Implications For Decision Making
The practical significance of Simpson's paradox surfaces in decision making situations where it poses the following dilemma: Which data should we consult in choosing an action, the aggregated or the partitioned? In the Kidney Stone example above, it is clear that if one is diagnosed with "Small Stones" or "Large Stones" the data for the respective subpopulation should be consulted and Treatment A would be preferred to Treatment B. But what if a patient is not diagnosed, and the size of the stone is not known; would it be appropriate to consult the aggregated data and administer Treatment B? This would stand contrary to common sense; a treatment that is preferred both under one condition and under its negation should also be preferred when the condition is unknown.
On the other hand, if the partitioned data is to be preferred a priori, what prevents one from partitioning the data into arbitrary sub-categories (say based on eye color or post-treatment pain) artificially constructed to yield wrong choices of treatments? Pearl shows that, indeed, in many cases it is the aggregated, not the partitioned data that gives the correct choice of action. Worse yet, given the same table, one should sometimes follow the partitioned and sometimes the aggregated data, depending on the story behind the data; with each story dictating its own choice. Pearl considers this to be the real paradox behind Simpson's reversal.
As to why and how a story, not data, should dictate choices, the answer is that it is the story which encodes the causal relationships among the variables. Once we extract these relationships and represent them in a graph called a causal Bayesian network we can test algorithmically whether a given partition, representing confounding variables, gives the correct answer. The test, called "back-door," requires that we check whether the nodes corresponding to the confounding variables intercept certain paths in the graph. This reduces Simpson's Paradox to an exercise in graph theory.
Read more about this topic: Simpson's Paradox
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