Proportional Dissimilarity Matrices for Gene Co-expression

Gene co-expression networks [12 (link), 13 (link)] are generally based on a pairwise distance or dissimilarity matrix which is often a function of correlation and thus not appropriate for relative data. Proportionality is appropriate, but ϕ does not satisfy the properties of a distance—most obviously, it is not symmetric unless β = 1:
$\begin{array}{ccc}\hfill \varphi (logx,logy)& =\hfill & 1+{\beta }^2-2\beta \left|r\right|\hfill \\ \hfill \varphi (logy,logx)& =\hfill & 1+\frac{1}{{\beta }^2}-2\frac{1}{\beta }\left|r\right|.\hfill \end{array}$
We are most interested in pairs of variables where β and r are near 1 and want to preserve the link between ϕ(log x, log y), β and r. Hence, our approach to forming a dissimilarity matrix is simply to work with ϕ(log x_i, log x_j) where i < j, in effect, the lower triangle of the matrix of ϕ values between all pairs of components. This symmetrised form of ϕ was then used to lay out a network of the 145 mRNAs that were involved in 424 pairwise relationships with ϕ < 0.05. We used the symmetrised form of ϕ as the basis of the cluster analysis and heatmap expression pattern display (e.g., S10 Fig.) described by Eisen et al. [14 ].