C$, with many types of significance measures. However, this means that $Q\rightarrow C$ is superfluous or redundant with respect to $X\rightarrow C$. Now $\delta_2=0$ and the speciousness depends on $\delta_1$, whether $\delta_1\leq 0$ or $\delta_1>0$ but the deviation from zero is insignificant. If we mark $Q=XZ$ and the corresponding frequency by $n_{xz}$, the exact test for the speciousness of $XZ\rightarrow C$ given $X\rightarrow C$ reduces to $$p_B=\frac{\sum_{i\geq n_{xzc}}\left(n_{xz} \atop i
    \right)\left(n_{x\neg z} \atop n_{xc}-i\right)}{\left(n_x \atop n_{xc} \right)}.$$ An interesting coincidence is that this is the same as the test for the significance of productivity in [@webbml]. With mutual information, the test is simply $MI_S=MI(XZ\rightarrow C|X)$ (assuming $\delta_1>0$).

In the second case, rule $X\rightarrow C$ is a specialization of rule $Q\rightarrow C$, i.e., $Q\subset X$. Once again, it will turn out that $X\rightarrow C$ has to be more significant than $Q\rightarrow C$ to make it specious, given an appropriate significance measure (see Section \[detecting\]). This means that one type of specious rules are generalizations of non-redundant rules (rules $X\rightarrow C$ which are better than any $Y\rightarrow C$, $X\subset Y$ with the given significance measure). Now $\delta_1=0$, and the speciousness depends solely on $\delta_2$. If we mark $X=QZ$ and the corresponding frequency 