# Is the method of mean substitution for replacing missing data out of date?

Is the method of mean substitution for replacing missing data out of date? Are there more sophisticated models that should be used? If so, what are they?

-
this site might give answer to your question. The link What is MI gives a list of various resources. –  mpiktas May 23 '11 at 11:47

Barring the fact that it's not necessary to shoot mosquitoes with a cannon (i.e. if you have one missing value in a million data points, just drop it), using the mean could be suboptimal to say the least: the result can be biased, and you should at least correct the result for the uncertainty.

There are some other options, but the one easiest to explain is multiple imputation. The concept is simple: based upon a model for your data itself (e.g. obtained from the complete cases, though other options are available, like MICE), draw values from the associated distribution to 'complete' your dataset. Then in this completed dataset you don't have anymore missing data, and you can run your analysis of interest.

If you did this only once (in fact, replacing the missing values with the mean is a very contorted form of this), it would be called single imputation, and there is no reason why it would perform better than mean replacement.

However: the trick is to do this repeatedly (hence Multiple Imputation), and each time do your analysis on each completed (=imputed) dataset. The result is typically a set of parameter estimates or similar for each completed dataset. Under relatively loose conditions, it is OK to average your parameter estimates over all these imputed datasets.

The advantage is that there also exists a simple formula to adjust the standard error for the uncertainty caused by the missing data.

If you want to know more, you probably want to read Little and Rubin's 'Statistical Analysis with Missing Data'. This also holds other methods (EM,...) and more explanation on how/why/when they work.

-
+1 I would assume that single imputation performs slightly better than mean substitution because you incorporate additional information ($0$ predictors vs $p$ predictors). However, I fully agree that MI is the way to go. –  Bernd Weiss May 23 '11 at 12:25

You did not tell us very much about the nature of your missing data. Did you check for MCAR (Missing Completely at Random)? Given that you cannot assume MCAR, mean substitution can lead to biased estimators.

As a non-mathematical starting point, I can recommend the following two references:

1. Graham, Hohn W. (2009): Missing Data Analysis: Making It Work in the Real World.
2. Allison, Paul (2002): Missing data. (see section "Imputation", p. 11)
-
@ Bernd the Graham reference is extremely good, it helped me a lot with getting the hang of multiple imputation. –  richiemorrisroe May 23 '11 at 12:38

If your missing values are randomly distributed, or your sample size is small, you might be better off just using the mean. I would first split the data into two parts: 1 with the missing values and the other without and then test for the difference in means of some key variables between the two samples. If there is no difference, you have some support for substituting the mean, or just deleting the observations entirely.

-Ralph Winters

-
But using the mean implies you're predicting the value at that point. That's not what's going on, what's going on is an attempt to recover a random value. It seems that since you have an estimate of the variance as well that you should use both (i.e., a random draw from the distribution). –  John May 23 '11 at 16:11
In addition, mean substitution will reduce the variance of your estimates which will throw all of your standard errors and confidence intervals for the rest of your analyese. –  richiemorrisroe May 23 '11 at 19:04
Yes. I was merely suggesting that the populations of the missing vs. non-missing data be examined before blindly diving into MI, which can take up a lot of computational power at the expense of minimum gains. –  Ralph Winters May 23 '11 at 20:19