  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 recommendation always holds and why this is sometimes true. We
examine deterministic imputation (i.e., single imputation with fixed values)
and stochastic imputation (i.e., single or multiple imputation with random
values) 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.
