Statistical methods used to combine the effective reproduction number, [Formula: see text], and other related measures of COVID-19 in the UK

Thomas Maishman, Stephanie Schaap, Daniel S Silk, Sarah J Nevitt, David C Woods, Veronica E Bowman

Research output: Contribution to journalArticlepeer-review


In the recent COVID-19 pandemic, a wide range of epidemiological modelling approaches were used to predict the effective reproduction number, R(t), and other COVID-19-related measures such as the daily rate of exponential growth, r(t). These candidate models use different modelling approaches or differing assumptions about spatial or age-mixing, and some capture genuine uncertainty in scientific understanding of disease dynamics. Combining estimates using appropriate statistical methodology from multiple candidate models is important to better understand the variation of these outcome measures to help inform decision-making. In this paper, we combine estimates for specific UK nations/regions using random-effects meta-analyses techniques, utilising the restricted maximum-likelihood (REML) method to estimate the heterogeneity variance parameter, and two approaches to calculate the confidence interval for the combined estimate: the standard Wald-type and the Knapp and Hartung (KNHA) method. As estimates in this setting are derived using model predictions, each with varying degrees of uncertainty, equal-weighting is favoured over the standard inverse-variance weighting to avoid potential up-weighting of models providing estimates with lower levels of uncertainty that are not fully accounting for inherent uncertainties. Both equally-weighted models using REML alone and REML+KNHA approaches were found to provide similar variation for R(t) and r(t), with both approaches providing wider, and therefore more conservative, confidence intervals around the combined estimate compared to the standard inverse-variance weighting approach. Utilising these meta-analysis techniques has allowed for statistically robust combined estimates to be calculated for key COVID-19 outcome measures. This in turn allows timely and informed decision-making based on all available information.

Original languageEnglish
Pages (from-to)1757-1777
Number of pages21
JournalStatistical Methods in Medical Research
Issue number9
Early online date3 Jul 2022
Publication statusPublished - Sept 2022


  • Basic Reproduction Number
  • COVID-19/epidemiology
  • Humans
  • Pandemics
  • Uncertainty
  • United Kingdom/epidemiology

Cite this