in a twolevel regression model, how can I compensate for missing values in a level-2 covariate which is only measured at level-2 (varies only between-groups)? FIML only works for dependent variables, correct?
You can mention the variances of the covariates on between. Then they will be treated as dependent variables and distributional assumptions will be made about them but observations with missing on them won't be deleted. You must include all covariates.
It is full-information maximum likelihood - you are just moving the approach up a level, but it's perfectly analogous. I am not aware of papers on it. If you do multiple imputations you would get similar results.
Is there any literature or advice on whether to include all covariates with missing data (by specifying their variances/covariances) as implied by Linda's response, vs. only bringing those covariates into the model that have missingness? I would assume their are potential gains (in a sense similar that imputations improve with additional variables) and loses (reduced power to dramatically increased numbers of estimated parameters). I have tried both on some applied data with similar results, but much longer run time when include all covariates regardless of whether they had missing data.
Clarification: The first sentence should read "Is there any literature or advice on whether to include all covariates (by specifying their variances/covariances) as implied by Linda's response, vs. only bringing those covariates into the model that have missingness?"
Does this hold for categorical/binary covariates? Older posts frequently recommend against this and suggest multiple imputation (MI) as an alternative, especially when the dependent variable is categorical. More recent posts frequently indicate bringing binary covariates into the model is ok, citing results in the MI literature. Is this primarily justified on the MI literature, or is there other work indicating the practice is reasonable with binary covariates (regardless of the distribution of the dependent variable)?
Yes although you do make distributional assumptions about them we have not found this to be problematic. You do the same thing with most multiple imputation programs also. Mplus allow variables to be specified as categorical for imputation.