The role of weights in regression modeling and imputation

Abstract

When fitting observations from a complex survey, the standard regression model
assumes that the expected value of the difference between the dependent variable and its model-based prediction is zero, regardless of the values of the explanatory variables. A rarely failing extended regression model assumes only that the model error is uncorrelated with the model’s explanatory variables. When the standard model holds, it is possible to create alternative analysis weights that retain the consistency of the model-parameter estimates while increasing their efficiency by scaling the inverse-probability weights by an appropriately chosen function of the explanatory variables.

When a regression model is used to impute for missing item values in a complex
survey and when item missingness is a function of the explanatory variables of the regression model and not the item value itself, near unbiasedness of an estimated item mean requires that either the standard regression model for the item in the population holds or the analysis weights incorporate a correctly specified and consistently estimated probability of item response. By estimating the parameters of the probability of item response with a calibration equation, one can sometimes account for item missingness that is (partially) a function of the item value itself.