A Unified Three-Stage Weighting Framework for Causal Inference and Mediation Analysis under Case-Control Sampling
Abstract
Population causal effects and their decomposition into direct and indirect effects cannot be estimated from case-control data without the outcome prevalence in the source population, which is rarely known. Because case-control sampling alters the population only through the outcome, the population is a mixture of the case and control distributions whose mixing weight is the prevalence. We show that this weight is identified by the population mean of any function of a single covariate that differs between cases and controls, and characterize what a reference distribution of the covariates identifies. This yields three prevalence estimators: moment calibration by the generalized method of moments, and two density-ratio fixed-point estimators using an individual-level reference sample or a synthetic sample drawn from the marginal distributions. Design weights based on the estimated prevalence reconstruct the population, and inverse probability weighting gives the total effect, the pure direct and indirect effects, and the mediated interaction. For the calibration estimator, we derive the joint asymptotic distribution of all stages and an overidentification test of the external information; we also quantify the sensitivity of the effects to the prevalence and extend the approach to frequency-matched designs. Simulations and an NHANES application illustrate the methods.
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BibTeX
@misc{islam2026unified,
title = {{A Unified Three-Stage Weighting Framework for Causal Inference and Mediation Analysis under Case-Control Sampling}},
author = {Tarikul Islam and Mahbub A. H. M. Latif},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.26543},
eprint = {2606.26543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.26543},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Unified Three-Stage Weighting Framework for Causal Inference and Mediation Analysis under Case-Control Sampling
AU - Tarikul Islam
AU - Mahbub A. H. M. Latif
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2606.26543
UR - https://arxiv.org/abs/2606.26543
AB - Population causal effects and their decomposition into direct and indirect effects cannot be estimated from case-control data without the outcome prevalence in the source population, which is rarely known. Because case-control sampling alters the population only through the outcome, the population is a mixture of the case and control distributions whose mixing weight is the prevalence. We show that this weight is identified by the population mean of any function of a single covariate that differs between cases and controls, and characterize what a reference distribution of the covariates identifies. This yields three prevalence estimators: moment calibration by the generalized method of moments, and two density-ratio fixed-point estimators using an individual-level reference sample or a synthetic sample drawn from the marginal distributions. Design weights based on the estimated prevalence reconstruct the population, and inverse probability weighting gives the total effect, the pure direct and indirect effects, and the mediated interaction. For the calibration estimator, we derive the joint asymptotic distribution of all stages and an overidentification test of the external information; we also quantify the sensitivity of the effects to the prevalence and extend the approach to frequency-matched designs. Simulations and an NHANES application illustrate the methods.
ER -