How To Get Along With Sum 1: Generalized Linear Models for Compositional Data Seminar

Multivariate measurements in the form of a composition, i.e., a vector of proportions or percentages that sum to 1, are of interest in a wide variety of application fields. A classic application area is geology, where rock samples are weighed into their constituent parts (e.g., different minerals). Other examples include elections (percent votes for different parties), geography (percent of land area used for different purposes), social research (percent of time spent on paid work, caring, leisure, etc) and of course statistics (probabilities on a finite sample space). We consider regression analysis, where interest is in the dependence of a composition upon explanatory variables. The most standard approach is from well-known work by J Aitchison: use standard linear models, with multivariate normal assumptions, for a reduced vector of /log-ratios/ of components. Aitchison's approach overcomes the inherent difficulty that the components sum to 1, by analysing only their relative values (on the log scale). In our work reported here, an alternative approach is developed: a natural generalized linear model which avoids data-transformation. This new approach extends the seminal work of Wedderburn (1974, Biometrika): it overcomes the usual difficulties with interpretation of linear models for log-measurements, and difficulties also when there are zeros in the data. (Joint work with a PhD student, Fiona Sammut)