*I’m analyzing data from a survey and would like to handle the missing values by multiple imputation. Should the survey weights be used as a covariate in the imputation model? *— Signed, Weighting for Your Response

Dear Weighting,

This is a very interesting question. As always, the answer is, “It depends.” It depends on what other covariates are in your imputation model.

Survey weights are typically used to correct for unequal probabilities of selection, so that estimates for the population are not unduly influenced by subjects from groups that were oversampled. Find out how the survey was designed. Most likely, you will discover that individuals’ probabilities of selection were largely determined by a few key variables such as age, race or ethnicity, geographic region, socioecenomic status, cluster size, and so on. If all the variables that determine the probability of selection are already present as covariates in your imputation model, then the survey weight won’t add any more information. You should try to include those design variables if possible. If some of them are unavailable, however, you might want to consider using the weight as a covariate.

Here’s a trick to determine if you have already included most of the important design variables. Using ordinary linear regression, regress the survey weight on the design variables that you have; if the R2 is high (and there’s no hard and fast rule here), then you probably don’t need to think about including the weight in your imputation model. If the R2 is low, perhaps you should include it.

If you decide to use the survey weight as a covariate in you imputation model, you may want to consider including it as a nominal variable. The theory of propensity scores (Rosenbaum & Rubin, 1983) tells us that if we classify subjects on the basis of their selection probabilities into five strata, more than 90% of the selection bias will be removed. Divide your subjects into five groups of equal size, using the quintiles of the weight distribution as your cut points. Create four dummy indicators to distinguish among these five categories. Then include the four dummies in your imputation model. That should do the trick.

**Reference**

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. *Biometrika, 70, *41-55.