if for all rules $r_1$, $r_2$ $$M_c(r_1|r_2)\leq M_c(r_2|r_1) \Rightarrow M(r_1)\leq M(r_2).$$ Order-isomorphic measures $M$ and $M_c$ are called [*distance-preserving*]{}, if $M_c(r_1|r_2)-M_c(r_2|r_1)=M(r_1)-M(r_2)$ and [*ratio-preserving*]{} if $\frac{M_c(r_1|r_2)}{M_c(r_2|r_1)}=\frac{M(r_1)}{M(r_2)}.$

It turns out that many commonly used measures are order-homomorphic or even order-isomorphic. For example, exact hypergeometric point probabilities are ratio-preserving ($P$s in Eq. \[probratio\] are replaced by hypergeometric probabilities) and mutual information and log-likelihood ratio (2$MI$) are distance-preserving. For the $\chi^2$-measure, order-homomorphism may not hold exactly but asymptotically [@lewis1962]. It is an open problem whether Fisher’s $p$-value (cumulative hypergeometric probability) is order-homomorphic with Birch’s $p$-value, like the corresponding point probabilities. If this could be proved, they would likely be the most robust measures for finding genuine rules.

Efficient speciousness detection is based on the following simple yet powerful theorem:

Let $(M, M_c)$ be a pair of order-homomorphic measures, where $M$ is a marginal and $M_c$ a conditional goodness measure for dependency rules. We assume that $M$ and $M_c$ are increasing (decreasing) by goodness, i.e., high (low) values indicate significant dependence. In addition, we assume that for any rule $r_1$ which is specious by another rule $r_2$ holds $M_c(r_1|r_2)<M_c(r_2|r_1)$ and $M_c(r_1|r_2)\leq \theta$ (for decreasing measures $M_c(r_1|r_2)>M_c(r_2|r_1)$ and $M_c(r_1|r_2)\geq \theta$), where $\theta$ is a predefined threshold. 