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Evaluating causal effects in a primary population of interest with unmeasured confounders is challenging. Although instrumental variables (IVs) are widely used to address unmeasured confounding, they may not always be available in the primary population. Fortunately, IVs might have been used in previous observational studies on similar causal problems, and these auxiliary studies can be useful to infer causal effects in the primary population, even if they represent different populations. However, existing methods often assume homogeneity or equality of conditional average treatment effects between the primary and auxiliary populations, which may be limited in practice. This paper aims to remove the homogeneity requirement and establish a novel identifiability result allowing for different conditional average treatment effects across populations. We also construct a multiply robust estimator that remains consistent despite partial misspecifications of the observed data model and achieves local efficiency if all nuisance models are correct. The proposed approach is illustrated through simulation studies. We finally apply our approach by leveraging data from lower income individuals with cigarette price as a valid IV to evaluate the causal effect of smoking on physical functional status in higher income group where strong IVs are not available. |
