Using generalized linear models to model compositional response data

Abstract

This work proposes a multivariate logit model which models the influence of explanatory variables on continuous compositional response variables. This multivariate logit model generalizes an elegant method that was suggested previously by Wedderburn (1974) for the analysis of leaf blotch data in the special case of J = 2, leading to our naming this new approach as the generalized Wedderburn method. In contrast to the logratio modelling approach devised by Aitchison (1982, J. Roy Stat. Soc. B.), the multivariate logit model used under the generalized Wedderburn approach models the expectation of a compositional response variable directly and is also able to handle zeros in the data. The estimation of the parameters in the new model is carried out using the technique of generalized estimating equations (GEE). This technique relies on the specification of a working variance-covariance structure. A working variance-covariance structure which caters for the specific variability arising in compositional data is derived. The GEE estimator that is used to estimate the parameters of the multivariate logit model is shown to be invariant to the values of the correlation and dispersion parameters in the working variance-covariance structure. Due to this invariance property and the fact that the estimating equations used under the generalized Wedderburn method are linear and unbiased, the GEE estimator achieves full efficiency across a wide class of potential dispersion and correlation matrices for the compositional response variables. As for any other GEE estimator, the estimator used in the generalized Wedderburn method is also asymptotically unbiased and consistent, provided that the marginal mean model specification is correct. The theoretical results derived in this thesis are substantiated by simulation experiments, and properties of the new model are also studied empirically on some classic datasets from the literatur