  Two-phase sampling designs are frequently employed in epidemiological studies
and large-scale health surveys. In such designs, certain variables are
exclusively collected within a second-phase random subsample of the initial
first-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 sample
size. Model-assisted calibration methods have been used to improve the
efficiency of second-phase estimators. However, no existing methods provide
appropriate calibration auxiliary variables while simultaneously considering
the complex sample designs present in both the first- and second-phase samples
in regression analyses. This paper proposes to calibrate the sample weights for
the second-phase subsample to the weighted entire first-phase sample based on
score functions of regression coefficients by using predictions of the
covariate of interest, which can be computed for the entire first-phase sample.
We establish the consistency of the proposed calibration estimation and provide
variance estimation. Empirical evidence underscores the robustness of the
calibration on score functions compared to the imputation method, which can be
sensitive to misspecified prediction models for the variable only collected in
the second phase. Examples using data from the National Health and Nutrition
Examination Survey are provided.
