# Missing data problem

I'm doing a logistic regression with 100 observations (of which 20 events), and a few predictors. One of the (binary) predictors is expected (theoretically) to be a quite strong predictor, but its value is missing for 30 records. By theory, we expect this variable to not be correlated with any of the other predictors, and its missingness to be MCAR. I have identified three possible strategies:

1. Do multiple imputation, based only (or mostly) on the outcome as there is no (expected) relationship with the other variables
2. Do a descriptive analysis with univariate and multivariate models based on complete data (maybe a model without the variable on the full data and one without on the subset of 70)
3. Add a category 'missing' to the variable and perform analyses as usual. This is a quite clean approach but I've read that it can introduce bias even with MCAR

What would be the best approach?

## 2 Answers

Dropping observations with missing data is not a good idea, even under MCAR, because of the loss of statistical power that will result.

Setting a category for missingness is also a bad idea, as it will introduce bias, as you indeed say.

A much better idea is to use multiple imputation. Use all the the variables, even if they have low correlations with the missing variable. If you are using the mice package in R, then the whole analysis can be conducted very easily, but do inspect the predictor matrix prior to simply using the default, in case there are relationships that do not make sense.

• Thanks! Given that theoretically there is no association between the variables, how would you decide what relationships make sense and which don't? – Sanderr Jan 24 at 14:15
• @Sanderr If there is very little correlation then mice will simply not include them in the predictor matrix anyway. If you have good reason for "knowing" that they should not predict the variable with missingness, but there happens to be some (spurious) correlation in the sample which induces mice to include something in the predictor matrix, then just specify your own predictor matrix with only the outcome predicting the missing values in the covariate. – Robert Long Jan 24 at 14:29

If you're interested in inference and the missing variable is only a control, just drop it. You assume that it's uncorrelated with the others, so it won't change their estimates. It might widen the standard errors a bit, but you can also got another model dropping the missing observations. Fitting that one is a good test of your assumptions: the coefs should be similar.

If the missing variable is of central interest, just drop the observations that don't have it. You're assuming MCAR, after all.

If doing pure prediction: multiple imputation will work, fine, but consider averaging the predictions, rather than averaging the coefficients. You don't need to worry about rubin's rules for standard errors on coefs.

• Thanks for your answer. The goal is to find which variables are good predictors of the outcome. I agree that dropping the observations with a missing value would work but I am a little concerned with the increased standard errors by doing this (I would lose 1/3 of events). – Sanderr Jan 24 at 13:12