# When is it a good idea to just use the average for imputation?

Suppose we have a data set test:

1
8
12
14
.
.
19


The  .  denotes missing values. When would it be better to use the average of the non-missing values to impute the missing values rather than assuming that the data comes from a normal distribution?

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Don Rubin wrote an influential paper proving that there is no single imputation method that will produce unbiased inferences (where "single imputation" means the imputing of only one value for a missing observation). However, in the same paper he pointed out that it may well be possible to create multiple imputations whose mean is an unbiased estimate of the missing value, and whose contributions to increased variance in subsequent analysis is a reasonable estimate of the added uncertainty due to data missingness.

This is his paper:

Rubin, D. B. (1976). Inference and missing data. Biometrika, 63(3):581–592.

And this an update to it: Rubin, D. B. (1996). Multiple imputation after 18+ years. Journal of the American Statistical Association, 91(434):473–489.

And this a gentle introduction to the topic of multiple imputation:

Schafer, J. L. (1999). Multiple imputation: a primer. Statistical Methods in Medical Research, 8:3–15.

There are a variety of statistical software packages that support multiple imputation (e.g. mice in R, or ice in Stata, or indeed Stata's built-in multiple imputation capabilities in recent versions).

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I understand Rubin's point to be that you can get an unbiased point estimate using careful single imputation, but the standard errors will be wrong. However, in many cases mean imputation will have other problems, in particular distorting patterns of association with other variables. –  Maarten Buis Jun 25 at 18:58
@MaartenBuis Thank you, I have tried to correct that in my revision... does that work? –  Alexis Jun 25 at 19:24
Not really. The point of (multiple) imputation is not to estimate missing values, those are assumed to be lost forever. However, you often do know other things about those individuals/firms/cows, i.e. other variables are observed for those observations. With (multiple) imputation you want to make the most effective use of that observed data, which you would throw away if you just ignored all observations with at least one missing value. –  Maarten Buis Jun 26 at 7:59
Suggestion: ... no single imputation method that will produce unbiased inference. This means that with single imputation the standard errors, $p$-values and confidence intervals will be off in the sense that they will ignore the uncertainty introduced by the imputation. –  Maarten Buis Jun 26 at 8:05

It is never a good idea to do this, but, if there is very little missing data then it will do relatively little harm, will be much easier to implement and, depending on your final audience, may be a lot easier to explain. However, a relatively sophisticated audience may object to the single mean imputation.

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One can also perform sensitivity analyses by, say, bracketing results based on mean imputation with results including reasonable minimum and maximum imputations. –  Alexis Jun 25 at 19:28