add info on limitations of ICAR

thesis/Chapters/Chapter2.qmd

@@ -48,14 +48,15 @@ An example in @elliottSpatialEpidemiologyMethods2001 chooses an exponential deca
 
 #### Space as discrete units {-}
 
-A more popular prior is the conditional autoregressive (CAR) prior, also known as a Gaussian Markov random field, which was first introduced by @besagSpatialInteractionStatistical1974.
+A more popular prior is the conditional autoregressive (CAR) prior, also known as a Gaussian Markov random field (GMRF), which was first introduced by @besagBayesianImageRestoration1991.
 These form a joint distribution as in @eq-MVN, but the covariance is usually defined instead in terms of the precision matrix
 $$
 \mathbf{P} = \pmb{\Sigma}^{-1} = \tau(\mathbf{D} - \rho \mathbf{A}),
 $$ {#eq-CAR-prec}
 where $\tau$ controls the overall precision of the effects, $\mathbf{A}$ is the spatial adjacency matrix formed by the small areas, $\mathbf{D}$ is a diagonal matrix with entries equal to the number of neighbours for each spatial unit, and the autocorrelation parameter $\rho$ describes the amount of correlation.
 This can be seen as tuning the degree of spatial dependence, where $\rho = 0$ implies independence between areas, and $\rho = 1$ full dependence.
 The case with $\rho = 1$ is called the intrinsic conditional autoregressive (ICAR) model.
+There sometimes exists further over-dispersion in the residuals that cannot be modelled by purely spatially-structured random effects.
 @besagBayesianImageRestoration1991 proposed the model (hereafter called BYM)
 $$
 S_i = U_i + V_i,
@@ -73,6 +74,8 @@ Policy is decided at these geographies, so there is reason to believe these boun
 Note, although these models group by geographical region, these models are not _spatial_ as they do not contain any information on the relative position of the areas.
 
 Of the two specifications that are spatial, either as a continuous process or discrete units, the Markov random field priors are often preferred for computational reasons, as we can exploit the sparseness of the adjacency matrix in our inference algorithms rather than computing the covariance between each pair of spatial units as in the general case of @eq-MVN.
+There are concerns, however, that the GMRF representation of space as an adjacency matrix, which was originally proposed for a regular lattice of pixels in image analysis, is reductive for more complicated spatial problems.
+Despite this, in an epidemiological context, @duncanSpatialSmoothingBayesian2017 found the standard ICAR model with binary, first-order neighbour weights outperformed models with a variety of different weighting schemes, including matrix weights based on higher-order degrees of neighbours, distance between neighbours, and distance between covariate values.
 
 In applications to disease mapping, spatial models are the natural choice when the disease exhibits a spatial pattern.
 This is the case for mortality from infectious diseases, particularly on short timescales like Covid-19 [@konstantinoudisRegionalExcessMortality2022].

thesis/Chapters/Chapter7.qmd

@@ -272,7 +272,7 @@ Both mortality from ischaemic heart disease and strokes have continued to follow
 These reflect improvements in reducing and controlling risk factors such as high blood pressure and high blood cholesterol, organisational changes to the NHS such that acute CVD episodes are treated in specific centres, improvements in the treatment of CVD including coronary angiographies and stent insertion, and public health campaigns such as FAST (Face drooping, Arm weakness, Speech difficulties, Time) so the general public know when to seek emergency help for a stroke.
 
 Although the management of CVDs has improved over the past decades, the burden of mortality has shifted towards dementias.
-This reflects that, beyond age and family history, the main risk factors for dementias are the same as for CVDs (smoking, obesity, diabetes, high blood pressure, high cholesterol).
+This reflects that, beyond age and family history, the main risk factors for dementias are the same as for CVDs (smoking, obesity, diabetes, high blood pressure, high cholesterol) [@yuEvidencebasedPreventionAlzheimer2020].
 Some part of this trend may also be due to increased diagnosis and coding of deaths as dementias, with doctors increasingly assigning mental and neurological conditions as the underlying cause of death rather than simply "dying of old age".
 
