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I'm quite familiar with MCAR, MAR and MNAR (NMAR) data but I have just come across a new (for me) term: Structurally Missing Data (SMD).

According to this page, Structurally missing data is data that is missing for a logical reason. They give an example where they ask people if they have kids and then they ask what the age of the youngest kid is. Some people answer "no" to the first question and then leave the second one blank (as there is no kids, so no age).

But here there is a relationship between having kids and the age of youngest. Since there is a relationship, this makes the data MAR.

So, my question is, is SMD a subset of MAR, or is there a reason why it is not?

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No, I would consider Structurally Missing Data to be a separate category, with distinct methods of dealing with it in analyses.

It is definitely not Missing at Random. By definition, it is non-random, being instead logically associated with specific values of a different variable. Let's use a lightly modified version of the example at the link: consider the variables Has_children? (yes/no) and age_of_youngest_child. If a person has no children, then age_of_youngest_child is undefined, not omitted. The missing values for age_of_youngest_child are associated logically with a specific value in Has_children?.

Note that MAR and MCAR are frequently solved by multiple imputation, while Structurally Missing Data cannot be.

EDIT (h/t to kjetil b halvorsen for the suggestion in the comments):

As to how to analyse data such as this, the key is to put the nested variable into the model as an interaction term only, with no main effect. This is explain in much more detail at How do you deal with "nested" variables in a regression model?

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    $\begingroup$ I'm not sure I agree with "If it were MAR, then the missing values would be associated with both levels of Has_children? with equal probability." But I'm not an expert on that. The key distinction here, in my mind, is that the age of the youngest child is not missing, it's undefined. It has a specific value, but that value is problematic for further analysis. As opposed to missing, where there is a value but it's been omitted. Correct? $\endgroup$ – Wayne Sep 19 '19 at 13:35
  • $\begingroup$ @Wayne Agreed. And you're right, I was mixing up MAR and MCAR, so that statement was incorrect. Thanks for catching that, editing now. $\endgroup$ – mkt - Reinstate Monica Sep 19 '19 at 13:54
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    $\begingroup$ Note, that for your example, how to deal with it in modeling is given here: stats.stackexchange.com/questions/372257/… you could include that in your answer! $\endgroup$ – kjetil b halvorsen Oct 5 '19 at 16:48
  • $\begingroup$ @kjetilbhalvorsen Good suggestion, thanks! $\endgroup$ – mkt - Reinstate Monica Oct 7 '19 at 18:02

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