Postgraduate research topics

We will be exploring (by simulation) the performance of GEEs for the analysis of unbalanced multivariate ANOVA designs.
No knowledge of GEEs is required, however a basic competence with R programming is essential.

Graphics systems do not generally provide primitives for drawing general smooth curves. The most commonly used technique is to appoximate the curve by a connected piecewise straight-line "polyline".

In R the curve defined by the function f over the interval [a,b] could be drawn as follows.

x = seq(a, b, length = 1000)
y = f(x)
plot(x, y, type = "n")
lines(x, y)

If the function f is discontinuous this approach fails (e.g. consider the function f(x) = 1/x on the interval [-1,1]).

The purpose of this topic is to explore techniques for
automatically detecting and locating function discontinuities and drawing the function over the intervals between the discontinuities.

The project could be taken further by developing methods for detecting and locating sharp "corners" in a function's graph. This amounts to finding the discontinuities in the derivative f'(x).

To take on this project you will need to be a competent R programmer (i.e. have completed or be doing 782) and be familar with notions like "discontinuity", "derivative", "smooth" etc.
 

The purpose of this project is to build a R library of functions for carrying out frequency-domain analysis of time series. This includes filtering, spectrum and cross-spectrum analysis. The basis of such
methodology is the fast-fourier transform (re)invented by John Tukey and collaborators. There are some existing time-series methods in R, but they are not particularly useful and are in need of extension.

To take on this project you will need to be a competent R programmer (i.e. have completed or be doing 782) and be familiar with the ideas of frequency-domain time series analysis.

The Gaussian copula is a multivariate model that says each variable can be independently transformed so that the distribution is multivariate Normal. This model is popular in finance and in bioinformatics. The Gaussian copula model implies that transforming each variable to Normal will increase the correlation, and the actual change in correlation provides a possible diagnostic. The project would involve both simulations and data analysis, and would require good computing and graphics skills.

Meta-analysis seeks to combine the results of several studies related to a particular research question. This is in order to obtain better estimates than could be obtained from a single study . Typically a common measure of effect size is used and a weighted average is calculated . The weighting takes account of differences between the various studies for example in sample size and demographic composition. Using data from several New Zealand longitudinal studies on children, the student will investigate and carry out the meta-analysis of selected health, educational and behavioural outcomes.

Balding (Likelihood-based inference for genetic correlation coefficients. Theoretical Population Biology, 2003, 63: 221-230) proposed a Bayesian method for estimating the coancestry coefficient. The coancestry coefficient can be regarded as a measure of low-level relatedness between individuals within a population, or alternatively, as a genetic distance between populations. Balding, Ayres and others encapsulated this method in a C programme called BayesFst (http://www.reading.ac.uk/Statistics/genetics/software.html). Part of this project will involve taking Professor Balding’s code and writing an R package to call it and simplify its use. The other part of this project will involve understanding the population genetics behind the programme and applying it to some data. One goal is to use multicore technology to run multiple chains of the MCMC sampler simultaneously. This project is suitable for students with very good programming skills and preferably with knowledge of C or a similar language.

Model selection for prediction is fairly well understood in independently-sampled data, with the Akaike Information Criterion (AIC) giving the best predictive model and the Schwartz Information Criterion (BIC) choosing the true model given enough data. Alastair Scott and I have ideas for how to extend these model-selection criteria to complex samples. The project would involve mostly simulations, but with some example analyses using real survey data. Needs good computing skills, STATS 330, should plan to take STATS 740.

An R package for the analysis of designed experiments

R has become the software of choice for many statistical applications. Yet, surprisingly, an important area of statistics in which there has been a lack of development of R functions is for the analysis of data from designed experiments. While the lm() and aov() functions in R, for example, can be used for this purpose, both have their limitations particularly if experimental design has a lot of structure. The goal of this project is to write a suite of R functions that enable users to define, in a straightforward manner, the structure of even the most complex experiments and which generate meaningful results for the purposes of designed experiments.
Pre-requisites: Completion of the 300-level experimental design paper and must be adept at writing functions in R.

It is now possible to measure the entire DNA sequence of a gene in a large number of individuals, and tests have been developed for assessing the importance of rare genetic variants. In some DNA resequencing studies that involve collaboration between multiple cohorts it is necessary to do the analysis separately within each cohort and combine the results. We need to know how much information is lost by this separate analysis and pooling. The project would mostly involve simulations, but there may be a real data example available towards the end. Needs good computing skills, STATS 310. Knowing what a gene is would be helpful but not completely necessary.

The R package xtable is capable of producing elaborately structured tables in LaTeX format, but only limited arrangements in HTML format. This project has the aim of allowing the production of far more sophisticated tables in HTML format using xtable. The student will also investigate other R packages.
Requirements: Should be enrolled in or have completed STATS 779. Completion of STATS 220 will also be advantageous.

