Missing at random is sometimes called ignorable, and the thing that bothers me so much about this is that it's neither random nor ignorable in the way we normally think about things. If you think about how it is random, then it is actually a little easier to imagine. If the missingness is dependent on the data you have in hand, which is the definition of MAR, then if you are to condition the data, to actually use the causes of missingness that are at hand in your dataset. If you use those in your missing data model, then you are conditioning on those variables, and there is no other kind of missingness left over. Anything that is left over is random. It's not really missing at random, but if you call it conditionally missing at random, you can see what's going on. After you condition on the data you already have, then any missingness that is left over is missing completely at random, so that's a nice way to think about it.

I think MCAR is a lot more common than we normally think. For example, this second kind of MCAR comes up if the cause of missingness happens to be uncorrelated with the variables in your model. Think about that. In this case, I have an example I often use. In our study, around 7th grade, 8th grade, and then in 9th grade, the kid is gone, out of the study. We can't find the kid. What happened is that the parents moved to take a new job in a different city, so they're gone. Well, if you think about it, some parents move because their kids are using drugs, and some parents are moving for totally different reasons. Some kids are using drugs; some kids are not using drugs when the parents move. What that is basically saying is that parents moving doesn't tell you anything about whether the kid is using drugs or not, which was our main dependent variable. In that case, the cause of missingness (parents moving to take a new job) was not correlated with drug use. It actually acts just like missing completely at random, and you don't even have to include it in the model, and it causes no biases.

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