Doubly robust integration of nonprobability and probability survey data
Doubly robust (DR) estimators of the population mean of an outcome have been proposed for integrating outcome and covariate data from a nonprobability sample with covariate data from a probability survey. These estimators combine inverse probability weighting with mass imputation. We review these DR estimators and extend them to allow estimation of a domain (subpopulation) mean, possibly using data from individuals outside the domain to improve estimation when the domain is small. For situations where an outcome is observed in both samples, we describe estimators that efficiently combine one of these DR estimators with a Horvitz-Thompson or Hajek estimator that uses only the probability survey data. We investigate the asymptotic relative efficiency of these combined estimators compared to their two component estimators, and carry out a simulation study to assess relative efficiency in finite samples. The relative efficiency depends on the ratio of the variances of the two component estimators and on how predictive the covariates are of the outcome. We illustrate the use of all the estimators by analysing data from the Natsal-3 probability survey and four contemporaneous online nonprobability surveys. Our extensions will enable greater uptake and practical use of these combination methods in the survey statistics field.