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There are quite a number of questions about missing data, but I can't find one related to its consequences on validity in the context of regression. Please correct me if this is not the case. The article on Wikipedia is quite vague on the issue too, with phrases like "with different impacts on the validity of conclusions from research", without really specifying what it means by "conclusions".

Question: it is clear to me that under missing completely at random (MCAR), results from a regression are both internally (i.e. consistency) and externally (inference to wider population) valid (minding no other problems of course). What about MAR and NMAR?

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I think maybe you are confused because you are looking for some kind of firm estimation of the consequences. But here is the rub with missing data. The data is missing so in many cases one cannot say what it's affect on your regression coefficients, SEs etc. would be.

Having data that has some points that are pleasantly MCAR will reduce the power of your model as there is a loss in statistical efficiency. But that is all.

Having data that has some points that are MAR is a bit annoying. Listwise deletion (also termed complete case analysis) may lead to results that are biased, but as the term suggests, one should be able to make good judgemental use of the observations and avoid such bias. Obviously the problem is to be able to say unequivocally that the data is MAR and for this you need 'separability': where the parameters for the missingness process are distinct from the model explaining the primary dependent. Most methods for dealing with incomplete data rest on the MAR assumption.

Having data that has some points which are MNAR is is a complete and very common b*tch. Here, we are trying to say something about the world (by modelling it, in say your LR) but we have observed too little of it to say anything. Going ahead and modelling the complete data will mean you cannot infer anything about the wider population, because your results will be biased - you don't have external validity. You won't know how because the information is, well, missing. You cannot make causal interpretations (internal validity) because you are missing data and the missingness indicators are dependent on that unobserved data. Without collecting more data or, in some cases, imputing the missing data... you are only able to talk about your incomplete sample. We have to go beyond the data and the model.

Sensitivity analysis is a good idea as if you wish to continue to model the data. You could think of doing a logistic regression using the missingness indicators as binary variables as a starting point to understand your patterns of missingness.

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