Compositional data analysis: where we are and where should we be heading?

We take stock of the present position of compositional data analysis, of what has been achieved in the last 20 years, and then make suggestions as to what may be sensible avenues of future research. We take an uncompromisingly applied mathematical view, that the challenge of solving practical problems should motivate our theoretical research; and that any new theory should be thoroughly investigated to see if it may provide answers to previously abandoned practical considerations. Indeed a main theme of this lecture will be to demonstrate this applied mathematical approach by a number of challenging examples

Compositional Data in Biomedical Research

Modern methods of compositional data analysis are not well known in biomedical research. Moreover, there appear to be few mathematical and statistical researchers working on compositional biomedical problems. Like the earth and environmental sciences, biomedicine has many problems in which the relevant scienti c information is encoded in the relative abundance of key species or categories. I introduce three problems in cancer research in which analysis of compositions plays an important role. The problems involve 1) the classi cation of serum proteomic pro les for early detection of lung cancer, 2) inference of the relative amounts of di erent tissue types in a diagnostic tumor biopsy, and 3) the subcellular localization of the BRCA1 protein, and it's role in breast cancer patient prognosis. For each of these problems I outline a partial solution. However, none of these problems is \solved". I attempt to identify areas in which additional statistical development is needed with the hope of encouraging more compositional data analysts to become involved in biomedical research

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing one or more parameters in their definition. Methods that can be linked in this way are correspondence analysis, unweighted or weighted logratio analysis (the latter also known as "spectral mapping"), nonsymmetric correspondence analysis, principal component analysis (with and without logarithmic transformation of the data) and multidimensional scaling. In this presentation I will show how several of these methods, which are frequently used in compositional data analysis, may be linked through parametrizations such as power transformations, linear transformations and convex linear combinations. Since the methods of interest here all lead to visual maps of data, a "movie" can be made where where the linking parameter is allowed to vary in small steps: the results are recalculated "frame by frame" and one can see the smooth change from one method to another. Several of these "movies" will be shown, giving a deeper insight into the similarities and differences between these methods