Semester 1

The main aim of this thesis is to develop an automatic MCMC estimation procedure for STAR-GARCH models that can be applied to any data set that could normally be modeled by any of the sub-classes of models that the STAR-GARCH model generalises. This will remove the need for linearity testing and model specification. The project will achieve three specific objectives:
(i) to formulate a fully Bayesian analysis of univariate STAR models, followed by univariate and multivariate STAR-GARCH models respectively;
(ii) to perform posterior analysis obtained in (i) by creating an efficient MCMC algorithm and using simulation studies, implemented by novel computer code in R;
(iii) to apply the MCMC estimation algorithm to real world data to examine regime-switching and varying volatility in the data.

The log-linear models have been a significant area of research in the field of categorical data analysis since the 1950s. However, until the mid-1970s, log-linear models only considered the modelling of nominal variables and did not make any assumption about the ordering of categories of an ordinal variable. Therefore, the log-linear models have been modified to incorporate the structure of any ordinal variable. This issue is especially relevant in most fields of social science. The ordinal log-linear models are amid the most widely used and powerful techniques to model association among the ordinal variables in categorical data analysis. Traditionally, the parameters from such models are estimated using iterative algorithms (such as the Newton-Raphson method, and iterative proportional fitting), but issues such as choice of poor initial values and contingency tables of larger dimensions can reduce the convergence rate as well as highly increase the number of iterations required for the algorithms to converge.

More recent advances have suggested a method of non-iterative estimation that gives numerically similar estimates as that of the iterative methods for the estimation of linear- by-linear association parameter in an ordinal log-linear model for a two-way table. This presentation will highlight the iterative and non-iterative techniques commonly used to estimate the linear-by-linear association parameter from two-dimensional ordinal log- linear models. It will provide an overview of how the growing number of non-iterative estimation techniques fit into the problem. Several possibilities to extend the research on the non-iterative estimates in order to validate their further use are discussed. The presentation will also highlight the research undertaken so far to achieve this objective. This includes considering the two fundamental estimates for the analysis of the association between two categorical variables forming a contingency table and to determine their asymptotic characteristics. A computational study is carried out for contingency tables of varying sizes to show that these two estimates are asymptotically unbiased. It is also shown that both estimates are asymptotically normally distributed. On the basis of the standard errors, their relative efficiency has been established for 13 commonly analysed contingency tables that appear throughout the literature.

Keywords: Ordinal log-linear models, non-iterative estimation, linear-by-linear association parameter, orthogonal polynomials, Newton Raphson, Iterative proportional fitting.

When analysing data that is made available from many government and corporate organisations, often it is de-identified so that the information on specific individuals is unknown. This ensures that strict privacy policies from individuals are maintained. In such cases, aggregate is all that is available to the analyst. Therefore the analysis of aggregate data has received increasing attention in the statistical, and allied, discipline over the last decade. This data is often summarised in the form of stratified 2x2 tables. Despite its relative youth in the statistical research, there is a wealth of literature in ecological inference (EI) that considers the association structure between categorical variables given only the aggregate information. However, current EI techniques suffer from major shortfalls in the assumptions that are required to be met. Recently, the development of aggregate association index (AAI) allows the analyst to quantify the overall extent of association between two dichotomous variables such that the assumptions underlying EI techniques are not violated. This presentation considers a number of issues concerning the analysis of association for aggregate with the AAI at the heart. One issue is the impact of the sample size on the magnitude of the AAI; two different strategies are suggested to cope with this issue. An alternative approach to dealing with the sample size issue is to consider a new index, referred to as the aggregate informative index (AII). The AII quantifies the extent of information that lies at the marginal level of data for 2x2 contingency tables. The F-statistic is also considered to evaluate the statistical significance of the information. We shall use Selikoff’s (1981) asbestosis data to demonstrate the applicability of AII and F-statistics. On the other hand, an adjustment is proposed to minimize the effect of sample size on the AAI directly, leading to the adjusted AAI. This index is considered in the context of Fisher’s (1935) criminal twin data.

Key words: Aggregate association index, aggregate data, ecological inference, F-statistic, 2x2 contingency tables

Missing data are a common problem in longitudinal research and if not dealt with correctly can lead to biased parameter estimates and variances. Incorrect methods for dealing with missing data are still common place. This research will be based on analysis of the old cohort (70-75yrs at baseline) of the Australian Longitudinal Study of Women’s Health (ALSWH; www.ALSWH.org.au ). Sleep data from five surveys will be used to extract latent sleep patterns. The relationship between sleep and other health measures such as morbidities and mortality will be investigated.

