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I have a bird which largely travels in packs with other birds. Though, not always.

Anyways, for most birds, I have a position covariate (longitude and latitude), but not for this one particular bird.

I want to do imputation, but before I can decide on multiple imputation and so on, I need to actually find something to impute with.

So, since I have the locations of other birds, does it make sense to fit a regression model on those birds, using another covariate (e.g., "time"), and then predict for my missing data-bird for imputation?

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You can try it. But it sounds like you don't have the data to answer the question you're interested in. Your problem is that you have a bunch of data on ordinary cases, and you want to infer something about a case that you don't have the same data about, but you know a priori is not ordinary. Statistics is largely based on the idea that you can make inferences using the assumption that things you haven't observed are, in some systematic way, similar to things you have observed. If that assumption doesn't hold, statistics doesn't have much to offer you.

Concretely, if you do what you're proposing here, the statistics will presumably infer that the stray bird behaves similarly to the rest of the birds, which you've already learned isn't the case.

A possible way out of this predicament is if you have some other information about the odd bird that distinguishes it from the other birds. For example, is the odd bird bigger than the other birds? Or does it have a different combination of sex and age? If you have a characteristic (or set of characteristics) like this, you can use the other birds to fit a model predicting position from the characteristic, then apply that model to the odd bird. Extrapolation is risky, though.

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