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Stat Med. 2015 Nov 10;34(25):3298-317. doi: 10.1002/sim.6553. Epub 2015 Jun 10.

Combining fractional polynomial model building with multiple imputation.

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Hub for Trials Methodology Research, MRC Clinical Trials Unit at UCL, Institute of Clinical Trials and Methodology, Aviation House, 125 Kingsway, London, WC2B 6NH, U.K.
Medical Statistics Department, London School of Hygiene & Tropical Medicine, Keppel St, London, WC1E 7HT, U.K.
MRC Biostatistics Unit, Institute of Public Health, Robinson Way, Cambridge, CB2 0SR, U.K.
NHS Blood and Transplant, John Radcliffe Hospital, Oxford, OX3 9BQ, U.K.


Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald-type statistics and weighted likelihood-ratio tests are proposed and evaluated in simulation studies. The Wald-based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only.


fractional polynomials; missing data; multiple imputation; multivariable fractional polynomials

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