Missing at random vs missing not at random: What if it is both? (Does one imply the other?)
Missing at random (MAR) means the NA frequency of the variable is never depended on the value of the variable itself.
Therefore in your example the data would be Missing not at random (MNAR)!
Why is this distinction important?
Because when the data is MNAR we have to identify the relationship between missingnes and the value however if the data was truly MAR or MCAR we could ignore the NAs or impute them with simple methods like mean-imputation.
Correctly identifying wether data is MCAR,MAR or MNAR is the only way to correctly identify how to deal with it!
But how can something be MAR and not MNAR?
Imagine you are asking senior citizens their birth month but also measure whether they have Alzheimers or similar memory impairments.
Birth month would likely be MAR as the fact whether it is missing correlates with the Alzheimer variable yet I have no information about the actual birth month from this fact.
I can predict from other variables whether the data is not missing but not what the actual value of it is!