Statistical methods used to combine the effective reproduction number, R(t), and other related measures of COVID-19 in the UK

In the COVID-19 pandemic, a range of epidemiological models have been used to predict the number of new daily infections, $I$, daily rate of exponential growth, $r$, and effective reproduction number, $R(t)$. These models differ in their approaches (e.g. mechanistic or empirical) and/or assumptions about spatial or age mixing, and some capture uncertainty in scientific understanding of disease dynamics, and/or have different simplifying assumptions. Combining estimates from multiple models to better understand the variation of these outcome measures is important to help inform decision making. We incorporate estimates of these outcome measures from a number of candidate models for specific UK nations/regions using meta analysis. Random effects models have been implemented to accommodate differing modelling approaches and assumptions between candidate models. Restricted maximum likelihood (REML) is used to estimate the heterogeneity variance parameter, with two approaches to calculate the confidence interval for the combined effect: standard Wald-type intervals and the Knapp and Hartung (KNHA) method. Approaches using REML alone and REML+KNHA provided similar ranges of variation for $R(t)$ and $r$. However, differences were observed when combining estimates on $I$, with the REML+KNHA approach providing more conservative confidence intervals. This is likely due to the limited number of candidate models contributing estimates for this outcome measure, coupled with the large variability observed between model estimates. Utilising these meta-analysis techniques has allowed for statistically robust combined estimates to be calculated for key COVID-19 outcome measures, allowing an overall assessment of the current response measures with associated uncertainty. This in turn allows timely and informed decision making based on all available information.