A comparison can have the same direction within every background stratum but reverse after pooling strata with different group compositions. For example, let unexposed event risks be in a low-risk stratum and in a high-risk stratum, and multiply the odds by for exposure within each stratum. The exposed risks are then approximately and . If 80% of unexposed subjects but only 1% of exposed subjects belong to the high-risk stratum, the pooled unexposed risk is while the pooled exposed risk is about . Every conditional odds ratio is above one, yet the pooled exposed risk is smaller. The differing background distributions create the reversal; this example does not identify a causal effect without assumptions on how exposure was assigned.
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Simpson's paradox is a phenomenon in statistics where a trend that appears in several different groups of data reverses or disappears when the groups are combined. This paradox can lead to misleading conclusions if the data is not properly analyzed, as the overall relationship may not reflect the relationships within the individual groups. The key concept behind Simpson's paradox is that the aggregation of data can mask or confound relationships due to lurking variables or different underlying distributions.