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I have a continuous variable with some values missing at random (MAR) that has to be categorized for further analysis. As far as I know it is not a good idea to categorize the continuous variable after multiple imputation, therfore, it has to be categorized first and then the categorical variable has to be imputed.

The question is: Does it make sense to use the underlying continuous variable as an auxiliary variable in multiple imputation?

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What you want is passive imputation, which is available in all software packages that implement multiple imputation with chained equations (MICE). Passive imputation occurs when a variable is imputed, and then a second variable is transformed from that variable (and possibly others). For example, if BMI (which involves height and weight) was missing for several individuals because their heights and weights were missing, passive imputation of BMI would involve imputing height and weight as usual and creating the BMI variable as transformations of those rather than imputing BMI directly (i.e., as if it weren't a transformation of height and weight). This way, BMI can be used in the imputation of other variables, but it always retains the appropriate relationship with height and weight.

In your case, you'd want to impute the continuous variable and passively impute the categorical variable by specifying the categorization scheme in the passive imputation formula. This way, you could include both the continuous and categorical variable in the imputation of other missing variables in your dataset, and, since the categorical variable is the one to be eventually used in the final analysis, the analysis will be "congenial" in that the relationship between the categorical variable and the analysis variables will be consistent (if correctly modeled).

Note it's generally a bad idea to categorize continuous variables, and there are many effective analyses that can handle continuous variables that are as or more interpretable than those for categorical variables.

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