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PeerGynt comments on Simpson's Paradox - Less Wrong
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Good post, thanks. One comment:
First, I assume you mean "aggregated", otherwise this statement doesn't make sense.
Second, I don't believe you. I say it's always smarter to use the partitioned data than the aggregate data. If you have a data set that includes the gender of the subject, you're always better off building two models (one for each gender) instead of one big model. Why throw away information?
There is a nugget of truth to your claim, which is that sometimes the partitioning strategy becomes impractical. To see why, consider what happens when you first partition on gender, then on history of heart disease. The number of partitions jumps from two to four, meaning there are fewer data samples in each partition. When you add a couple more variables, you will have more partitions than data samples, meaning that most partitions will be empty.
So you don't always want to do as much partitioning as you plausibly could. Instead, you want to figure out how to combine single partition statistics corresponding to each condition (gender, history,etc) into one large predictive model. This can be attacked with techniques like AdaBoost or MaxEnt.
The OP's assertion is true. Stratifying on certain variables can introduce bias.
Consider that you have a cohort of initially healthy men, and you are trying to quantify the causal relationship between an exposure (eg eating hamburgers) and an outcome (eg death). You have also measured a third variable, which is angina pectoris (cardiovascular disease).
Assume that the true underlying causal structure, which you are unaware of, is that hamburgers cause cardiovascular disease, which subsequently causes death.
Now look at what happens if you stratify on cardiovascular disease: In the strata consisting of men who don’t have cardiovascular disease, you will find no cases of death. This will lead you to conclude that in men who don’t have cardiovascular disease, eating hamburgers does not cause death. This is false, as eating hamburgers will cause them to develop cardiovascular disease and then die.
What you have done in this situation, is stratify on a mediator, thereby “blocking” the pathway running through it. There are also many other situations in which adjusting for a variable introduces bias, but it gets more complicated from here.
For further information on this I suggest reading an upcoming book called “Causal Inference”, by James Robins and Miguel Hernan, who taught me this material. The first ten chapters are available for free online at http://www.hsph.harvard.edu/faculty/miguel-hernan/files/hernanrobins_v1.10.9.pdf .