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Chapter 2 presents the theory behind the development of a multivariate logit model to be used with continuous compositional data. Estimation of the model parameters is car- ried out using the technique of generalized estimating equations with a working variance- covariance structure that reflects the properties of compositional variables. Different ways in which standard errors may be estimated are also explored and a new model-based variance estimator which ‘borrows strength across subjects’ (Liang and Zeger, 1986) is de- veloped. Measures which are appropriate for testing the quality of fit of the multivariate logit model for compositional data are also presented.

As mentioned in Section 1.2.2, the standard methodology used to model compositional response variables is that devised by Aitchison (1982, 1986). Chapter 3 will provide more detail on Aitchison’s regression method and it will show how Aitchison’s regression model relates to a multiplicative regression model that is introduced in Chapter 2. Despite be- ing two different methods, Aitchison’s method and the generalized Wedderburn method have some striking similarities of form. The formal similarities of the two approaches will be presented in this chapter together with an in-depth study of the properties of esti- mators obtained under the two approaches. An efficiency comparison between the GEE estimator used under the generalized Wedderburn method and the ML estimator used under Aitchison’s method is carried out using a small simulation study, under various sample sizes, coefficients of variation and correlation coefficients, with compositional data being generated through multivariate lognormally distributed latent variables. The gen- eralized Wedderburn method and Aitchison’s method are then compared on two widely used datasets from the compositional data literature, the Arctic Lake dataset (e.g. Aitchi- son, 1986; Tsagris et al., 2011; Maier, 2014) and the Foraminiferal dataset (e.g. Aitchison, 1986; Palarea-Albaladejo et al., 2007; Scealy and Welsh, 2011; Tsagris, 2015).

Chapter 4 makes some comparisons with Dirichlet models. Since the Dirichlet regression model may be specified using the same logit model that is estimated by the generalized Wedderburn approach, in Chapter 4 we first present some theoretical background on the Dirichlet regression model. A Dirichlet regression model is then fitted to the Arctic Lake dataset and the resulting fit is compared to that obtained by the generalized Wedderburn method. The estimates obtained from fitting the Dirichlet regression model to the Arctic Lake dataset are then used to generate data for a simulation study which compares the efficiency of the GEE estimator, used under the generalized Wedderburn approach, with

the ML estimator used in the Dirichlet regression model.

Chapter 5 contains a brief introduction to an early development version of thecglm pack- age. This R package may be used to fit the newly proposed generalized Wedderburn method and Aitchison’s multivariate regression model to compositional data. It also pro- vides basic tools for model summary and model criticism.

Finally, Chapter 6 contains a summary of the material presented in this thesis together with some concluding comments and suggestions for further studies.

Chapter 2

A Multivariate Generalized Linear

Model

2.1

Introduction

Amongst researchers of compositional data analysis, the method which is most likely to be used to model the influence of predictors on compositional response variables is that of logratio-transforming the data, assuming the distribution of the transformed data to be the multivariate normal distribution and then proceeding with using ordinary least squares estimation. However, as mentioned in Chapter 1, the logratio methodology fails when dealing with zero-valued responses. Also, the logratio methodology models the mean of the logratios, rather than the mean of the compositional response variables directly, so interpretation of regressions based on logratios is rather indirect.

In this work, a latent multiplicative regression model (MRM) is first introduced. This model is based on the consideration that in modeling compositional response variables, treating the effects and errors as multiplicative on the untransformed components is more suitable than treating them as additive. Also, rather than modeling transformed data, the MRM transforms the model expectations, in the already-familiar way that generalized linear models represent an alternative to data transformation prior to linear modelling. The fact that a multiplicative model is used to model compositional data is based on the analogy of the operation of perturbation 1 _{(Aitchison, 1986), which is a multiplicative}

operation in the simplex, with the operation of translation2_{, the latter being an additive} 1

Definition 2.1.1. The perturbation between any twoJ-part compositions Y∗ andYis defined by

Y∗⊕Y=C(Y1∗Y1, . . . , Y

∗

JYJ)

where ⊕is the notation that is typically used to denote the perturbation operation and C(·) denotes the closure operation that has been defined in (1.1).

