detecting such specious rules, one has to test the significance of the observed deviations from conditional independence. If the deviations are not significant with a desired level, then the rule can be considered specious.

Statistical and information-theoretic tests
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Birch’s exact test [@birch] and mutual information are two robust methods for evaluating significance of partial dependencies and detecting specious rules. Birch’s exact test defines the exact hypergeometric probability that dependency $Q\rightarrow C{=}c$ is at least as strong as observed, if $Q$ and $C$ were actually conditionally independent given $X$ (null hypothesis). In the original article, Birch gives only the hypergeometric point probability $P(n_{qc}~|~n,
n_x,n_{xc},n_{xq},n_{q},n_c)$ that the frequency of $QC{=}c$ is exactly $n_{qc}$, when data size ($n$) and frequencies of $X$, ($X,C{=}c$), ($X,Q$), $Q$ and $C{=}c$ ($n_x$, $n_{xc}$, $n_{xq}$, $n_{q}$, and $n_c$) are fixed. The corresponding cumulative probability ($p$-value) is obtained by summing up all point probabilities where the dependency between $Q$ and $C{=}c$ is at least as strong as observed (which is equivalent to $N_{qc}\geq n_{qc}$ under the null hypothesis of conditional independence). Now, the test for the speciousness of rule $Q\rightarrow C{=}c$, given $X\rightarrow C{=}c_x$ ($c_x=c$ or $c_x=1-c$) becomes $$\begin{gathered}
\label{bircheq}
p_B=P(N_{qc}\geq n_{qc}~|~n,n_x,n_{xc},n_{xq},n_{q},n_c)\\
=\frac{\sum_{i+j\geq n_{qc}}\left(n_{xq} \atop i
    \right)\left(n_{x\neg q}) 