Instrumental Variable Estimators for Binary Outcomes
The estimation of exposure effects on study outcomes is almost always complicated by non-random exposure selection - even randomised controlled trials can be affected by participant non-compliance. If the selection mechanism is non-ignorable then inferences based on estimators that fail to adjust for its effects will be misleading. Potentially consistent estimators of the exposure effect can be obtained if the data are expanded to include one or more instrumental variables (IVs). An IV must satisfy core conditions constraining it to be associated with the exposure, and indirectly (but not directly) associated with the outcome through this association. Here we consider IV estimators for studies in which the outcome is represented by a binary variable. While work on this problem has been carried out in statistics and econometrics, the estimators and their associated identifying assumptions have existed in the separate domains of structural models and potential outcomes with almost no overlap. In this paper, we review and integrate the work in these areas and reassess the issues of parameter identification and estimator consistency. Identification of maximum likelihood estimators comes from strong parametric modelling assumptions, with consistency depending on these assumptions being correct. Our main focus is on three semi-parametric estimators based on the generalised method of moments, marginal structural models and structural mean models (SMM). By inspecting the identifying assumptions for each method, we show that these estimators are inconsistent even if the true model generating the data is simple, and argue that this implies that consistency is obtained only under implausible conditions. Identification for SMMs can also be obtained under strong exposure-restricting design constraints that are often appropriate for randomised controlled trials, but not for observational studies. Finally, while estimation of local causal parameters is possible if the selection mechanism is monotonic, not all SMMs identify a local parameter.
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