I want to do a (multinomial) logistic regression to predict 5 different physical activity classes based on different variables extracted for each subject. However, I have one variable (i.e., time after disease onset) in my dataset with missing values for some of the subjects, i.e. for the healthy control subjects. These missing values are thus not missing at random, but don't make sense for healthy controls. How would you deal with this variable? I could remove this variable completely from the model, but it does have a strong predicting power. Or would you simply set a to an arbitrary value, e.g. -100?
First, add another binary variable "healthy", if it is not there already. Then set all missing values of "time" to zero.
The resulting model should do what you want and be interpretable. The coefficient of "healthy" will describe the difference in probability of a healthy person compared with someone who just got infected. Any person with nonzero "time" will be "healthy" equal to zero.
If you fit an alternative-specific MNL model (also known as McFadden's model), you will estimate a different effect of the predictor for each modality of your dependent variable (minus one as reference/baseline category). Then it will be possible to constrain the effect of "time" to be null for the "healthy" group.