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I have a survey data, in which there are some missing data (not answered questions). I threw away those where the whole page(s) questions were missed, but there are still some with unanswered questions here or there.

How to examine if these missing data are missing at random or not. Is there any hypothesis test I can run?

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    $\begingroup$ Technically if you are using MI you still keep whole pages of missing data, such as when the participant forgot to turn the survey page over. Which is a statistical rule of thumb: you must never print two sided surveys. $\endgroup$ – AdamO Jun 29 '16 at 17:11
  • $\begingroup$ The questions on the same pages are highly correlated, so they are not missing at random. $\endgroup$ – ziweiguan Jun 29 '16 at 20:27
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    $\begingroup$ That is not what missing at random means. $\endgroup$ – AdamO Jun 29 '16 at 20:44
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Here is one way to test the missingness-at-random assumption.

Suppose the question on participant's income has some missing entries. Run a logistic regression with income as your response and everything else as predictors. Your response would be 1 if it's missing, 0 otherwise. The p-value of the predictors should give you an idea whether this MAR assumption is any good.

Do the same for all other columns with missing data.

EDIT : There is a huge literature behind this issue. I'm risking possibly misleading simplification here. See Ch 25 of this,

Gelman, A., & Hill, J. (2006). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.

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  • $\begingroup$ In the process explained by you, how would you interpret the p-value if being significant <0.05 (significance level). So you mean to say if many predictors p-values >0.05 indicates that they dont contribute to the missing nature of this predictor and so can be treated as Missing at Random?? $\endgroup$ – ShyamSundar R Sep 6 '20 at 11:54
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A little bit of terminology:

  • Missing completely at random: Missingness does not depend on any observed or unobserved variables.
  • Missing at random: Missingness does not depend on unobserved variables, but it may depend on observed variables.
  • Missing not at random: Missingness depends on unobserved variables (it may or may not also depend on observed variables).

The answer given by horaceT shows a way to test whether your data is missing at random, but there is a strong assumption here: you have to assume that your data is not missing not at random (sorry for the double negative!). In other words, your null hypothesis is "missing completely at random", and the alternative hypothesis is "missing at random".

The reason for this is clear: you cannot test if missingness depends on unobserved variables, because, well, you didn't observe/measure them. This subtlety is important, because it affects how you interpret your results.

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  • $\begingroup$ @M Turgeon, Would like to mention that: Missing not at random depends only on unobserved variables. ref. ncbi.nlm.nih.gov/books/NBK493614 $\endgroup$ – Will Zhang May 28 at 15:56
  • $\begingroup$ @WillZhang whether missingness depends or not on observed variables is irrelevant to the definition of MNAR, the key point is that it depends on unobserved variables. I'll edit my response to reflect that point. $\endgroup$ – M Turgeon May 29 at 17:03

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