Using calibration weighting to adjust for nonresponse under a plausible model

When we estimate the population total for a survey variable or variables, calibration forces the weighted estimates of certain covariates to match known or alternatively estimated population totals called benchmarks. Calibration can be used to correct for sample-survey nonresponse, or for coverage error resulting from frame undercoverage or unit duplication. The quasi-randomization theory supporting its use in nonresponse adjustment treats response as an additional phase of random sampling. The functional form of a quasi-random response model is assumed to be known, its parameter values estimated implicitly through the creation of calibration weights. Unfortunately, calibration depends upon known benchmark totals while the covariates in a plausible model for survey response may not be the benchmark covariates. Moreover, it may be prudent to keep the number of covariates in a response model small. We use calibration to adjust for nonresponse when the benchmark model and covariates may differ, provided the number of the former is at least as great as that of the latter. We discuss the estimation of a total for a vector of survey variables that do not include the benchmark covariates, but that may include some of the model covariates. We show how to measure both the additional asymptotic variance due to the nonresponse in a calibration-weighted estimator and the full asymptotic variance of the estimator itself. All variances are determined with respect to the randomization mechanism used to select the sample, the response model generating the subset of sample respondents, or both. Data from the U.S. National Agricultural Statistical Service's 2002 Census of Agriculture and simulations are used to illustrate alternative adjustments for nonresponse. The paper concludes with some remarks about adjustment for coverage error.