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.