that it can be avoided if either $Q$ or $C$ is independent from $X$. In the example of treatment and recovery, the paradox would have been avoided, if the same proportion of men and women had been allocated to the new treatment, thus removing the dependency between the sex and treatment. However, this is seldom possible in retrospective studies or exploratory data analysis and examples of Yule-Simpson’s paradox occur in real world data. One solution to the problem could be dependency rule analysis with speciousness detection. If dependency rules were searched from the treatment–recovery data, it would immediately be clear that the strongest and most significant dependencies are [*Male $\rightarrow$ Treatment*]{} and [*Male $\rightarrow$ Recovery*]{}, while [*Treatment $\rightarrow$ Recovery*]{} is clearly weaker and relatively insignificant. Still, an exhaustive search would probably output it, together with many other specious rules, because they do not check interrelations of dependencies.

Detecting specious associations {#detecting}
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Theoretical results
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Detecting specious dependency rules reduces to the following core problem: Given two rules, $r_1$: $Q\rightarrow C{=}c$ and $r_2$: $X\rightarrow C{=}c_x$, $c,c_x\in \{0,1\}$, is $r_1$ specious by $r_2$, $r_2$ by $r_1$, or neither? In essence, this is the same as asking: How probable it is to 