# When is multiple imputation useful for multilevel models?

I am working on a longitudinal data set, with each person being measured 8 times on each dependent variable. Some of the dependent variables are continuous; some are counts (mostly with means between 50 and 100) and one is dichotomous. There are 3 independent variables: The main one is continuous and ther are two binary covariates. I intend to use multilevel models. I am not (at least for now) interested in relationships among the dependent variables.

About 75% of the subjects have complete data. The missing data patterns vary widely and are not monotone, but most of the missing data is at the later time points.

From substantive considerations, it seems very likely that the data are MAR, and the means on the variables are mostly pretty similar (with some outliers due to small sample sizes in some patterns).

My usual approach here would be to use MLMs without doing imputation, but a little research has shown me that some researchers recommend multiple imputation when there is missing data even in MLMs with MAR missingness.

I prefer to use SAS but can also use R, but software is not my main concern.

I'd be interested in any recommendations or pointers to review articles.

EDIT: There is no missing data on the independent variables, only the outcomes

• Do you have missing only on the outcomes or do you also have missing on the independent variables, Peter? Apr 10, 2020 at 19:53
• Only on the outcomes. Apr 10, 2020 at 22:09
• Then @DimitrisRizopoulos's answer is right on! Apr 10, 2020 at 22:12

## 1 Answer

In general, mixed-effects models will provide you with valid inferences under MAR, provided that the random-effects structure is appropriately specified. Therefore, no (multiple) imputation is required. Namely, the model specifies the distribution of the complete data outcome data $$Y_i$$ for all time points. Under MAR, we can predict/impute the missing outcome data $$y_i^m$$ using the observed data $$y_i^o$$. This is done by exploiting the correlation structure between $$y_i^m$$ and $$y_i^o$$ provided by the specification of the complete data distribution.

Now, if you have doubts that the chosen random-effects structure captures the correlations in the outcome data sufficiently well, then you could consider applying multiple imputation. This is because in multiple imputation (assuming here that you do that with chained equations), you regress the outcome at each time point on the outcomes at all the other time points, specifying thus a more flexible correlation structure.