Enhancing Bayesian small area level methods with applications in health

James Hogg - PhD Final Seminar

Supervisors: Kerrie Mengersen (QUT), Susanna Cramb (QUT), Peter Baade (CCQ) and Jessica Cameron (CCQ)

8th April, 2024

Acknowledgement of Traditional Owners

QUT acknowledges the Turrbal and Yugara, as the First Nations owners of the lands where QUT now stands. We pay respect to their Elders, lores, customs and creation spirits. We recognize that these lands have always been places of teaching, research and learning.

QUT acknowledges the important role Aboriginal and Torres Strait Islander people play within the QUT community.

⛔

A single estimate representing all Australia

(e.g., rate of cancer)

Here is a map of Australia, colored to represent a single estimate. In this instance, think of it as representing the national rate of cancer. This single estimate is quite useful because it enables us to draw comparisons with data from previous years and even with similar data from other countries. Through such estimates and comparisons, we can develop policies to reduce the national rate of cancer.

However, it would be more helpful to have the rate of cancer at the community level. This shift from a single estimate to

✅

Many estimates representing all communities

\(=\) small area estimates

many estimates allows us to conduct more effective comparisons, particularly now between communities.

community level are called small area estimates

Such detailed data, which allows us to understanding the spatial variation of cancer, is invaluable for supporting more informed decision-making.

Motivation

• Small area estimates \(\to\) more effective comparisons
• Small area analysis
  • Reporting/analyses of individual level data by aggregating into meaningful groups

Tricky in Australia!

Tricky in Australia!

• Highly heterogeneous population
• Sparsity \(\to\) Very little data

“This can be very small sample sizes, small populations, or even small counts of a health outcome.

Sparsity

Leads to …

• Information only reliable or available \(\to\) large geographical regions (e.g., states)
• Reliable estimates mostly in major cities

Sparsity

Leads to …

• Information only reliable or available \(\to\) large geographical regions (e.g., states) (Low spatial resolution)
• Reliable estimates mostly in major cities (Poor reach)

Resolution

Reach

Resolution

Geographical and population size of the areas

Reach

Proportion of areas with estimates

1. “Low resolution may be estimates for states, while high would be postcodes”
2. “Low reach would be having estimates for 10% of areas, while high reach would be estimates for all regions”

Like the variance bias tradeoff, we can trade high reach for low resolution and vice versa. But ideally we want both.

Why we do need high resolution and reach?

• Reduces biases (e.g., Modifiable areal unit problem (MAUP) (Fotheringham & Wong, 1991), ecological bias, etc)
• Allows for more detailed investigation of spatial patterns
• Supports equity

Tricky in Australia!

• Highly heterogeneous population
• Sparsity \(\to\) Very little data

Sparsity

Worse with survey data

• International studies:
  • Small area sample sizes greater than 50 (Parker et al., 2020; Vandendijck et al., 2016)
• Current Australian surveys:
  • Small area sample sizes 5 to 13

“To obtain high resolution with these small counts, we need to use more fancy methods, specifically models…”

Small area modelling

• Disease mapping (Cramb et al., 2020)
  • Registry or administrative data
  • Generates maps of risk for diseases

• Small area estimation (SAE) (Rao & Molina, 2015)
  • Survey data
  • Produce reliable estimates when sample sizes are small

• Bayesian inference is ubiquitous with both

Small area modelling in Australia

Increasingly relevant

Health outcomes exhibit variations based on geography (e.g., remoteness and socioeconomic status (AIHW, 2018, 2019; Duncan et al., 2019; Patterson et al., 2014; PHIDU, 2018))

1. “Despite the difficulties we already mentioned, small area modelling is increasingly relevant.”
2. “We also know that when the Australian Institute of Health and Welfare, the premier federal health agency, writes about something…”

Thesis aim

Develop and apply Bayesian small area methods to health data in Australia

🏹 Sparsity issues

📈 Improve spatial resolution

🤔 Support informed decision making

Cancer (Pillar I)

Burden of disease (Pillar II)

Cancer (Pillar I)

Burden of disease (Pillar II)

Background

• Cancer accounts for 19% of Australia’s total health burden ¹ (AIHW, 2019)
• Prevention strategy
  • Reduce prevalence of behaviours known to increase the risk of cancer (i.e., cancer risk factors).
    • Smoking, obesity, poor diet, insufficient physical activity, excessive sun exposure, alcohol, etc
• Prevalence of cancer risk factors varies geographically (Abdel-Rahman, 2021; AIHW, 2018; Berkowitz et al., 2016; CDC, 2021; Hermes & Poulsen, 2012; PHIDU, 2018; Ritchie & Roser, 2017)

1. Health burden: the impact of living with illness and injury and dying prematurely

Social Health Atlas of Australia ¹

😊 Prevalence estimates of cancer risk factors for small areas

😨 Estimates suffer from resolution and reach issues

1. Developed by Public Health Information Development Unit (PHIDU) and Australian Bureau of Statistics (ABS)

Research Problem

The need to improve the spatial resolution and reach of small area level cancer risk factor estimates in Australia to enable more targeted cancer prevention strategies.