 These contrasting trends in mortality from ischaemic heart disease and dementias could explain the finding that these causes of death have largely driven the heterogeneity of the slowdown in life expectancy gains since around 2010 at varying rates across districts, suggesting that CVD risk factors within the population have influenced the inequality in progress in recent years.

thesis/Frontmatter/abbreviations.tex

@@ -6,6 +6,7 @@
     \textbf{CVD} & \textbf{C}ardio\textbf{v}ascular \textbf{D}isease\\
     \textbf{GBD} & \textbf{G}lobal \textbf{B}urden of \textbf{D}isease\\
     \textbf{GHE} & \textbf{G}lobal \textbf{H}ealth \textbf{E}stimates\\
+    \textbf{GMRF} & \textbf{G}aussian \textbf{M}arkov \textbf{R}andom \textbf{F}ield\\
     \textbf{ICAR} & \textbf{I}ntrinsic \textbf{C}onditional \textbf{a}uto\textbf{r}egressive\\
     \textbf{ICD} & \textbf{I}nternational \textbf{C}lassification of \textbf{D}iseases\\
     \textbf{IMD} & \textbf{I}ndex of \textbf{M}ultiple \textbf{D}eprivation\\

thesis/Frontmatter/acknowledgements.tex

@@ -1,11 +1,25 @@
 Thanks be to James Bennett.
 
-% Majid Ezzati, Seth Flaxman.
-% Eric Johsnon
-% Kyle Foreman, Robbie Parks.
-% Barbara Metzler, Emily Muller.
-% Ricky Nathvani, Honor Bixby, Sierra Clark, Victor Lhoste.
-% Sam Acors
-% Solange.
-% Parents, Ros
-% Geoff Hardern.
+% Firstly, I would like to thank my supervisor Majid Ezzati for his guidance and mentorship.
+% I have learnt so much from you over the past years, and it's inspiring to see the level of care and attention you pay towards your team.
+% I am incredibly grateful for the energy you have put into our work.
+% It's an honour join the long and distinguished list of your former PhD students.
+
+% The stalwart of the group, James Bennett, has been far more than just a supervisor, but also a patient sounding board, a guide to the light when no model will converge, and a great friend.
+% I hope I didn't bother you too much, but to work so closely with you over the past years has been a pleasure.
+% My supervisor, Seth Flaxman, has been a fountain of ideas and energy, and every conversation has stretched my thinking.
+% Robbie Parks and Kyle Foreman have looked out for me throughout the past years and I'd like to thank them for pushing me towards new opportunities whenever they can.
+
+% I am part of an incredibly talented and welcoming group of researchers at Imperial.
+% Ricky Nathvani, Sierra Clark, Honor Bixby, Perviz Asaria, Bin Zhou, and the wider group of NCD-RisC collaborators have provided a constant stream of insights and entertainment.
+% My fellow PhD students, Barbara Metzler, Emily Muller, and Victor Lhoste, have survived the journey with me and made the experience so much fun.
+% Beyond the group, I'd like to thank Adam Howes for helpful, technical conversations and Sam Acors for unhelpful, non-technical conversations.
+% There would be no results in this thesis without the dedicated IT support from Eric Johnson and the invaluable probabilistic modelling advice from the oracle-like Chris Paciorek.
+
+% Thank you to my mother, Claire, and father, Aly, for my incredible upbringing and the foundation you have laid for me.
+% Thanks to Rosalind as well.
+% Of everyone, my biggest thanks go to Solange, for keeping me sane, happy and loved throughout.
+% I am very lucky to have you.
+
+% This thesis is dedicated to my grandfather, Geoffrey Hardern, who sadly passed during my PhD studies.
+% It was his dream for all of his grandchildren to go to university, and I hope he'd be proud of how long I have spent there.

