Kendall's tau coefficient¶

Description¶

Kendall’s tau is a non‑parametric rank correlation coefficient introduced in 1938. It quantifies the ordinal association between two variables based on the number of concordant vs discordant pairs. The coefficient ranges from –1 (perfect inversion) to +1 (perfect agreement), with 0 indicating no association. The version defined by Yanai et al. (2005) is not a rank‑correlation measure but a tissue‑specificity index for genes, bounded between 0 (housekeeping) and 1 (tissue‑specific)

Formulas¶

Let \(x_{g,c}\) denote the expression of gene \(g\) in tissue \(c\), xith \(k\) total tissues. We define the normalized expression :

\[ \hat{x}_{g,c}=\frac{x_{g,c}}{\displaystyle\max_{1\leq i \leq k}x_{g,i}} \]

For each gene \(g\), compute :

\[\tau_g=\frac{\displaystyle\sum_{i=1}^{k} (1 - \hat{x}_{g,i})}{k-1}\]

This \(\tau_g\) varies from 0 (ubiquitously expressed) to 1 (specific to a single tissue).

Sources¶

Wikipedia - Kendall rank correlation coefficient

Wikipedia - Kendall tau distance

Itai YANAI et al. « Genome-wide midrange transcription profiles reveal expression level relationships in human tissue specification ». In : Bioinformatics 21 (mar. 2005)

“Applying Deep Learning algorithm to perform lung cells annotation”, A. Collin, 2020

Code¶

Computing the Kendall tau distance