# When to Include the Outcome in Your Imputation Model: A Mathematical Demonstration and Practical Advice

Missing data is a common challenge when analyzing epidemiological data, and imputation is often used to address this issue. This talk will 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 a fixed value) 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. Likewise, we mathematically demonstrate that including the outcome variable in imputation models when using deterministic methods is not recommended, and doing so will induce biased results. A discussion of these results along with practical advice will follow.