JDS Journal of Data Science 1683-86021680-743X1680-743X School of Statistics, Renmin University of China JDS1234 10.6339/26-JDS1234 Computing in Data Science An Estimation Framework for Combining Probability and Non-probability Samples https://orcid.org/0000-0002-0720-4319 ElkasabiMahmoudmelkasabi@rti.org1∗ https://orcid.org/0000-0002-0778-9847 LewisTaylor1 https://orcid.org/0000-0001-8894-1240 WilliamsMatthew1 Center for Official Statistics, RTI International, Durham, NC 27713, USA Corresponding author. Email: melkasabi@rti.org. 202686202600120 Supplementary Material

The online supplementary material contains annotated R syntax and results to illustrate estimation from non-probability samples and combining estimates from probability and non-probability samples.

8920252952026 2026 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2026 Open access article under the CC BY license.

Survey researchers are increasingly adopting hybrid sampling designs to address the limitations of traditional probability sampling, especially when studying rare or hard-to-reach populations. Challenges such as high screening costs, low statistical efficiency, and operational constraints make purely probability-based approaches impractical in many contexts. This article uses public data from the National Health and Nutrition Examination Survey to demonstrate how one can make population estimates from a hybrid sampling strategy that combines data from a stratified, multistage probability sample with data from a non-probability sample within the same primary sampling units as the probability sample. We outline a framework and discuss methods for analyzing data from a hybrid sample such as this, where covariates and survey outcomes are observed in both the probability and non-probability samples. We present a case study to illustrate the framework. We provide the case study R code in the supplementary material.

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