The difference between statistics of independence between input and residual: 
$C_{IR} = I ( R_y , X ) − I ( R_x , Y ) $

University of Michigan - Ann Arbor

Several cause-effect pair filters (not involving any learning) have been proposed as causation coefficients, based on various data generative assumptions. Here are some examples:
 - R-squared $C_{R2}$
 - Input/Residual Independence $C_{IR}$
 - Entropy $C_H$
 - Slope $C_S$
 - Conditional Distribution Standard Deviation $C_{CDS}$

Examples of Cause-Effect Pair Filters as Causation Coefficients

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

The difference between the R-square statistic measuring goodness of fit in both direction: 
$C_{R2} = R_y^2-R_x^2$

R-squared (Cause-Effect Pair Filter)

Input/Residual Independence (Cause-Effect Pair Filter)

The difference in (discretized) entropy of the marginal distributions: 
$C_H = H_x − H_y$ , with $H_x = \sum_{k=1}^m p_k \log p_k$
using m = 20 bins

Entropy (Cause-Effect Pair Filter)

The difference in average slope in both direction: 
$C_S = S_y − S_x$ , with $S_y = \sum_{k=1}^{m-1} p_k \Delta y_k/\Delta x_k$ 
using m = 20 intervals

Slope (Cause-Effect Pair Filter)

The difference between conditional distributions average standard deviation, in both directions: 
$C_{CDS} = CDS ( X | Y ) − CDS ( Y | X )$

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