Generalised Median Polish Based on Additive Generators

Contingency tables often arise from collecting patient data and from lab experiments. A typical question to be answered based on a contingency table is whether the rows or the columns show a significant difference. Median Polish (MP) is fast becoming a prefered way to analyse contingency tables based on a simple additive model. Often, the data need to be transformed before applying the MP algorithm to get better results. A common transformation is the logarithm which essentially changes the underlying model to a multiplicative model. In this work, we propose a novel way of applying the MP algorithm with generalised transformations that still gives reasonable results. Our approach to the underlying model leads us to transformations that are similar to additive generators of some fuzzy logic connectives. In fact, we illustrate how to choose the best transformation that give meaningful results by proposing some modified additive generators of uninorms. In this way, MP is generalied from the simple additive model to more general nonlinear connectives. The recently proposed way of identifying a suitable power transformation based on IQRoQ plots [1] also plays a central role in this work.