Median polish

Median polish utilizes the medians obtained from the rows and the columns of a two-way table to iteratively calculate the row effect and column effect on the data. The results are not meant to be sensitive to the outliers, as the iterative procedure uses the medians rather than the means.

Model for a two-way table

Suppose an experiment observes the variable Y under the influence of two variables. We can arrange the data in a two-way table in which one variable is constant along the rows and the other variable constant along the columns. Let i and j denote the position of rows and columns (e.g. y_ij denotes the value of y at the ith row and the jth column). Then we can obtain a simple linear regression equation:

${\displaystyle {\mathbf{y}}_{ij}=b_0+b_1{x}_i+b_2{z}_j+{\epsilon }_{ij},}$

where b₀, b₁, b₂ are constants, and x_i and z_j are values associated with rows and columns, respectively.

The equation can be further simplified if no x_i and z_j values are present for the analysis:

${\displaystyle {\mathbf{y}}_{ij}=b_0+c_i+d_j+{\epsilon }_{ij},}$

where c_i and d_j denote row effects and column effects, respectively.