Judea Pearl's "Simpson Machine", introduced in his paper Understanding Simpson's paradox, is an example of a causal structure where stepwise inclusion of control variables into a regression model switches the sign of an estimated causal association in every step. This example illustrates not only how certain causal structures lead to Simpson's Paradox, but also that deciding which variables to include in a control set is impossible without considering the underlying causal structure.
This web-page allows you to play around with the Simpson machine and generate example data to help you understand Simpson's paradox and the relationship between causal structure, adjustment, and bias. You can generate Simpson machines of different sizes, adjust for different variables to see the effect on the causal effect estimate, and generate R code of the simulator to generate and analyse the data yourself.
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