measure such that the signs of terms are reversed in the case of negative conditional dependency (i.e., $MI(Q,C|X=x)$ becomes $-MI(Q,C|X=x)$ if $\delta_c(Q,C|X=x)<0$). The modified (signed) measure is denoted $MI_S$. Once again, one should decide a minimum $MI_S$ value which is required for non-speciousness.

Similarly, one can design a $\chi^2$-based test for speciousness (like Mantel-Haenszel’s test and its variations that asymptotically approximate $p_B$), but it has the same problem as $MI$ in separating positive and negative conditional dependencies.

Relationship to redundancy, equivalence and Yule-Simpson’s paradox {#specialcases}
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Speciousness of specialized and generalized rules {#specase1}
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Speciousness is closely related to the concepts of productivity [@webbml] and redundancy [@kingfkais] and the question of whether a more general rule $X\rightarrow C{=}c$ or its specialization $Q\rightarrow C{=}c$ is superfluous. There are two possible cases, where $X\rightarrow C{=}c$ makes $Q\rightarrow C{=}c$ specious and either $X\subset Q$ or $Q\subset X$ (where $\subset$ denotates a proper subset). For simplicity, we represent the results only for positive consequent $C{=}1$.

In the first case, rule $X\rightarrow C$ is a generalization of rule $Q\rightarrow C$, i.e., $X\subset Q$. In the next section we will show that this phenomenon can occur only if dependency $X\rightarrow C$ is more significant than $Q\rightarrow 