sense, the core question of speciousness is whether Yule-Simpson’s paradox occurs in the population. Assuming positive dependency $Q\rightarrow C{=}c$, the paradox can be expressed formally by three conditions

-   $P(C{=}c|Q)>P(C{=}c|\neg Q)$ i.e. $\delta(Q,C{=}c)>0$;

-   $P(C{=}c|Q,X)\leq P(C{=}c|\neg Q,X)$ i.e. $\delta_c(Q,C{=}c|X)\leq 0$;

-   $P(C{=}c|Q,\neg X)\leq P(C{=}c|\neg Q,\neg X)$ i.e. $\delta_c(Q,C{=}c|\neg X)\leq 0$.

In this case, $Q\rightarrow C{=}c$ is specious by definition. However, it may easily go unnoticed in the traditional data analysis involving only pair-wise dependencies. Furthermore, it is possible that the confounding variable $X$ is not present in the data.

A classical example demonstrates a specious dependency between a new treatment and recovery (see e.g., [@lindley]). In the example, a positive marginal dependency between treatment and recovery is observed when the patients are considered as a whole. However, when the data is stratified by the sex of the patients, it turns out that the new treatment is negatively associated with recovery both for women and men. The explanation is that men were more often selected to have the new treatment and they also had a higher recovery rate. One can easily show (by setting $\delta_1\leq 0$ and $\delta_2\leq 0$) that Yule-Simpson’s paradox can occur only if $$\delta(Q,C)\leq \frac{\delta(X,Q)\delta(X,C)}{P(X)P(\neg X)}.$$ This means 