Meta-Analysis of entropy and mutual information
In most cases, decision-makers are not interested in evaluating the
performance of a diagnostic test in a single study. Instead, they would
like to know the totality of evidence generated in a series of studies
evaluating the particular test. In such cases, a meta-analysis of
summary statistics is employed.
Meta-analysis is initiated with the computation of a summary statistic
for each study 20. In our case, this summary statistic
is the value of MI associated with the diagnostic test under
investigation. The next step in meta-analysis is to compute the weighted
average of MI, where the weights used are typically the inverse of the
MI variance, which is related to sample size 20.
According to Roulston 21, the variance of the entropy
is given by
\begin{equation}
\text{Var}\left(H\left(D\right)\right)=\left[\left(\operatorname{}p+H\left(D\right)\right)^{2}+\left(\operatorname{}\left(1-p\right)+H\left(D\right)\right)^{2}\right]\cdot\frac{p\left(1-p\right)}{N}\ \nonumber \\
\end{equation}which is valid for study sample size greater than 10.
Solving for the variance of MI we derive the expression:
\begin{equation}
{\text{Var}\left(I\left(D,T\right)\right)=\left(\operatorname{}\left(p_{11}+p_{12}\right)+\operatorname{}\left(p_{11}+p_{21}\right)-\operatorname{}p_{11}+I\left(D,T\right)\right)^{2}\left(\frac{p_{11}\left(1-p_{11}\right)}{N}\right)\backslash n}{+\left(\operatorname{}\left(p_{11}+p_{12}\right)+\operatorname{}\left(p_{12}+p_{22}\right)-\operatorname{}p_{12}+I\left(D,T\right)\right)^{2}\left(\frac{p_{12}\left(1-p_{12}\right)}{N}\right)\backslash n}{+\left(\operatorname{}\left(p_{21}+p_{22}\right)+\operatorname{}\left(p_{11}+p_{21}\right)-\operatorname{}p_{21}+I\left(D,T\right)\right)^{2}\left(\frac{p_{21}\left(1-p_{21}\right)}{N}\right)\backslash n}{+\left(\operatorname{}\left(p_{21}+p_{22}\right)+\operatorname{}\left(p_{12}+p_{22}\right)-\operatorname{}p_{22}+I\left(D,T\right)\right)^{2}\left(\frac{p_{22}\left(1-p_{22}\right)}{N}\right)}\nonumber \\
\end{equation}See, table 1 for definitions of \(p_{11},\ p_{12},\ p_{21},\) and\(p_{22}\). Unabridged derivations are presented in the appendix.
Numerical examples of these derivations are shown in Table 2.