In its purest form, multiple imputation is a technique that compensates for item nonresponse using prediction modeling. Although developed in a Bayesian framework, its advocates claim the technique has good "frequentist" properties. With weighted survey data, however, this is generally true only when the item missingness is completely at random. We present a way to conduct a weighted multiple imputation under which resulting estimates are doubly protected from nonresponse bias; that is to say, if either the assumed prediction model or the response (propensity) model is correct, the resulting estimator is nearly unbiased in some sense. Unfortunately, the multiple-imputation-variance estimator will itself be nearly unbiased only when both models hold. Unlike multiple imputation, available imputation and variance-estimation techniques requiring only one of the two models to be true generally focus on a single survey item at a time.
