Is Simpson's Paradox always an example of confounding? Or is it possible to have a Simpson's paradox effect without an extra variable lurking in the background?
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2$\begingroup$ This baseball batting average "paradox" en.wikipedia.org/wiki/Simpson%27s_paradox#Batting_averages can be explained based on differing numbers of at bats (divisor for batting average) in the seasons. I don't consider number of at bats to be a "confounding" variable (but maybe other people do?), but it plays a key role in Simpson's paradox being able to occur in this example. $\endgroup$– Mark L. StoneJul 1, 2016 at 21:51
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1$\begingroup$ Why wouldn't number of bats being a confounding variable? It certainly can be described as such and, if so, implies that that instance of Simpson's paradox could be described as an instance of confounding. Isn't this the case? $\endgroup$– GeorgeJul 1, 2016 at 22:07
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1$\begingroup$ There's no clear relationship between number of at bats and batting average. For instance, in this (first) example, Jeter had a higher batting average in the year he had more at bats, but Justice had a lower batting average in the year he had more at bats. As I said, I suppose it depends on how confounding variable is defined and interpreted. $\endgroup$– Mark L. StoneJul 1, 2016 at 22:22
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$\begingroup$ I think you may have slightly misunderstood what a confounding variable is. Simpson's Paradox does require something in the background that can be represented as a variable, but that doesn't mean it's a confounding variable; a confounding variable influences both the independent and dependent variables, not just the independent variable. See @RobertF 's example below, which may clarify this point. $\endgroup$– jbowmanSep 12, 2021 at 18:09
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3 Answers
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You could imagine forming subclasses based on X, and the relationship between X and Y within each subclass opposes the relationship between X and Y across the sample. You could conceive of the subclasses as a confounder, but if you've artificially imposed them and they come from nothing but the already measured X variable, then no additional substantive confounding variable would have to be introduced.
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No, Simpson's paradox is not always about confounding. In fact, I would say there is no reason to be surprised by sign reversals if you already know the covariate you adjust for is a confounder, you should check this answer here. You can have sign reversal adjusting for colliders or mediators, and without causal knowledge, you cannot know which estimate will give you the correct answer. If you want to play with simulations showing several sign reversals each time you include a covariate for adjustment, you can check the Simpson Machine in Dagitty's website.
answered Dec 13, 2018 at 5:20
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Here's a simple visual example of Simpson's Paradox where there is no confounding:
Observing the relationship between the two variables Sex and Medical Cost, there would appear to be a strong causal relationship:
However if you add a third variable, Age, in the causal diagram:
it becomes clear that the relationship between Sex and Cost is not significant, rather there is a strong linear relationship between Age and Cost.
Meanwhile, there clearly should be no causal relationship between Age and Sex in the diagram, hence Age is not a confounder. To be clear, in this example Sex would no longer be in a causal relationship with Cost, which would by definition mean confounding is not possible if there are only two variables in the path diagram.
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1$\begingroup$ Are you sure this example works? In the diagram you drew age clearly is a confounder because it is correlated with BOTH sex (the males in the diagram are older than the females) and cost (older people have higher costs). If it were not actually correlated with sex then the male and female "clouds" in the chart would exactly overlap and there would be no Simpson's paradox. $\endgroup$ Sep 13, 2021 at 0:54
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$\begingroup$ @GrahamWright My intention was to pick variables that clearly have no causal link (that is, changing someone's age doesn't affect their sex, or vice versa), but for whatever reasons in the sample that was collected there are clearly fewer older females and younger males. Perhaps this wasn't the best example. My understanding is that confounding requires a causal relationship between the confounder variable Z and the variables X and Y, not just an association. $\endgroup$– RobertFSep 13, 2021 at 2:42
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1$\begingroup$ Your scatter plot seems to clearly says that in THESE data not controlling for age will give you the wrong answer, no matter what causal structure is at play, right? If age were somehow correlated to gender without being causally connected through it (either directly or through some other mediating variable) and you analyzed these data without controlling for age you would detect a relationship between gender and cost that is entirely due to the correlation between age and gender. Isn't that Simpson's paradox, at least in practice? $\endgroup$ Sep 13, 2021 at 11:59
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$\begingroup$ @GrahamWright Correct - or at least it's more probable that age is linked to cost rather than sex if you model a linear regression on the data (the coefficient for sex should have p > .05 if you include age in the model). If in fact there was a causal relationship between Age and Sex that explained the age discrepancies in males & females, then yes we'd be still be seeing Simpson's Paradox, but this time with a confounder. $\endgroup$– RobertFSep 13, 2021 at 15:00