Regression with MI of missing values: estimating residuals

Question

Imagine I want to regress y on x1, ..., xk. I have missing values in the dependent variable, as well as in some of the independent variables, and have used multiple imputation to create D imputed datasets.

So I run the regression using each of the datasets, combine the estimated parameters and standard errors using Rubin's rules, etc. I can also obtain predicted values of y.

My question is: can I obtain predicted residuals? I would have thought that since I can predict yhat, it would be acceptable to use yhat to predict residuals ... particularly for observations without missing values of y, but also when there D imputed values of y, just by averaging them first.

However, the Stata manual states that this is not appropriate:

"The MI predictions should be treated as a final result; they should not be used as intermediate results in computations. For example, MI estimates of the linear predictor cannot be used to compute residuals as is done in non-MI analysis. Instead, completed-data residuals should be calculated for each imputed dataset ... "

Is there any reason why I should not calculate residuals in the way I described, or by predicting separate residuals for each imputed dataset, and then averaging those? Thanks.

http://www.stata.com/manuals13/mimipredict.pdf

Answer 1

I'm not exactly sure why Stata says this. However, one thing to consider is that most properly, each predicted yhat is an estimate - it's the mean yhat from each imputation. Moreover, you have some cases with missing y - what is your residual in this case? It's not clear. I suspect that may be the main problem. Any interpretation about the residuals you make in that case may be biased, because you're excluding the people with missing Ys.

If you're trying to perform regression diagnostics, some guidance I've seen is that you want to examine plots for a few of your imputed datasets.

As an aside, if you have some observations with a missing dependent variable, why are you using those observations at all? In my field (outcomes research in healthcare, e.g. randomized trials of drugs), the advice we get is usually to drop them. If they are missing on the DV, then nothing can help you. The guys who set up the experiment or data collection needed to minimize missing data in the first place. One exception is when your DV is a multi-item scale, and some people have one or missing items in the scale; here, it's often quite reasonable to impute the missing items, then calculate the scale (but if you have a lot of people missing the entire scale, or many people miss the same item, or many people miss a number of items, then you have to ask what happened).