There is a highly complex pattern of missing data within the ALSWH data set. This complexity consists of dropout due to death/ill-health, loss to follow-up, other dropout (random/non-random), survey non-response (random/non-random), missing covariates data as well as missing responses (random/non-random), item non-response (random/non-random) and mixed variable metrics. No current method for dealing with missing data fully addresses the complexity of missing data present in the ALSWH data set.

The novel contribution of this research will therefore be the development of an analytical approach that deals with this complexity. This research will combine and build on the current literature involving pattern mixture models (PMM). It will combine features of the mixture-PMM, Bayesian estimation, and the Multiple Imputation PMM (MI PMM), and extend them by allowing for both Missing at Random (MAR) and Not Missing at Random (NMAR) dropout, as well as allowing for MAR and NMAR intermittent missing. Finally, it may be of use to combine the developed PMM with a joint model, using multi-state modelling or survival analysis to model the dropout due to death/ill- health. No previous work has attempted to tackle comprehensively such a complex array of missing data.

Keywords: longitudinal, missing, PMM, latent class, missing covariates, monotone, non-monotone

For Bayesian estimation of the binomial parameter, when the aim is to "let the data speak for themselves", the uniform or Bayes-Laplace prior appears preferable to the reference/Jeffreys prior recommended by objective Bayesians like Berger and Bernardo.

Here confidence intervals tend to be "exact" or "approximate", aiming for either minimum or mean coverage to be close to nominal. The latter criterion tends to be preferred, subject to "reasonable" minimum coverage. I will first re-iterate examples of how the highest posterior density (HPD) credible interval based on the uniform prior appears to outperform both common approximate intervals and Jeffreys prior based intervals, which usually represent credible intervals in review articles.

Second, an important aspect of the recommended interval is that it may be seen to be invariant under transformation when taking into account the likelihood function. I will also show, however, that this use of the likelihood does not always lead to excellent, or even adequate, frequentist coverage.

Third, this approach may be extended to nuisance parameter cases by considering an "appropriate" likelihood of the parameter of interest. For example, quantities arising from the 2x2 contingency table (e.g. odds ratio and relative risk) are important practical applications, apparently leading to intervals with better frequentist performance than that found for HPD or central credible intervals. Preliminary results suggest the same for "difficult" problems such as the ratio of two Normal means ("Fieller-Creasy") and the binomial N problem.

Jim’s research deals with four aspects of the analysis of some very large samples or indeed population data on surname frequencies:

• Models of the number of surnames which occur k times: ‘frequencies-of-frequencies’;
• Estimating the number of surnames in a population from the frequencies in one or more samples: the ‘number of species’ problem in another guise;
• Surname extinction and mutation; and
• Rank-frequency distributions.

There has been a recent revival in interest in the mathematics of surname frequencies, much of which has been published in the physics literature (for a good reason!). The earliest papers were in the 1840s, followed by two important papers in 1875 and two in 1931, after which the topic was largely forgotten until about 2001.

After outlining his proposed sequence of thesis chapters and the other items covered in the written report prepared for the Confirmation Committee, Jim will outline what he sees as the problem with current methods of identifying the form of distribution for rank-frequency distributions fitted to discrete data (such as word counts) and parameter estimation methods. The erroneous identification and estimation methods date from George Zipf’s 1932 book on word frequencies and a misunderstanding of a paper by Benoit Mandelbrot.

Statistical models are an essential part of data analysis across many diverse fields. However it is important to critically assess any fitted model, confirming that the model really is compatible with the data, before meaningful conclusions are possible. Generalized linear models (GLMs) provide a flexible modelling framework encompassing many commonly used models including the normal linear model, logistic regression model and Poisson regression model.

This talk will describe how the smooth testing concept (originally proposed by Neyman (1937) and further developed by Rayner et al. (2009) among others) can be used to test the distributional assumption in a GLM. This will summarize some of the key points from the recently submitted PhD thesis including results of power and size studies, the practical interpretation of smooth tests in the model development context, and the development of an R package which implements the smooth test in a form that can be easily applied to models fitted using the standard glm() function within the R statistical computing environment.