Modelling non-standard longitudinal outcomes using fast Bayesian inference | Courses

Overview

Repeated observations over time on individual patients allows the opportunity to analyse factors that influence the changes over time in a variable of interest. For example, we may be interested in how a particular treatment received influences the level of a particular blood biomarker, or a count of the number of seizures experienced. Many longitudinal outcomes are commonly analysed using generalised linear mixed models. Bayesian methods such as the popular Markov Chain Monte Carlo (MCMC) are a common way of estimating parameters in such models.

About this opportunity

Most current software for modelling longitudinal outcomes is limited to a few standard types of outcomes, such as Gaussian models for continuous data, Bernoulli models for binary data and Poisson models for counts. However, other types of outcomes are possible including categorical and ordinal outcomes or binary and count variables that display overdispersion (more variability than would be expected given the standard Bernoulli or Poisson models). Models exist within the statistical literature to account for these various features of outcome data.

However, in large datasets many of these models can be computationally prohibitive to fit, with MCMC approaches taking a very long time to fit. One potential solution to this is through techniques such as Variational Bayes inference. These approaches aim to develop approximate solutions to the desired posterior distribution by using products of simpler distributions, and can often provide very accurate solutions much more quickly than MCMC.

This project will aim to develop variational approximations for various non-standard longitudinal outcomes. Possible types of outcomes include ordinal and multinomial models for categorical longitudinal outcomes, discrete Weibull models for overdispersed counts, and beta-binomial models for overdispersed binary variables.

In each case models will be tested using extensive simulation studies, and through analysis of real large datasets, to allow for a thorough assessment of the speed gains, and the accuracy of the variational approach. Examples of clinical applications for this work include the use of ordinal mixed models for analysing changes in diabetic retinopathy screening grading over time, and the use of discrete Weibull models to assess overdispersion in counts of seizures over time in epilepsy patients. The project could also consider inclusion of survival data with non-standard longitudinal outcomes within a joint modelling framework. A useful contribution of the project will be the production of R code to allow users to fit these advanced models.

Who is this opportunity for?

This project is open to self-funded UK and international applicants.

How to apply

1. Contact supervisors

Please email your CV and cover letter to the primary supervisor, Dr David Hughes, in the first instance: dmhughes@liverpool.ac.uk.
2. Prepare your application documents

You may need the following documents to complete your online application:
- A research proposal (this should cover the research you’d like to undertake)
- University transcripts and degree certificates to date
- Passport details (international applicants only)
- English language certificates (international applicants only)
- A personal statement
- A curriculum vitae (CV)
- Contact details for two proposed supervisors
- Names and contact details of two referees.
3. Apply

Finally, register and apply online. You'll receive an email acknowledgment once you've submitted your application. We'll be in touch with further details about what happens next.

The application deadline is 8 April 2026.

Fees and funding

Your tuition fees, funding your studies, and other costs to consider.

Tuition fees

UK fees (applies to Channel Islands, Isle of Man and Republic of Ireland)

Full-time place, per year - £5,006

International fees

Full-time place, per year - £31,250

Fees stated are for the 2025-26 academic year.

Additional costs

We understand that budgeting for your time at university is important, and we want to make sure you understand any costs that are not covered by your tuition fee. This could include buying a laptop, books, or stationery.

Find out more about the additional study costs that may apply to this project, as well as general student living costs.

Funding your PhD

This opportunity is for students with their own funding. Funding should cover course fees, living expenses and research expenses (bench fees).

If you're a UK national, or have settled status in the UK, you may be eligible to apply for a Postgraduate Doctoral Loan worth up to £30,301 to help with course fees and living costs.

There’s also a variety of alternative sources of funding. These include funded research opportunities and financial support from UK research councils, charities and trusts. Your supervisor may be able to help you secure funding.

We've set the country or region your qualifications are from as United Kingdom.

Scholarships and bursaries

We offer a range of scholarships and bursaries that could help pay your tuition fees and living expenses.

Duncan Norman Research Scholarship

If you’re awarded this prestigious scholarship, you’ll receive significant funding to support your postgraduate research. This includes full payment of your PhD fees and a cash bursary of £17,000 per year while you study. One award is available in each academic year.

John Lennon Memorial Scholarship

If you’re a UK student, either born in or with strong family connections to Merseyside, you could be eligible to apply for financial support worth up to £12,000 per year for up to three years of full-time postgraduate research (or up to five years part-time pro-rata).

Sport Liverpool Performance Programme

Apply to receive tailored training support to enhance your sporting performance. Our athlete support package includes a range of benefits, from bespoke strength and conditioning training to physiotherapy sessions and one-to-one nutritional advice.

Discover our full range of scholarships and bursaries.

Contact us

Have a question about this research opportunity or studying a PhD with us? Please get in touch with us, using the contact details below, and we’ll be happy to assist you.

Dr David Hughes

Modelling non-standard longitudinal outcomes using fast Bayesian inference.