New Methods for Modelling Dependent Data
Creators: name: Tho, Zhi Yang
Publisher: name: The Australian National University
Year: 2024

description: The analysis of dependent data is becoming increasingly important, as these data provide unique opportunities to solve diverse scientific problems of interest across disciplines such as ecology, meteorology, epidemiology and political science. This thesis aims to develop and investigate various novel statistical methods to improve the modelling and utilisation of the underlying dependence structures of dependent data, in particular correlated multivariate or multi-response data and spatial data. Chapter 2 proposes a new joint mean and correlation regression model for general multivariate data, which allows simultaneous analysis of the relationship between the mean vector with observed covariates, and the association between the correlation matrix (between responses) and similarity measures of additional predictor information. The ability of the proposed model to handle a wide variety of discrete and (semi-)continuous responses is demonstrated through its application to two multivariate abundance datasets in ecology consisting of overdispersed counts of Carabidae ground beetle species in Scotland, and presence-absence records of alpine plant species in France. The joint estimator developed for the mean regression coefficients and correlation regression parameters is demonstrated to be consistent and asymptotically normally distributed, with differing rates of convergence under a general setting where the number of responses tend to infinity. Chapter 3 focuses on modelling the conditional dependence structure in multivariate binary data, motivated by a real application aiming to understand how the dependence between different U.S. senators' votes is determined by similarities in their attributes. An Ising similarity regression model that regresses pairwise interaction coefficients in the Ising model against a set of similarity measures available/constructed from covariates is proposed. A regularised pseudo-likelihood estimator using the adaptive lasso penalty is further developed to facilitate the selection of relevant similarity measures; estimation and selection consistency are established under a setting where the number of similarity measures and responses tend to infinity. Chapter 4 proposes a cokrig-and-regress (CNR) method to overcome the spatial misalignment problem between the response and the covariates in spatial linear mixed models. This makes use of the spatial dependencies within each covariate and between covariates through a generalised Kronecker product spatial cross-covariance structure. A parametric bootstrap procedure is also employed to bias-correct the CNR estimates of the spatial covariance parameters, and for uncertainty quantification. Applying CNR to the spatially misaligned air pollution and weather data in China reveals a number of non-linear relationships between PM2.5 concentration and several meteorological covariates, as well as the importance of accounting for both the additional variability due to spatial misalignment correction and potential cross-covariances between covariates when it comes to statistical inference for spatially misaligned data.