  The two-phase sampling design is a cost-effective sampling strategy that has
been widely used in public health research. The conventional approach in this
design is to create subsample specific weights that adjust for probability of
selection and response in the second phase. However, these weights can be
highly variable which in turn results in unstable weighted analyses.
Alternatively, we can use the rich data collected in the first phase of the
study to improve the survey inference of the second phase sample. In this
paper, we use a Bayesian tree-based multiple imputation (MI) approach for
estimating population means using a two-phase survey design. We demonstrate how
to incorporate complex survey design features, such as strata, clusters, and
weights, into the imputation procedure. We use a simulation study to evaluate
the performance of the tree-based MI approach in comparison to the alternative
weighted analyses using the subsample weights. We find the tree-based MI method
outperforms weighting methods with smaller bias, reduced root mean squared
error, and narrower 95\% confidence intervals that have closer to the nominal
level coverage rate. We illustrate the application of the proposed method by
estimating the prevalence of diabetes among the United States
non-institutionalized adult population using the fasting blood glucose data
collected only on a subsample of participants in the 2017-2018 National Health
and Nutrition Examination Survey.
