Decomposition of pearson residuals of three-variables contingency cube

This paper shows the meaning of Pearson residuals when a contingency table is three dimensional. While information granules of statistical independence of two variables can be viewed as determinants of 2 times 2- submatrices, those of three variables consist of several combinations of linear equations which will become odds ratio when they are equal to 0. Interstingly, the property on the symmetry of two dimensional tables is lost, and the lost of symmetry gives some meaning of Pearson residuals of three dimensional tables.