Bayesian spatio-temporal methods for small-area estimation of HIV indicators

Note

If you’re interested in my advice on doing a PhD, see this blog post!

Tip

I have made the completed corrections for this thesis available. There are various other enhancements I would ideally like to make, but as they say “a good thesis is a done thesis”.

PhD in Modern Statistics and Statistical Machine Learning at Imperial College London.

Available as: HTML or PDF.

Supervised by: Seth Flaxman and Jeff Eaton.

Progress towards ending AIDS as a public health threat by 2030 is not being made fast enough. Effective public health response requires accurate, timely, high-resolution estimates of epidemic and demographic indicators. Limitations of available data and statistical methodology make obtaining these estimates difficult. I developed and applied Bayesian spatio-temporal methods to meet this challenge. First, I used scoring rules to compare models for area-level spatial structure with both simulated and real data. Second, I estimated district-level HIV risk group proportions, enabling behavioural prioritisation of prevention services, as put forward in the UNAIDS Global AIDS Strategy. Third, I developed a novel deterministic Bayesian inference method, combining adaptive Gauss-Hermite quadrature with principal component analysis, motivated by the Naomi district-level model of HIV indicators. In developing this method, I implemented integrated nested Laplace approximations using automatic differentiation, enabling use of this algorithm for a wider class of models. Together, the contributions in this thesis help to guide precision HIV policy in sub-Saharan Africa, as well as advancing Bayesian methods for spatio-temporal data.

Chapters

	Title	GitHub repository	Journal
1	Introduction
2	The HIV/AIDS epidemic
3	Bayesian spatio-temporal statistics
4	Models for areal spatial structure	beyond-borders	In preparation!
5	A model for risk group proportions	multi-agyw	PLOS Global Public Health
6	Fast approximate Bayesian inference	naomi-aghq	In preparation!
7	Conclusions
A	Appendix to models for areal spatial structure
B	Appendix to a model for risk group proportions
C	Appendix to fast approximate Bayesian inference

Citation

If you would like to cite this work, please use:

@phdthesis{howes23,
  author = {Howes, Adam},
  school = {Imperial College London},
  title = {Bayesian spatio-temporal methods for small-area estimation of HIV indicators},
  year = {2023}
}

Slides

Slides for my thesis defense are available here. They may be useful to provide a brief overview of the research. For more detailed slides, see this presentation.

Frequently asked questions

How can I read the thesis?

Thanks for being interested! You can read either the HTML or PDF version. I know, an overwhelming choice. Depending on my appetite, it may be improved post-defense in March 2024.

How did you format this thesis?

I used the R package thesisdown, inspired by bookdown. So far it has been working relatively seamlessly, so I’d recommend it! I’ve found gt together with knitr::is_html_output and knitr::is_latex_output works well to present tables nicely in multiple formats.

Are there any resources you’d recommend for an introduction to this area of research?

I’d recommend something like Spatial and Spatio-temporal Bayesian models with R-INLA by Marta Blangiardo and Michela Cameletti. I have a repository with further miscellaneous recommended resources, if you are interested.

I’m a statistician: which parts of the thesis might interest me?

If you use spatial random effects to model areal spatial structure, Chapter 4. This chapter uses a variety of model comparison techniques (scoring rules, coverage assessments, information criteria) that may also be of interest. If you’re interested in modelling multinomial data using the multinomial-Poisson transformation and structured random effects, Chapter 5. If you have a complicated model which is not compatible with R-INLA, but would still like to use INLA-like methods, Chapter 6. If you have such a model, get in touch! All of the methods are compatible with any model written in the (very general) Template Model Builder R package (TMB).

I’m a HIV epidemiologist: which parts of the thesis might interest me?

Primarily Chapter 5 will be of interest to you. The Global AIDS Strategy sets out goals for prioritisation of prevention programming for adolescent girls and young women according to risk behaviour and epidemic setting. To enable implementation of the strategy, I estimated risk group specific population sizes, prevalence and incidences at a district-level. I also used these estimates to evaluate the extent to which risk varies by age, behaviour, and geographic area. This work forms that basis for the “sub-national HIV estimates in priority populations” UNAIDS tool. You may also be interested in the analysis in Chapter 6 applying the Naomi small-area estimation model to data from Malawi.

Enhancements

Here are a collection of enhancements I think would improve this thesis:

Chapter 2

• The sections about challenges and statistical approaches used to overcome those challenges could be 1) better connected to the work done in the thesis, and 2) better integrated with existing literature. Doing so is relatively challenging 1) because this chapter precedes proper introduction of the methods used in this thesis, and is instead meant to provide a high-level overview, and 2) because the statistical approaches described e.g. “borrowing information” are relatively general and would be difficult, though not impossible, to credit to any particular works.

