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I have data with a continuous longitudinal outcome and one of the covariates is a categorical longitudinal variable. Both of them have missingness and were collected at the same time. So this means if the outcome is missing, the covariate is also missing.

My questions are the following:

  1. If there is missingness in the covariate, is direct likelihood still unbiased?
  2. If I will do multiple imputation using chained equations (fcs in SAS or mice in R), should I impute first the missing longitudinal covariate before the longitudinal outcome? Is it fine to do this way? Or it is better to impute them together in one go?
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For your two major questions:

If there is missing in the covariate, is direct likelihood still unbiased?

In short, it can be, but that will depend on how much missingness is present and how much of that is structurally dependent. In other words, if $x$ reliably predicts missingness in $y$, there is probably something going on there that you need to investigate (a discussion on when missingness like this occurs can be seen in this paper). The missingness maps in the plot_pattern() function for the MICE package are really nice for checking that. Depending on the issue, you will either need to impute the data or find a way to model the missingness directly. The book I link below has strategies for that purpose.

If I will do multiple imputation using chained equations (fcs in SAS or mice in R), should I impute first the missing longitudinal covariate before the longitudinal outcome? Is it fine to do this way? Or it is better to impute them together in one go?

It is better to impute all of it together. The idea behind MICE is that you are essentially estimating in the data based off the relationships captured in the data frame. Even things like the format of the data can impact this estimation. The more variables you omit, the less information it has to go off of, which makes it more prone to inaccuracy. How inaccurate that becomes depends on many factors (e.g. sample size, number of variables, level of missingness). In general, you just want more available information to work with.

It may be useful to read up on the mice vignettes and the book from the creator, titled Flexible Imputation of Missing Data. The book from what I remember speaks very specifically about how to impute missing data in mixed models.

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    $\begingroup$ (+1) Also worth spending some time with is the work of Craig Enders and colleagues, who have been hammering away at missing data issues in longitudinal and multilevel contexts for some time now. appliedmissingdata.com $\endgroup$
    – Erik Ruzek
    Commented Jan 3 at 19:32

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