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Missing data is a common challenge when analyzing epidemiological data, and
imputation is often used to address this issue. Here, we investigate the
scenario where a covariate used in an analysis has missingness and will be
imputed. There are recommendations to include the outcome from the analysis
model in the imputation model for missing covariates, but it is not necessarily
clear if this recommmendation always holds and why this is sometimes true. We
examine deterministic imputation (i.e., single imputation where the imputed
values are treated as fixed) and stochastic imputation (i.e., single imputation
with a random value or multiple imputation) methods and their implications for
estimating the relationship between the imputed covariate and the outcome. We
mathematically demonstrate that including the outcome variable in imputation
models is not just a recommendation but a requirement to achieve unbiased
results when using stochastic imputation methods. Moreover, we dispel common
misconceptions about deterministic imputation models and demonstrate why the
outcome should not be included in these models. This paper aims to bridge the
gap between imputation in theory and in practice, providing mathematical
derivations to explain common statistical recommendations. We offer a better
understanding of the considerations involved in imputing missing covariates and
emphasize when it is necessary to include the outcome variable in the
imputation model.