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Simpson's paradox is introduced on wikipedia using 'metrics' of success rates and regression coefficients, for the first (success rate of kidney stone treatments): How to resolve Simpson's paradox?, and the second: Coefficient changes sign when adding a variable in logistic regression, and also, this paper Understanding Simpson’s Paradox linked from the first question.

Quoting from the paper:

If we partition the data into subpopulations, each representing a specific value of the third variable, the phenomena appears as a sign reversal between the associations measured in the disaggregated subpopulations relative to the aggregated data, which describes the population as a whole.

It seems to me this quote still applies if we replace "sign reversal between the associations measured" such that:

If we partition the data into subpopulations, each representing a specific value of the third variable, the phenomena appears as a change in the interpretation of any metric relative to the aggregated data, which describes the population as a whole.

That is, the important bit about the paradox is actually the "partition the data into subpopulations" and not the specific metric. As with the example on kidney stones, the second quote makes sense if the metric is the success rate of a treatment.

In this question,

  1. I am wondering whether the second quote is an appropriate generalisation of Simpson's paradox beyond success rates and regression coefficients.
  2. Has anyone encountered other examples (not related to success rates or regression coefficients) you would consider as a type of Simpson's paradox?

For example, when evaluating two competing models using log-likelihood mean model log-likelihood per observation, would it be possible that model A is better than model B on disjoint subsets $X$ and $Y$ of the data, but worse than model B on the whole dataset $X \cup Y$.

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It seems to me that in order for the second statement to be definitively true, we need to have some restrictions on 'metric'. For example, it can't be the count of the data points--there's no way for there to be more men than women in both of two disjoint subgroups and more women than men in the combined group.

I think your specific example--log likelihood of model A and model B--also falls under this category, if by "model $i$" you mean a specific distribution.

Any parameter that depends on ratios or distances between points--like the mean, or the standard deviation of the groups, or so on--is subject to the reversal. Since that likely describes most reasonable metrics, I suspect the statement is mostly true.

If, instead of fixed models, you mean a distribution-fitting procedure, then it seems clear that you could have a procedure A that performs better on the subgroups but worse than a procedure B that performs better on the total group. The intuition here is that models embed prior beliefs about the structure of the data, and you could separate a dataset in such a way that a particular structure works better for subsets and a different structure works better for the whole.

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    $\begingroup$ Thank you for your answer, it has shed some more light on the intuition behind a "general" paradox. I just wanted to note that your linked paper and mine are the same, but I couldn't understand how to generalise the paradox beyond regression coefficients from reading the paper alone. $\endgroup$ – Alex Aug 30 '16 at 1:19
  • $\begingroup$ also, I agree with your statement that the metric needs to be restricted, specifically the counts, and model log-likelihood (which will be additive over subgroups), cannot provide a paradox. $\endgroup$ – Alex Aug 30 '16 at 1:22

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