Analysis of contingency tables based on generalised median polish with power transformations and non-additive models.

Frank Klawonn,Akiko Kukita,Frank Pessler,Balasubramaniam Jayaram,Katja Crull

doi:10.1186/2047-2501-1-11

Abstract

Contingency tables are a very common basis for the investigation of effects of different treatments or influences on a disease or the health state of patients. Many journals put a strong emphasis on p-values to support the validity of results. Therefore, even small contingency tables are analysed by techniques like t-test or ANOVA. Both these concepts are based on normality assumptions for the underlying data. For larger data sets, this assumption is not so critical, since the underlying statistics are based on sums of (independent) random variables which can be assumed to follow approximately a normal distribution, at least for a larger number of summands. But for smaller data sets, the normality assumption can often not be justified.Robust methods like the Wilcoxon-Mann-Whitney-U test or the Kruskal-Wallis test do not lead to statistically significant p-values for small samples. Median polish is a robust alternative to analyse contingency tables providing much more insight than just a p-value.Median polish is a technique that provides more information than just a p-value. It explains the contingency table in terms of an overall effect, row and columns effects and residuals. The underlying model for median polish is an additive model which is sometimes too restrictive. In this paper, we propose two related approach to generalise median polish. A power transformation can be applied to the values in the table, so that better results for median polish can be achieved. We propose a graphical method how to find a suitable power transformation. If the original data should be preserved, one can apply other transformations – based on so-called additive generators – that have an inverse transformation. In this way, median polish can be applied to the original data, but based on a non-additive model. The non-linearity of such a model can also be visualised to better understand the joint effects of rows and columns in a contingency table.

Highlights

Contingency tables often arise from collecting patient data and from lab experiments
The rows and columns of a contingency table correspond to two different categorical attributes
As an alternative, one can apply reversible transformations based on additive generators, leading to non-additive median polish

Summary

Introduction

Contingency tables often arise from collecting patient data and from lab experiments. The rows and columns of a contingency table correspond to two different categorical attributes. The rows correspond to six different groups The columns in this case reflect replicates. A typical question to be answered based on data from a contingency table is whether the rows or the columns show a significant difference. The underlying assumption for the t-test is that the data in the corresponding rows or columns originate from normal distributions. Median polish [6] – a technique from robust statistics and exploratory data analysis – is another way to analyse contingency tables based on a simple additive model. This backtransformation requires special transformations related to additive generators With such back-transformation the median polish result can be interpreted on the original data values as non-additive model.

G2 G3 G4 G5 G6 Column effect

Findings

Conclusions