give the definition in the population level and explain how statistical significance testing can be used to detect specious rules from sample data.

Let $X$ and $Q$ be sets of binary attributes and $C$ a single binary attribute. Dependency rule $Q\rightarrow C{=}c$ ($c\in\{0,1\}$) expressing positive dependency between $Q$ and $C{=}c$ is specious if there is another rule $X\rightarrow C{=}c_x$ ($c_x\in \{0,1\}$) such that $\delta_c(Q,C{=}c|X)\leq 0$ and $\delta_c(Q,C{=}c|\neg X)\leq 0$ in the population.

Figure \[spectypes\] shows four alternative ways that rule $X\rightarrow C{=}c$ or $X\rightarrow C{\neq}c$ can make $Q\rightarrow C{=}c$, $c\in \{0,1\}$ specious. Note that in a and d, the specious rule is $Q\rightarrow C$ and in b and c, it is $Q\rightarrow \neg C$.

![image](fourcases){width="\textwidth"}

In the sample level, speciousness of $Q\rightarrow C{=}c$ can be detected by studying the conditional leverages $\delta_1=\delta_c(Q,C|X)$ and $\delta_2=\delta_c(Q,C|\neg X)$ and statistical significance of the conditional dependency. If both $\delta_1\leq 0$ and $\delta_2\leq 0$, then the observed dependency $Q\rightarrow C{=}c$ is completely due to variable $X$ and either disappears or changes its direction when conditioned. However, sample data is only an incomplete presentation of the underlying population, and it is possible that both conditional leverages are weakly positive, even if the rule is specious. For 