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The Stata Journal
Volume 11 Number 3: pp. 345-367



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Nonparametric bounds for the causal effect in a binary instrumental-variable model

Tom M. Palmer
MRC Centre for Causal Analyses in Translational Epidemiology
School of Social and Community Medicine
University of Bristol, UK
tom.palmer@bristol.ac.uk
Roland R. Ramsahai
Statistical Laboratory
University of Cambridge, UK
Vanessa Didelez
School of Mathematics
University of Bristol, UK
Nuala A. Sheehan
Departments of Health Sciences and Genetics
University of Leicester, UK
Abstract.  Instrumental variables can be used to make inferences about causal effects in the presence of unmeasured confounding. For a model in which the instrument, intermediate/treatment, and outcome variables are all binary, Balke and Pearl (1997, Journal of the American Statistical Association 92: 1172–1176) derived nonparametric bounds for the intervention probabilities and the average causal effect. We have implemented these bounds in two commands: bpbounds and bpboundsi. We have also implemented several extensions to these bounds. One of these extensions applies when the instrument and outcome are measured in one sample and the instrument and intermediate are measured in another sample. We have also implemented the bounds for an instrument with three categories, as is common in Mendelian randomization analyses in epidemiology and for the case where a monotonic effect of the instrument on the intermediate can be assumed. In each case, we calculate the instrumental-variable inequality constraints as a check for gross violations of the instrumental-variable conditions. The use of the commands is illustrated with a re-creation of the original Balke and Pearl analysis and with a Mendelian randomization analysis. We also give a simulated example to demonstrate that the instrumental-variable inequality constraints can both detect and fail to detect violations of the instrumental-variable conditions.
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