There are a number of packages dealing with the generalized lambda distribution on CRAN (and one which is no longer on CRAN). This project will use existing work to bring some of this disparate code into a coherent form using the same structure as my other packages for distributions.
Requirements: STATS 310, and have enrolled in or completed either STATS 779 or STATS 782.

Financial data typically is heavy-tailed and skewed. This project will examine the fitting of multivariate data using marginals which are heavy-tailed and copulas.
Requirements: Should be enrolled in or have completed STATS 779. Some knowledge of financial statistics will also be advantageous.

Length of hospital stay is measured as the number of days from admission to discharge. As a data variable, it has particular characteristics: it is a count rather than continuous; and it has an atypical distribution skewed towards many short stays and with a long tail containing outliers with very long stays. Using New Zealand hospital admissions data, the student will investigate and carry out analyses of the determinants of length of stay, assessing the strengths and weaknesses of and comparing the results from various methods.

Peter Davis
pb.davis@auckland.ac.nz

Alastair Scott
a.scott@auckland.ac.nz

Each year in April / May / June, the electricity industry discovers the amount by which the winter demand for electricity has increased since the previous winter. Much of this increase is not predictable in advance, because it arises from new installations of equipment (e.g. for heating) that is used only in the winter months. It is of interest to estimate the size of the increase as early as possible in the winter, using data available at that time. Complications include weather (e.g. this winter may be starting out colder than last winter did) and public holidays (particularly Easter, the date of which varies from year to year).
Techniques: regression, time series, R.

(This might involve simulation studies and practical examples of the performance of different approaches to dealing with multiple case-mix factors (e.g ~30). Prognostic scores could be considered too).

Thomas Lumley
t.lumley@auckland.ac.nz

Alastair Scott
a.scott@auckland.ac.nz

Barry Milne
b.milne@auckland.ac.nz

Description: On the face of it BART should be a useful tool for estimating propensity scores. However initial experimentation with this approach suggested that it was inferior to logistic regression approaches in achieving covariate balance. Could further experimentation with settings within the BART method improve BART's performance (e.g number of depth of trees fitted) improve BART's performance or is there something else going on here? Could it be that sparsenesss in the data upsets BART or does the fact that covariate balance rather prediction, per se, is the aim of propensity score estimation mean that BART is less well suited to propensity score estimation than to pure prediction problems.

Thomas Lumley
t.lumley@auckland.ac.nz

Peter Davis
pb.davis@auckland.ac.nz

Description: Percolation describes a class of mathematical models for porous media. These models have been used to analyze such things as fractures in sea ice. Typical models are defined in terms of a single parameter p, and exhibit a ``phase transition’’ (as p increases from 0 to 1) at some critical value p_c.
We will investigate phase transitions of some dependent percolation models via probability theory and simulation.
Required background would be stats 320 or 325 and a 3rd year pure mathematics course.

This project will look at ways in which people display data about what is happening at different geographical locations to inform scope and design decisions for a data-displayed-on maps module for the iNZight package (http://www.stat.auckland.ac.nz/~wild/iNZight/) and build some prototypes. In addition to decisions about what gets displayed and how, thinking is needed about input-data formats that should be catered for.
Requirements: Stats 220 and Stats 380. [Some exposure to Geographic Information Systems would be useful background, but not necessary]

Alastair Scott
a.scott@auckland.ac.nz

Peter Davis
pb.davis@auckland.ac.nz

The accuracy with which phylogenetic relationships between species are estimated largely depends on models describing how substitutions accumulate among protein sequences over the course of evolution.
There is good evidence that the substitution process between amino-acids is _not_ Markovian. This project aims at designing computationally-tractable non-Markovian models of evolution and assess the fit of these models to real sequences.

Chris Triggs
cm.triggs@auckland.ac.nz

Stephanie Budgett/ Maxine Pfannkuch
s.budgett@auckland.ac.nz
m.pfannkuch@auckland.ac.nz

Stephanie Budgett/ Maxine Pfannkuch
s.budgett@auckland.ac.nz
m.pfannkuch@auckland.ac.nz

Renate Meyer
renate.meyer@auckland.ac.nz

Thomas Lumley
t.lumley@auckland.ac.nz

Outline: For non-Poisson point process models, the distribution of parameter estimates is analytically intractable and must be conducted via simulation methods. A fairly ``obvious'' method of constructing a
confidence interval for a parameter depends on the somewhat dubious assumption that the shape of distribution of the estimate is translation invariant. The basic idea of this project is to investigate, by simulation, just how well this method performs. I.e. when we compute, say, 95% confidence intervals what percentage coverage do we actually get? Also how does this performance vary with the model, with the true value of the parameter in question, with the sample size, and possibly with other relevant considerations?

Depending on time constraints and the student's inclinations we might also look for ways of improving the ``obvious'' method. For instance we might investigate whether the Effron and Tibshirani ``accelerated bias-corrected method'' of forming bootstrap confidence intervals can be applied in this context.

The only prerequisites for this project are a basic understanding of confidence intervals and a good facility for programming in the R language.