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Abstract:
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The standard regression model, whether linear, logistic, or Poisson, assumes that the expected value of the model error, the difference between the dependent variable and its model-based prediction, is zero no matter what the values of the explanatory variables. The standard model can fail for a given population. A rarely-failing extended regression model assumes only that the model error is uncorrelated with the model’s explanatory variables. Consistent estimates under either the standard or extended model given complex survey data with inverse-probability weights (broadly defined) can be determined by fitting a weighted estimating equation based on the extended model’s assumption of the model error being uncorrelated with the 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 item missingness is a function of the explanatory variables of the regression model and not the item value itself (i.e., item values are missing at random), 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. Weights can be adjusted to retain protection against bias when the standard model for the item value fails while increasing the efficiency of the estimated item mean when the standard model for holds among members of the population that would not have provided item values if surveyed.
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