Reference based multiple imputation – what’s the right variance and how to estimate it?

Reference based multiple imputation methods have become a popular approach for handling missing data in the analysis of randomised trials (Carpenter et al 2013). Very roughly speaking, they impute missing outcomes in patients in the active arm assuming that the missing outcomes behave as if the patient switched onto the control treatment. This is in contrast to what is now the standard approach, based on the missing at random assumption, which effectively imputes missing outcomes for patients in a given arm as if they remained on the same treatment as they were randomised to.

Soon after reference based MI methods were proposed, people started noticing that Rubin’s rules variance estimator, which is the standard approach for analysing multiply imputed datasets, overstated the variance of treatment effects compared to the true frequentist variance of the effect estimator (Seaman et al 2014). This means that if Rubin’s rules are used, the type 1 error will be less than the standard 5% level if the null hypothesis is true, and power is lower (sometimes substantially) than if the frequentist variance were used for inference.

In a new pre-print on arXiv I review the congeniality issue and the bias in Rubin’s variance estimator, and summarise some of the arguments made in favour and against using Rubin’s rules with reference based methods. In the end I personally conclude that the frequentist variance is the ‘right’ one, but that we should scrutinise further whether the referenced based assumptions are reasonable in light of the behaviour they cause for inferences. For instance, they lead to a situation where the more data are missing, the more certain we are about the value of treatment effect, which would ordinarily seem incorrect.

I also review different approaches for estimating the frequentist variance, should one decide it is of interest, including efficiently combining bootstrapping with multiple imputation, as proposed by Paul von Hippel and myself a paper (in press at Statistical Science) and available to view here.

I hope the paper stimulates further debate as to what the right variance is for reference based methods, and would very much welcome any comments on it.

19th July 2021 – a short talk about this work can be viewed here.

22nd September 2021 – this work has now been published in the journal Statistics in Biopharmaceutical Research, and is available open-access here.

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