The Asymptotic Distribution of the Weighted Altham's Index in Log-Ratio Analysis

Abstract

Log-ratio analysis is a well-known framework for investigating and modeling compositional data. This method utilizes log-ratio transformations. The vectors connecting points on the maps illustrate the logarithmic relationships between data values in corresponding rows or columns. Correspondence analysis also creates a map where the proximity of points and other geometric features of the map reflect relationships between rows, between columns, and between rows and columns. Indeed, both methods share a similar theory, allowing for a graphical display of the association between the variables. While it is possible to verify in correspondence analysis the significance of the association between the variables, as well as between each row and column category, it seems not to be possible to perform the same inferential analyses within the log-ratio analysis. The investigative capabilities of the log-ratio analysis are then limited to graphical visualisation alone. To overcome this drawback, we introduce the asymptotic distribution of the weighted Altham's index (at the heart of the weighted log-Rratio analysis) under a Poissonian model and develop confidence circles for each row and column category of this approach.