2

operation in the real space.

The motivation for the MRM modeling the mean on the original scale comes from Firth (1987, 1988), who has shown that modeling the mean on the original scale through a multiplicative model rather than on the log-transformed data might yield better efficiency of the estimators, as well as overcoming the aforementioned problems of the analysis of logarithms.

The latent multiplicative regression model (MRM) is presented in Section 2.2. A brief note on identifying the parameters of the MRM is presented in Section 2.3. Section 2.4 focuses on parameter estimation. Since only the first two moments of the latent variables underlying the compositional response variables and no further distributional assumption is made in the specification of the latent MRM, it will be shown how quasi-likelihood estimation may be used to estimate the parameters in the MRM. Details on the general technique of quasi-likelihood estimation and properties of the quasi-likelihood estimator are provided in Section 2.4.1. Section 2.4.2 then explains how quasi-likelihood methods may be applied to estimate the parameters in the MRM. A quasi-likelihood estimator is robust to the specification of a covariance structure but this robustness does not extend to the estimated variance-covariance matrix of the quasi-likelihood estimator. This draw- back is overcome through using the technique of generalized estimating equations (GEE). The technique of generalized estimating equations uses the mean-variance specification of quasi-likelihood estimation but it is also able to cater for any correlation that may arise between the observed variables by introducing a working correlation matrix. Section 2.4.3 shows how generalized estimating equations may be applied to estimate the parameters in the multiplicative regression model. It will also be shown that the generalized least squares estimator which is used to estimate the parameters of interest is invariant under differ- ent dispersion and correlation parameters. Independence estimating equations with equal dispersion parameters may thus be used to estimate the model parameters, which makes this system of estimating the parameters very appealing. The problem with using such a system, however, is that the sum of the estimated means is not constrained to be equal to 1. Compositional response variables are sum constrained, so their estimated means should be constrained accordingly. An alternative new system of estimating equations, referred to by the namehybrid is developed in Section 2.4.5. The hybrid system retains the invariance property of the generalized least squares estimator for the parameters of interest whilst

denoteJ−1-vectors whose components are the logratios defined as logYj

YJ , logpj pJ and logY ∗ j Y∗ J , for j= 1, . . . , J−1, respectively. Then P∗+W∗ = log Y1Y1∗−1 YJYJ∗−1 + log Y1∗ Y∗ J , . . . , " log YJ−1Y ∗−1 J−1 YJYJ∗−1 ! + log _Y∗ J−1 Y∗ J #! = ... = log Y1 YJ , . . . ,log YJ−1 YJ = W,

also imposing the sum constraint on the estimated means. In Section 2.5, the estimating equations obtained under the hybrid system whenJ = 2 are also shown to be the same as Wedderburn’s estimating equations (Wedderburn, 1974), which were used for the analysis of barley leaf data. Through this equivalence, the hybrid system for J = 2 inherits all the desirable properties of Wedderburn’s quasi-likelihood estimator. Based on this devel- opment, in Section 2.6 a generalization of Wedderburn’s system of estimating equations to J > 2 is sought and developed by constructing generalized estimating equations for a multivariate logit model. A working variance-covariance structure that is suitable for modeling compositional response variables is then identified in Section 2.7. Different ways in which standard errors may be estimated are explored in Section 2.8. A new estimator of the standard errors which ‘borrows strength across subjects’ (Liang and Zeger, 1986) is developed in Section 2.8.2. The final section first presents different measures that are used in testing quality of fit in a typical GEE analysis. Subsequently, measures which are appropriate for testing the quality of fit of our logit model for compositional data are presented.

2.2

The Latent Multiplicative Regression Model (MRM)