thesis/_thesis/Chapters/Chapter2.html

@@ -276,16 +276,16 @@ <h4 class="unnumbered anchored" data-anchor-id="space-as-a-continuous-process">S
 </section>
 <section id="space-as-discrete-units" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="space-as-discrete-units">Space as discrete units</h4>
-<p>A more popular prior is the conditional autoregressive (CAR) prior, also known as a Gaussian Markov random field, which was first introduced by <span class="citation" data-cites="besagSpatialInteractionStatistical1974">Besag (<a href="../references.html#ref-besagSpatialInteractionStatistical1974" role="doc-biblioref">1974</a>)</span>. These form a joint distribution as in <a href="#eq-MVN">Equation&nbsp;<span>2.1</span></a>, but the covariance is usually defined instead in terms of the precision matrix <span id="eq-CAR-prec"><span class="math display">\[
+<p>A more popular prior is the conditional autoregressive (CAR) prior, also known as a Gaussian Markov random field (GMRF), which was first introduced by <span class="citation" data-cites="besagBayesianImageRestoration1991">Besag et al. (<a href="../references.html#ref-besagBayesianImageRestoration1991" role="doc-biblioref">1991</a>)</span>. These form a joint distribution as in <a href="#eq-MVN">Equation&nbsp;<span>2.1</span></a>, but the covariance is usually defined instead in terms of the precision matrix <span id="eq-CAR-prec"><span class="math display">\[
 \mathbf{P} = \pmb{\Sigma}^{-1} = \tau(\mathbf{D} - \rho \mathbf{A}),
-\tag{2.2}\]</span></span> where <span class="math inline">\(\tau\)</span> controls the overall precision of the effects, <span class="math inline">\(\mathbf{A}\)</span> is the spatial adjacency matrix formed by the small areas, <span class="math inline">\(\mathbf{D}\)</span> is a diagonal matrix with entries equal to the number of neighbours for each spatial unit, and the autocorrelation parameter <span class="math inline">\(\rho\)</span> describes the amount of correlation. This can be seen as tuning the degree of spatial dependence, where <span class="math inline">\(\rho = 0\)</span> implies independence between areas, and <span class="math inline">\(\rho = 1\)</span> full dependence. The case with <span class="math inline">\(\rho = 1\)</span> is called the intrinsic conditional autoregressive (ICAR) model. <span class="citation" data-cites="besagBayesianImageRestoration1991">Besag et al. (<a href="../references.html#ref-besagBayesianImageRestoration1991" role="doc-biblioref">1991</a>)</span> proposed the model (hereafter called BYM) <span id="eq-BYM"><span class="math display">\[
+\tag{2.2}\]</span></span> where <span class="math inline">\(\tau\)</span> controls the overall precision of the effects, <span class="math inline">\(\mathbf{A}\)</span> is the spatial adjacency matrix formed by the small areas, <span class="math inline">\(\mathbf{D}\)</span> is a diagonal matrix with entries equal to the number of neighbours for each spatial unit, and the autocorrelation parameter <span class="math inline">\(\rho\)</span> describes the amount of correlation. This can be seen as tuning the degree of spatial dependence, where <span class="math inline">\(\rho = 0\)</span> implies independence between areas, and <span class="math inline">\(\rho = 1\)</span> full dependence. The case with <span class="math inline">\(\rho = 1\)</span> is called the intrinsic conditional autoregressive (ICAR) model. There sometimes exists further over-dispersion in the residuals that cannot be modelled by purely spatially-structured random effects. <span class="citation" data-cites="besagBayesianImageRestoration1991">Besag et al. (<a href="../references.html#ref-besagBayesianImageRestoration1991" role="doc-biblioref">1991</a>)</span> proposed the model (hereafter called BYM) <span id="eq-BYM"><span class="math display">\[
 S_i = U_i + V_i,
 \tag{2.3}\]</span></span> where <span class="math inline">\(U_i\)</span> follow an ICAR distribution, and <span class="math inline">\(V_i\)</span> are independent and identically distributed random effects. The addition of the spatially-unstructured component <span class="math inline">\(V\)</span> accounts for any non-spatial heterogeneity.</p>
 </section>
 <section id="space-as-a-nested-hierarchy-of-geographies" class="level4 unnumbered">
 <h4 class="unnumbered anchored" data-anchor-id="space-as-a-nested-hierarchy-of-geographies">Space as a nested hierarchy of geographies</h4>
 <p>The relationships between different levels of a hierarchy of geographical units are often incorporated into models as a nested hierarchy of random effects. These models account for when spatial units lie within common administrative boundaries. This is often a desirable property of the model for certain geographies, like states in the US, which are administrative. Policy is decided at these geographies, so there is reason to believe these boundaries may have a greater effect on health outcomes than spatial structure. <span class="citation" data-cites="finucaneBayesianEstimationPopulationLevel2014">Finucane et al. (<a href="../references.html#ref-finucaneBayesianEstimationPopulationLevel2014" role="doc-biblioref">2014</a>)</span> demonstrate how country-level blood pressure can be modelled by exploiting the hierarchy global, super-region, region and country. Note, although these models group by geographical region, these models are not <em>spatial</em> as they do not contain any information on the relative position of the areas.</p>
