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Two-phase sampling designs are frequently employed in epidemiological studiesand large-scale health surveys. In such designs, certain variables areexclusively collected within a second-phase random subsample of the initialfirst-phase sample, often due to factors such as high costs, response burden,or constraints on data collection or measurement assessment. Consequently,second-phase sample estimators can be inefficient due to the diminished samplesize. Model-assisted calibration methods have been widely used to improve theefficiency of second-phase estimators. However, none of the existing methodshave considered the complexities arising from the intricate sample designspresent in both first- and second-phase samples in regression analyses. Thispaper proposes to calibrate the sample weights for the second-phase subsampleto the weighted first-phase sample based on influence functions of regressioncoefficients for a prediction of the covariate of interest, which can becomputed for the entire first-phase sample. We establish the consistency of theproposed calibration estimation and provide variance estimation. Empiricalevidence underscores the robustness of calibration on influence functionscompared to the imputation method, which can be sensitive to misspecifiedprediction models for the variable only collected in the second phase. Examplesusing data from the National Health and Nutrition Examination Survey areprovided.

Using Model-Assisted Calibration Methods to Improve Efficiency of Regression Analyses with Two-Phase Samples under Complex Survey Designs