Chapter 3

• The writing throughout Chapter 3 is weak in places. Particularly the sections on 1) deterministic Bayesian inference methods (difficult to know how much to say given focus of later chapters on this material) 2) properties of spatio-temporal data, 3) aspects of the survey section.
• In Chapter 3 I follow other authors in using the notation $u_k(w_{ki})$ to refer to random effects. It would be good to connect up this functional notation with specifying random effects as $u_i$ rather than some function of some covariates.
• In Chapter 3, it would be nice to include a figure illustrating the DHS sampling procedure.

Chapter 4 and Appendix A

• The simulation study was run using 250 replicates. As you can see from the plots showing the mean and standard errors, this sample size was insufficient to distinguish between models in some cases. All the more so zooming into single areas. It would be relatively simple to increase the sample size here, but this wasn’t done in the interests of time.
• For the simulation study on the four vignette geometries, the lengthscale priors are mis-specified with respect to the true lengthscale. This seems like an odd choice. Likely these experiments should be rerun simulating data from a more suitable lengthscale than the value 2.5 used currently.
• In Chapter 4, it would be useful to frame the Besag model (and BYM2, if possible) in terms of an equivalent kernel. I believe that the technical vignette Paciorek (2008) does this.
• In Chapter 4, calculating the DIC and WAIC values for each of the fitted models would be informative as to the possible benefits of the other model comparison techniques used in the chapter. This would require writing a function to take a model fitted using TMB or aghq and output the model comparison criteria. Likely the best approach would be to use samples, as this is the most transferable way.

Chapter 5 and Appendix B

• Too little emphasis is placed on the HIV prevalence and HIV incidence results, as compared to the HIV risk group results. For example, continental choropleths could be produced for the these epidemiological quantities as well.
• Chapter 5 could benefit from more discussion of the statistical results and conclusions from the work. Some of this work is already done in my retrospective blog post about the work.

Chapter 6 and Appendix C

• A more thorough description of the approximations to the Laplace approximation used by Rue, Martino, and Chopin (2009) and Wood (2020). It would be instructive to implement these approximations for a simple example. Additionally, a more complete description of the “augmenting the latent field” issue (perhaps a simple example would be instructive here too).
• For all figures showing the use of a quadrature rule, it could be informative to compute and display the resulting integral estimate. When compared to a known truth, this would make demonstrate the value of e.g. adaption.
• Inclusion of some broader discussion of the value of automatic differentiation for INLA-like inference strategies. See the conversation I began here on the R-INLA Google group.
• Further detail about how automatic differentiation works might be helpful. This could include a simple example. See this nice blog post “symbolic differentiation in a few lines of code” by Rich FitzJohn.
• Although the epilepsy example shows that the INLA results from TMB are highly comparable to R-INLA, they are not exactly the same. As such it would be valuable to provide an explanation for the possible causes. There are things that R-INLA does that I have not talked about. The best source of information about this is Osgood-Zimmerman and Wakefield (2022).
• Running NUTS via tmbstan for Appendix A I found that some of the chains hung for a very long time. This is suggestive of the posterior geometries being tricky. In part this is confusing because I have previously run some of these models, implemented in rstan directly, without trouble. (See the tmbstan implementations here, and the rstan implementations here, all as part of the arealutils R package). It could be beneficial to 1) try to understand why it is that sometimes the chains hang 2) repeat the comparison using rstan versions of the models. The challenge in doing 2) is that then the guarantee that the models are the same is lost. That said, I have previously overcome this by taking parameters, evaluating their log-posterior under the TMB and rstan C++ templates, and making sure that they are identical up to a additive constant (this is on the log-scale, remember). For example, I did this in the case of the epilepsy GLM. Thinking about it more, this would be a valuable exercise in any case, to ensure that the implementations in the package are consistent. (I don’t think there is a way to evaluate the objective function corresponding to an R-INLA model, but if there were that would be great to do too.)
• For the figure in Chapter 6 showing the CCD grid, I think that these points (produced using rsm::ccd) should have associated weights (and therefore be different (aes(size = ...))) but I am unsure as to how to generate the weights. There is a section in Rue, Martino, and Chopin (2009) which could be useful.
• I ran Laplace marginals with emprirical Bayes for the Naomi ELGM but did not have enough time to present the results and integrate them into the discussion. This would be of interest to do.
• For the section about Naomi as an ELGM, it would be valuable to note 1) which features are possible in R-INLA (and exactly why or why not) 2) whether the feature is an important non-linearity of the model or more of a trivial notation issue.
• At one point I was interested in the kernel stein discrepancy as a way to measure distances between (samples from) distributions. I would have been interested to read more about these measures and their relation to MMD.

General

• Writing of the results and conclusions sections for Chapters 4 and 6 was relatively rushed. As such, it’s likely that a more thorough job could be done interpreting the results and connecting them to key takeaways.
• Rendering the PDF version, figures and tables tend to move around a lot, especially in the appendix. It would be good to have been control over this, but I know that this can be challenging in LaTeX.
• Note to check that references are displayed in a systematic way. Are there any better settings than the one that I have currently?