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Open Access Open Badges Research article

Combining estimates of interest in prognostic modelling studies after multiple imputation: current practice and guidelines

Andrea Marshall12*, Douglas G Altman1, Roger L Holder3 and Patrick Royston4

Author affiliations

1 Centre for Statistics in Medicine, University of Oxford, Oxford, UK

2 Warwick Clinical Trials Unit, University of Warwick, Coventry, UK

3 Department of Primary Care & General Practice, University of Birmingham, Birmingham, UK

4 Hub for Trials Methodology Research and UCL, MRC Clinical Trials Unit, London, UK

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Citation and License

BMC Medical Research Methodology 2009, 9:57  doi:10.1186/1471-2288-9-57

Published: 28 July 2009



Multiple imputation (MI) provides an effective approach to handle missing covariate data within prognostic modelling studies, as it can properly account for the missing data uncertainty. The multiply imputed datasets are each analysed using standard prognostic modelling techniques to obtain the estimates of interest. The estimates from each imputed dataset are then combined into one overall estimate and variance, incorporating both the within and between imputation variability. Rubin's rules for combining these multiply imputed estimates are based on asymptotic theory. The resulting combined estimates may be more accurate if the posterior distribution of the population parameter of interest is better approximated by the normal distribution. However, the normality assumption may not be appropriate for all the parameters of interest when analysing prognostic modelling studies, such as predicted survival probabilities and model performance measures.


Guidelines for combining the estimates of interest when analysing prognostic modelling studies are provided. A literature review is performed to identify current practice for combining such estimates in prognostic modelling studies.


Methods for combining all reported estimates after MI were not well reported in the current literature. Rubin's rules without applying any transformations were the standard approach used, when any method was stated.


The proposed simple guidelines for combining estimates after MI may lead to a wider and more appropriate use of MI in future prognostic modelling studies.