-<p>Of the two specifications that are spatial, either as a continuous process or discrete units, the Markov random field priors are often preferred for computational reasons, as we can exploit the sparseness of the adjacency matrix in our inference algorithms rather than computing the covariance between each pair of spatial units as in the general case of <a href="#eq-MVN">Equation&nbsp;<span>2.1</span></a>.</p>
+<p>Of the two specifications that are spatial, either as a continuous process or discrete units, the Markov random field priors are often preferred for computational reasons, as we can exploit the sparseness of the adjacency matrix in our inference algorithms rather than computing the covariance between each pair of spatial units as in the general case of <a href="#eq-MVN">Equation&nbsp;<span>2.1</span></a>. There are concerns, however, that the GMRF representation of space as an adjacency matrix, which was originally proposed for a regular lattice of pixels in image analysis, is reductive for more complicated spatial problems. Despite this, in an epidemiological context, <span class="citation" data-cites="duncanSpatialSmoothingBayesian2017">Duncan et al. (<a href="../references.html#ref-duncanSpatialSmoothingBayesian2017" role="doc-biblioref">2017</a>)</span> found the standard ICAR model with binary, first-order neighbour weights outperformed models with a variety of different weighting schemes, including matrix weights based on higher-order degrees of neighbours, distance between neighbours, and distance between covariate values.</p>
 <p>In applications to disease mapping, spatial models are the natural choice when the disease exhibits a spatial pattern. This is the case for mortality from infectious diseases, particularly on short timescales like Covid-19 <span class="citation" data-cites="konstantinoudisRegionalExcessMortality2022">(<a href="../references.html#ref-konstantinoudisRegionalExcessMortality2022" role="doc-biblioref">Konstantinoudis et al., 2022</a>)</span>. Nested hierarchies are a more suitable choice when administrative areas are meaningful and have an effect on the health outcomes of the population. For example, state-specific abortion laws in the USA could affect maternal mortality, and so a model should include an effect for each state.</p>
 </section>
 <section id="modelling-variation-beyond-space" class="level4 unnumbered">
@@ -398,9 +398,6 @@ <h4 class="unnumbered anchored" data-anchor-id="recent-public-health-strategy-an
 <div id="ref-bennettContributionsDiseasesInjuries2018" class="csl-entry" role="listitem">
 Bennett JE, Pearson-Stuttard J, Kontis V, Capewell S, Wolfe I, Ezzati M. 2018. Contributions of diseases and injuries to widening life expectancy inequalities in <span>England</span> from 2001 to 2016: A population-based analysis of vital registration data. <em>The Lancet Public Health</em> <strong>3</strong>:e586–e597. doi:<a href="https://doi.org/10.1016/S2468-2667(18)30214-7">10.1016/S2468-2667(18)30214-7</a>
 </div>
-<div id="ref-besagSpatialInteractionStatistical1974" class="csl-entry" role="listitem">
-Besag J. 1974. Spatial <span>Interaction</span> and the <span>Statistical Analysis</span> of <span>Lattice Systems</span>. <em>Journal of the Royal Statistical Society: Series B (Methodological)</em> <strong>36</strong>:192–225. doi:<a href="https://doi.org/10.1111/j.2517-6161.1974.tb00999.x">10.1111/j.2517-6161.1974.tb00999.x</a>
-</div>
 <div id="ref-besagBayesianImageRestoration1991" class="csl-entry" role="listitem">
 Besag J, York J, Mollié A. 1991. Bayesian image restoration, with two applications in spatial statistics. <em>Annals of the Institute of Statistical Mathematics</em> <strong>43</strong>:1–20. doi:<a href="https://doi.org/10.1007/BF00116466">10.1007/BF00116466</a>
 </div>
@@ -419,6 +416,9 @@ <h4 class="unnumbered anchored" data-anchor-id="recent-public-health-strategy-an
 <div id="ref-downingJointDiseaseMapping2008" class="csl-entry" role="listitem">
 Downing A, Forman D, Gilthorpe MS, Edwards KL, Manda SO. 2008. Joint disease mapping using six cancers in the <span>Yorkshire</span> region of <span>England</span>. <em>International Journal of Health Geographics</em> <strong>7</strong>:41. doi:<a href="https://doi.org/10.1186/1476-072X-7-41">10.1186/1476-072X-7-41</a>
 </div>
+<div id="ref-duncanSpatialSmoothingBayesian2017" class="csl-entry" role="listitem">
+Duncan EW, White NM, Mengersen K. 2017. Spatial smoothing in <span>Bayesian</span> models: A comparison of weights matrix specifications and their impact on inference. <em>International Journal of Health Geographics</em> <strong>16</strong>:47. doi:<a href="https://doi.org/10.1186/s12942-017-0120-x">10.1186/s12942-017-0120-x</a>
+</div>
 <div id="ref-dwyer-lindgrenInequalitiesLifeExpectancy2017" class="csl-entry" role="listitem">
 Dwyer-Lindgren L, Bertozzi-Villa A, Stubbs RW, Morozoff C, Mackenbach JP, van Lenthe FJ, Mokdad AH, Murray CJL. 2017a. Inequalities in <span>Life Expectancy Among US Counties</span>, 1980 to 2014: <span>Temporal Trends</span> and <span>Key Drivers</span>. <em>JAMA Internal Medicine</em> <strong>177</strong>:1003–1011. doi:<a href="https://doi.org/10.1001/jamainternmed.2017.0918">10.1001/jamainternmed.2017.0918</a>
 </div>