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Multiple imputation using auxiliary imputation variables that only predict missingness can increase bias due to data
Elinor Curnow1,2, Rosie P Cornish3,4, Jon E Heron3,4
1Department of Population Health Sciences, Bristol Medical School, University of Bristol, Bristol, UK. elinor.curnow@bristol.ac.uk.
Background:
Epidemiological and clinical studies often have missing data, frequently analysed using multiple imputation (MI). In general, MI estimates will be biased if data are missing not at random (MNAR). Bias due to data MNAR can be reduced by including other variables ("auxiliary variables") in imputation models, in addition to those required for the substantive analysis. Common advice is to take an inclusive approach to auxiliary variable selection (i.e. include all variables thought to be predictive of missingness and/or the missing values). There are no clear guidelines about the impact of this strategy when data may be MNAR.
Methods:
We explore the impact of including an auxiliary variable predictive of missingness but, in truth, unrelated to the partially observed variable, when data are MNAR. We quantify, algebraically and by simulation, the magnitude of the additional bias of the MI estimator for the exposure coefficient (fitting either a linear or logistic regression model), when the (continuous or binary) partially observed variable is either the analysis outcome or the exposure. Here, "additional bias" refers to the difference in magnitude of the MI estimator when the imputation model includes (i) the auxiliary variable and the other analysis model variables; (ii) just the other analysis model variables, noting that both will be biased due to data MNAR. We illustrate the extent of this additional bias by re-analysing data from a birth cohort study.
Results:
The additional bias can be relatively large when the outcome is partially observed and missingness is caused by the outcome itself, and even larger if missingness is caused by both the outcome and the exposure (when either the outcome or exposure is partially observed).
Conclusions:
When using MI, the naïve and commonly used strategy of including all available auxiliary variables should be avoided. We recommend including the variables most predictive of the partially observed variable as auxiliary variables, where these can be identified through consideration of the plausible casual diagrams and missingness mechanisms, as well as data exploration (noting that associations with the partially observed variable in the complete records may be distorted due to selection bias).
Insights
Including irrelevant auxiliary variables in multiple imputation (MI) can worsen bias when data are missing not at random (MNAR). Researchers should select auxiliary variables carefully based on their predictive power for the missing data, not just include all available ones.
Area of Science:
- Epidemiology
- Biostatistics
Background:
- Epidemiological and clinical studies frequently encounter missing data, often handled using multiple imputation (MI).
- Multiple imputation estimates can be biased if data are missing not at random (MNAR).
- Auxiliary variables can mitigate bias in MI for MNAR data, but selection strategies require careful consideration.
Purpose of the Study:
- To explore the impact of including auxiliary variables predictive of missingness but unrelated to the partially observed variable in MI models.
- To quantify the additional bias introduced by such auxiliary variables in linear or logistic regression models.
- To assess the effect on exposure coefficients when the partially observed variable is the outcome or exposure.
Main Methods:
- Algebraic quantification and simulation studies were used to assess bias.
- The study compared MI models with and without the inclusion of a specific type of auxiliary variable.
- Analysis involved both continuous and binary partially observed variables, applied to outcomes and exposures.
- Re-analysis of data from a birth cohort study illustrated the findings.
Main Results:
- Including an auxiliary variable predictive of missingness but unrelated to the partially observed variable can introduce substantial additional bias.
- This additional bias is particularly pronounced when the outcome is partially observed and missingness is driven by the outcome itself.
- Bias is even greater when both the outcome and exposure contribute to the missingness mechanism.
Conclusions:
- The common practice of including all available auxiliary variables in MI should be avoided when data may be MNAR.
- Auxiliary variables should be selected based on their strong predictive association with the partially observed variable.
- Identification of appropriate auxiliary variables requires careful consideration of causal diagrams, missingness mechanisms, and data exploration, accounting for potential selection bias.
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