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?

  • $\begingroup$ 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
  • 2
    $\begingroup$ That is not what missing at random means. $\endgroup$ – AdamO Jun 29 '16 at 20:44

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 observed or unobserved 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.


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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