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I've had these two explained multiple times. They continue to cook my brain. Missing Not at Random makes sense to be, and Missing Completely at Random makes sense...it's the Missing at Random that doesn't as much.
What gives rise to data that would be MAR but not MCAR?
Missing at random (MAR) means that the missingness can be explained by variables on which you have full information. It's not a testable assumption, but there are cases where it is reasonable vs. not.
For example, take political opinion polls. Many people refuse to answer. If you assume that the reasons people refuse to answer are entirely based on demographics, and if you have those demographics on each person, then the data is MAR. It is known that some of the reasons why people refuse to answer can be based on demographics (for instance, people at both low and high incomes are less likely to answer than those in the middle), but there's really no way to know if that is the full explanation.
So, the question becomes "is it full enough?". Often, methods like multiple imputation work better than other methods as long as the data isn't very missing not at random.
Amelia, mi, and mice. The similarities and differences are fascinating. (Amelia's over impute is quite interesting.)
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I'm not sure if this is correct, but the way I've tried to understand it is as if there is a 2x2 matrix of possibilities which isn't quite symmetrical. Something like:
Pattern / Data Explains Pattern
Yes No
Yes MAR MNAR
No -- MCAR
That is, if there is a pattern to a variable's missingness and the data we have cannot explain it we have MNAR, but if the data we have (i.e. other variables in our data set) can explain it we have MAR. If there is no pattern to the missingness, it's MCAR.
I may be way off here. Also, this leaves open the definition of "Pattern", and "Data explains". I think of "Data explains" as meaning other variables in your data set explain it, but I believe that your procedure can also explain it (e.g. a good example in another thread is if you have three measurement variables that measure the same thing and your procedure is if the first two measurements disagree by too much you take a third measurement).
Is this accurate enough for intuition, CV?
I was also struggling to grasp the difference, so maybe some examples could help.
MCAR: Missing completely at random, this is great. It means that the non-response is completely random. So your survey is not biased.
MAR: Missing at random, worse situation. Imagine you are asking for IQ and you have much more females participants than males. Lucky for you, IQ is not related to gender, so you can control for gender (apply weighting) to reduce bias.
MNAR: Not missing at random, bad. Consider having survey for level of income. And again, you have more females than males participants. In this case, this is a problem, because level of income is related to gender. Therefore your results will be biased. Not easily to get rid of.
You see, it is a "triangle" relationship between target variable (Y, such as income), auxiliary variable (X, such as age) and response behavior (R, the response group). If X is related to R only, good-ish (MAR). If there is relation between X and R and X and Y, its bad (MNAR).
