Single Proxy Control
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Negative control variables are sometimes used in non-experimental studies to detect the presence of confounding by hidden factors. An outcome is said to be a valid negative control outcome (NCO) or more broadly, an outcome that is a proxy for confounding to the extent that it is influenced by unobserved confounders of the exposure effects on the outcome in view, although not causally impacted by the exposure. Tchetgen Tchetgen (2013) introduced the control outcome calibration approach (COCA), as a formal NCO counterfactual method to detect and correct for residual confounding bias. For identification, COCA treats the NCO as an error-prone proxy of the treatment-free counterfactual outcome of interest, and involves regressing the NCO, on the treatment-free counterfactual, together with a rank-preserving structural model which assumes a constant individual-level causal effect. In this work, we establish nonparametric COCA identification for the average causal effect for the treated, without requiring rank-preservation, therefore accommodating unrestricted effect heterogeneity across units. This nonparametric identification result has important practical implications, as it provides single proxy confounding control, in contrast to recently proposed proximal causal inference, which relies for identification on a pair of confounding proxies. For COCA estimation we propose three separate strategies: (i) an extended propensity score approach, (ii) an outcome bridge function approach, and (iii) a doubly robust approach which is unbiased if either (i) or (ii) is unbiased. Finally, we illustrate the proposed methods in an application evaluating the causal impact of a Zika virus outbreak on birth rate in Brazil.
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