during the search and it was used as a baseline for comparison. No minimum frequency thresholds or other constraints were used, except with Accidents, which was computationally the most demanding data set. When the top-1000 rules were searched from Accidents, the complexity of rules was restricted to six attributes (max five attributes in the antecedent). For speciousness testing with $MI_S$, we used a cautious threshold $\theta=0.5$. Larger thresholds were tried if no type 3 specious rules were detected. Birch’s $p$-value was calculated for checking that $\theta$ was not too large (i.e., none of the pruned partial dependencies were significant).

For each specious rule $Q\rightarrow C{=}c$, we recorded its subtype according to the first $X\rightarrow C{=}c$ or $X\rightarrow C{\neq}c$ that made it specious. The subtypes were the same as in Algorithm \[specdetect\]. In addition, we calculated several statistics for comparing specious (still non-redundant) and non-specious rules (their frequency, strength of dependency, rule complexity) and evaluating the degree of speciousness (conditional leverages, $MI_S$, Birch’s $p$, strength of the mediating rule $X\rightarrow Q{=}q$, $q\in \{0,1\}$). Type 0 rules were excluded from statistics, because they are a special case, which cannot be considered specious per se.

Results
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Proportions of specious rules and their 