-regression prediction at design vector x, the estimated true-effect variance combines residual heterogeneity with uncertainty in the fitted mean:

Var_prediction = tau^2 + x' Cov(beta_hat) x

The critical-value distribution remains a methodological choice. Common software offers a default normal-or-t rule and a Riley alternative with one fewer residual degree of freedom. Neither approximation removes uncertainty from estimating tau-squared, and changing the package default would alter existing results.

Decision

meta_regression() accepts prediction_interval_method="default" | "riley" for mixed-effects models. The canonical resolved method is stored in result.method.prediction_interval_method.

The default remains normal_or_t_k_minus_p:

  • normal coefficient inference uses a standard normal prediction critical value;
  • either Hartung-Knapp mode uses a t critical value with k-p degrees of freedom.

The opt-in Riley method uses a t critical value with k-p-1 degrees of freedom regardless of the coefficient-inference distribution:

prediction = x' beta_hat
PI_Riley = prediction +/- t_(k-p-1) *
           sqrt(tau^2 + x' Cov(beta_hat) x)

Riley intervals require k-p >= 2. Requesting Riley for a common-effect model or a mixed model without enough residual degrees of freedom raises a domain error. The selection is preserved by deleted-study refits and reused by predict() and bubble().

Both rules predict the distribution of true effects in a new study at the specified moderator values. They do not add an unknown sampling variance for a future observed effect.

Validation

The committed fixed-version R metafor fixture records default and predtype="Riley" predictions for normal and Hartung-Knapp inference, multivariable moderator values, and a zero-tau-squared boundary. Unit tests also check the critical-value formula and invalid degrees of freedom. Property-based tests verify symmetry, unchanged mean-effect inference, and the Riley interval's greater width relative to the default rule.

Consequences

  • existing fits retain their numerical default;
  • the alternative is explicit, auditable, and reproducible;
  • Riley is documented as an alternative approximation rather than an automatic small-sample correction;
  • future prediction-interval rules require a separate statistical decision and independent reference coverage.