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Adjusted Wallace

The Wallace coefficient (W) directly indicates the agreement between partitions and therefore can be easily interpretated. Nevertheless we recommend considering the expected Wallace under independence (W_i) to make sure the agreement is not due to chance alone (Pinto et al., 2008).

Another way to take into account the expected Wallace under independence is to calculate an adjusted version of the Wallace coefficient, in the same way an adjusted Rand was calculated. The Adjusted Wallace (AW) is calculated according to the expression:

$AW_{A\rightarrow B}=\frac{W_{A\rightarrow B} - W_{i(A\rightarrow B)}}{1-W_{i(A\rightarrow B)}}$

Because an analytical formula is know to calculate the Wallace confidence interval (Pinto et al., 2008), the 95% confidence interval for the Adjusted Wallace can be calculated by the following expression:

$CI(AW_{A \rightarrow B})_{95%}=AW_{A \rightarrow B} \pm 2 \times \sqrt{var(AW_{A \rightarrow B})}$

$\approx AW_{A \rightarrow B} \pm 2 \frac{1}{SID_B} \qquad \sqrt{var(W_{A \rightarrow B})}$

where SID_B is the Simpson's Index of Diversity of clustering B.

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