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Degrees of NMAR

I've been working on a paper on nonresponse bias. As part of the setup, we describe the MAR and NMAR mechanisms first defined by Little and Rubin. In short, Missing-at-Random means we can fix the bias due to the missingness using the available data. While Not-Missing-at-Random means we can't repair the bias with the available data.

It can be hard to discuss the problem with this divide. We were looking at situations where the bias could be smaller and it could be bigger. The NMAR/MAR distinction doesn't capture that very well. There is another formulation that is actually pretty good for discussing different degrees of bias remaining after adjustment. It's due to the following article (I couldn't find it online):

Kalton, G. and D. Kasprzyk (1986). "Treatment of missing survey data." Survey Methodology 12: 1-16.
They define bias as having two components: A and B. One of the components is susceptible to adjustment and the other is not. In some situations, you can reduce the bias due to A and there will still be the bias due to B. In other situations, by adjusting for A you can actually increase the bias (i.e. when A and B are equal and in opposite directions).

In any event, I think this is useful for talking about bias as it gives us a way to talk about more or less biased that the NMAR/MAR distinction does not.


  1. This is a very interesting point, James. Kristen and Frauke's simulations in Sociological Methods & Research make a similar point. I wonder, however, about the implications. I think it would still be preferable to adjust for A even if it leads to greater bias, for at least three reasons. First, with respect to nonresponse variance, adjusting for A should decrease it. Especially if one considers a longitudinal survey where the key estimates of change. Second, adjusting for A may increase bias in estimates of one variable, but decrease it for estimates of other variables. Third, we rarely have much information for adjustments, so we try to use as much as we can. And seldom the ability to evaluate the impact on bias (i.e., a gold standard). Maybe the import of these findings is if there can be particular adjustment variables that can be identified to exacerbate bias in multiple surveys. I really look forward to seeing your article!

    1. My main point was just as a way of thinking about the issue. In the real world, it's probably too simple to say it's either biased or it's not. Usually, it's more or less biased. We need a way to talk about the more or less part.

      The simulations that I did were interesting and in a way comforting. These were done with real survey data. In most cases, the bias was either A or B. The two weren't mutually present. The end result was that you could either fix it almost completely, or you couldn't do anything about it. I didn't see any "pathological" situations where the adjustments made the bias worse.

    2. Interesting! I really look forward to seeing what you found in more detail. I completely agree on talking about more or less bias in estimates, but still see statements like this one in a report I am reading right now: "To the extent that non-response is associated with age, sex and region, the adjustment will remove non-response bias."

  2. Hi James,
    I am a frequent reader of your blog, thanks for your posts! I think this is an interesting point in many ways. First, the missing data field seems to be dominated by "missing data people", who generally don't care about the extent of bias at all. So it's good if we, being "survey people" bring bias into the story. Second, I think that once you talk about bias, and missing data corrections, it is important to look at the effect of the corrections on both bias and variances. And I agree with Andy that it is usually better to correct, even if one knows that missingness is NMAR. In that case, missingness usually exists of multiple components. A MAR part and a NMAR part. And it is usually worthwhile to fix the bit you can fix, even if that appears to worsen your overall bias.


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