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Suppose I am interested in fitting a linear regression model as follows:

Y = a + b1 * age(continuous) + b2 * sex + b3 * income

This model will be run in both the whole sample and subgroups (defined by age categories including 15–35 years, 36-45 years, and >=46 years).

There are missing values in age, sex, and income, and multiple imputation (e.g., MICE) is used to impute the missing values.

My question is: Besides including all the variables in the analysis model in the imputation model, should the categorised age variable be also included in the imputation model?

I am not a statistician. I read the paper "On the Bias of the Multiple-Imputation Variance Estimator in Survey Sampling". It seems that subgroup variable was referred to as domain.

Thank you in advance for your help!

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I think an easier and more correct solution would be to only estimate the model for the total sample and add interaction terms including the continuous age variable (instead of performing a subgroup analyses with discretized age). That way, you could study whether the regression coefficients for sex and/or income vary across age levels without having to categorize age (which is typically not recommended) and run multiple subgroup analyses. It would also simplify your imputation approach (it could be applied to continuous age, sex, and income in the overall sample).

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  • $\begingroup$ Excellent answer. Categorizing age will create edge artefacts and hide easily explained outcome variability. General note: imputation models should be at least as rich as the outcome model, so routinely include nonlinear terms in the imputation models. the R Hmisc package aregImpute function makes this easy, when you don't quite trust the linearity assumptions of mice. $\endgroup$
    – Frank Harrell
    Jul 30 at 13:15
  • $\begingroup$ Thank you for your expertise. In this hypothetical example, I aim to study if the effect of income on Y varies across different age groups (depending on the background knowledge: 15–35 years, 36-45 years, and >=46 years), in addition to adjusting for age (continuous) as a confounder. Would it be reasonable to include an interaction term in the analysis model and also in the imputation model: Analysis model: Y = a + b1 * age(continuous) + b2 * sex + b3 * income + b4 * age(categorized) + b5 * age(categorized) * income $\endgroup$
    – Willi Zhang
    Aug 2 at 8:09
  • $\begingroup$ To add further, I don't see the problem with conducting multiple subgroup analyses. I don't know if you meant the multiple testing problem. In my view, I don't see multiple testing is a real problem and is a simply pseudo-statistical term which has been discussed many years. The fundamental issue is the uncertainty around the background knowledge, not the p-value. With that being said, your answer is not actually related to my question. $\endgroup$
    – Willi Zhang
    Aug 2 at 8:25
  • $\begingroup$ There are several issues/problems related to the multiple subgroup analysis approach. First, by discretizing age, you loose information that is contained in age as a continuous variable and, as a result, you end up with reduced statistical power. Second, the analysis is inefficient because you have to run multiple different regression models instead of a single comprehensive one. Third, statistically comparing regression coefficients across the different "age subgroups" becomes difficult because the coefficients are obtained in separate analyses. $\endgroup$
    – Christian Geiser
    Aug 2 at 11:16
  • $\begingroup$ Take a look at the following paper for how to properly address missing data in an interaction model with continuous variables: Enders, C. K., Baraldi, A. N., & Cham, H. (2014). Estimating interaction effects with incomplete predictor variables. Psychological Methods, 19(1), 39–55. doi.org/10.1037/a0035314 $\endgroup$
    – Christian Geiser
    Aug 2 at 11:16

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