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Reading papers related to the imputation of missing values related to the -omics field, systematically imputation algorithms were less accurate when imputing MNAR compared to imputing MCAR. My intuition is the following: Missing values are classified as MNAR when there is a process behind the generation of the data that influence the missing values. Then to be able to impute those MNAR, it is not enough to find relationships between features, rather, it is more important to know the process which is behind the missing values to impute.

Is my intuition right? am I missing other essential points?

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There can be "a process behind the generation of the data that influence the missing values" while the data are nevertheless "missing at random" (MAR) in the technical sense (and thus suitable for multiple imputation). What's required for data to be MAR is that "the missingness can be explained by variables on which you have full information".

The problem with data "missing not at random" (MNAR) is that your data by themselves do not contain adequate information about the missingness. Data MNAR could be due to a relation between the probability of missingness and the "true" value itself, but they could also be due to a relation of missingness to some other variable that was not included in the data. That's also why it's impossible to prove that data are MAR; you never know about possible unknown unknowns.

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It's not that MNAR data are harder to impute, but rather that MNAR data violate the assumptions made by the multiple imputation procedure. As a result of these assumption violations, the pooled results may be biased (whereas pooled results using MCAR or MAR data would not be).

MNAR data produce biased estimates in multiple imputation and maximum likelihood because, in MNAR data, there is an unexplained relationship between a subject's "true" score on a variable and the probability that that score will be missing. The fact that this relationship can exist as long as it is explained shows why auxiliary variables are helpful.

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    $\begingroup$ The data generating mechanism is not even identified under MNAR. So I would disagree with your claim that MNAR does not make things harder to impute. It makes things fundamentally harder because, no matter how much data you collect, you will never be able to pin down what is going on. Under MAR there is no such foundational issue. $\endgroup$
    – guy
    Sep 25, 2018 at 22:00

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