Abstract
Variants of a widely discussed problem related (but not restricted) to causal inference are called Simpson’s paradox. In one version, the paradox is that while a new drug may be better than the old drug for both male and female patients, when the data are combined, for all individuals, the old drug appears better. In these cases, the odds ratio is used to determine which treatment is better. First, the paradox is illustrated, and a brief overview of some of the published arguments is given, which aim at explaining what is wrong. Most of these theories say that the paradox occurs as a result of properties of the data or of the data collection procedure. This chapter takes a different position. It is argued that the odds ratio may not be appropriate to measure effect size, because it fails to take into account how popular the compared treatments were, which is a relevant information collected in observational studies. A competing, consistent measure of effect (and a concept of effect) is developed, which never commits the paradox. Finally, the last section does not suggest neither the odds ratio nor the measure developed in the previous section to be used universally; rather, it is argued that for a good choice of the better treatment, additional aspects, not only the numbers of positive and negative responses, need to be taken into account.
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Notes
- 1.
In the biostatistics literature, a somewhat different phenomenon is called Simpson’s paradox, and the properties and the main results are different.
- 2.
This was only approximately true, in the sense that this was the case in the largest departments and where the difference in admission rates was not negligible. Other examples, to be discussed, showed exact Simpson’s paradoxes.
- 3.
Of course, the paradox will not go away, in general, just by changing the order of conditioning with all data sets. This is a particular feature of these data.
- 4.
Consistent, therefore, means that Simpson’s paradox never occurs.
- 5.
This was seen as a desirable property, when the odds ratio was used to measure the strength of association.
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Rudas, T. (2018). Simpson’s Paradox. In: Lectures on Categorical Data Analysis. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7693-5_9
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