But wait…

• The Social Health Atlas estimates from…
  • Most appropriate survey data available at the time: 2017-2018 National Health Survey (NHS)
• For small area modelling \(\to\) NHS is highly sparse survey data
• Modelling proportions with such sparse data gives rise to unique methodological challenges

NOTE:
Proportion \(=\) number of health outcomes / population
Proportion \(=\) prevalence

Research Problem

Lack of small area estimation methods for sparse survey data.

Objective 1

Small area estimation

• 1.1 Method: Develop a Bayesian SAE method for proportions, which specifically tackles data sparsity.
• 1.2 Application: Generate small area level proportion estimates for several cancer risk factors across Australia.

• Imagine we have high-resolution estimates for many cancer risk factors and many areas
• Challenge in decision making
  • Striking the right balance between insufficient or overwhelming quantities of data.

Indices

• Interest in indices (Ceely, 2020; Hedibert et al., 2012; Hogan & Tchernis, 2004; Nardo et al., 2008; Nethery et al., 2015; Norwood et al., 2020)
  • Single interpretable metric that describes a complex phenomenon
  • Reduce the dimensionality of multivariate data
• Current empirical methods ⛔
  • Restrictive
• Model-based methods ✅
  • Complex data characteristics

“Some examples include socioeconomic status, IQ, Forest Fire Danger Index

Objective 2

Index creation

• 2.1 Method: Develop a Bayesian model-based approach to index creation.
• 2.2 Application: Using cancer risk factor data, create the first area-level cancer risk factor indices for Australia.
  • High resolution and complete reach

Cancer (Pillar I)

Burden of disease (Pillar II)

Cancer (Pillar I)

Burden of disease (Pillar II)

Background

• Burden of disease (BOD) estimates inform health policy
• Fully Bayesian models are underused in full BOD studies ¹
  • Existing research has limitations (Bannick et al., 2019; Foreman et al., 2012; GBD 2015 Chronic Respiratory Disease Collaborators, 2017; MacNab, 2007, 2009; Steel et al., 2018)
• Full BOD studies in Australia (AIHW, 2022; DOHWA, 2020)
  • No Bayesian and no small area estimates

Not fully Bayesian, fail to acknowledge relevant applied literature or enforce strict data requirements. Moreover, the particularly relevant work by MacNab does not consider how Bayesian methodology could be employed in full BOD studies

1. Calculate burden from all disease (n > 200).

Objective 3

Burden of disease

• 3.1 Method: Explore and develop appropriate Bayesian spatiotemporal methods to model the wide range of health data used in full BOD studies.
• 3.2 Application: Using two very different diseases ¹, generate and map the first small area level BOD estimates in Australia.

1. Coronary heart disease (CHD) and asthma in Western Australia

Objectives, Chapters, Papers (Outputs)

Thesis format: Publication 📃

Chapter 3/Paper 1

A two-stage Bayesian small area estimation approach for proportions

Accepted for publication in the International Statistical Review

James Hogg, Jessica Cameron, Susanna Cramb, Peter Baade, Kerrie Mengersen. A two-stage Bayesian small area estimation approach for proportions.

Background

• Increasing demand \(\to\) robust small area estimates from smaller surveys
• Sparsity is key challenge when working with proportions
• Individual level and area level models pushed to extent and cannot address challenges

Use both?

Use both?

• Two-stage approaches have emerged (Das et al., 2022; Gao & Wakefield, 2023)
  • Hints at benefits \(\to\) no full investigation
• Mitigate weakness of either model and leverage their strengths

Contribution 🎉

A Bayesian two-stage method for proportions that tackles data sparsity challenges.

Thesis aim

Develop and apply Bayesian small area methods to health data in Australia

👉 Sparsity issues

Improve spatial resolution

Support informed decision making

The two-stage logistic normal (TSLN) approach

• Stage 1 (TSLN-S1)
  • Fit a individual-level logistic model to the survey data
• Stage 1 (S1) estimates
  • Using fitted values calculate area-level S1 estimates and sampling variances
• Stage 2 (TSLN-S2)
  • Smooth S1 estimates and impute estimates for non-sampled areas using an area-level Fay-Herriot (1979) model

The TSLN approach

Stage 1

Individual level logistic model \[ \begin{eqnarray} y_{ij} &\sim& \text{Bernoulli}(p_{ij})^{\tilde{w}_{ij}} \\ \text{logit}(p_{ij}) &=& \mathbf{x}_{ij} \boldsymbol{\beta} + e_i \\ e_i &\sim& N(0, \sigma_e^2) \end{eqnarray} \]

where:

• \(y_{ij}\) is the binary response for sampled individual \(j\) in small area \(i\)
• \(p_{ij}\) is the probabilitiy for \(\dots\)
• \(\tilde{w}_{ij}\) is the sampled scaled weight for \(\dots\)

Stage 1 (S1) estimates

The S1 estimate ¹ for area \(i\) is,
\[ \hat{\theta}_i^{\text{S1}} = \text{logit}\left( \frac{\sum_{j=1}^{n_i} w_{ij} p_{ij}}{n_i} \right) \] with \(\gamma_i^{\text{S1}}\) is the corresponding sample variance.

where:

• \(w_{ij}\) is the sample weight for sampled individual \(j\) in small area \(i\)
• Sample weights in small area \(i\) sum to \(n_i\) (the area sample size)

Stage 2

The area level FH model is composed of: a
measurement error model, \[ \hat{\theta}_i^{\text{S1}} \sim N\left( \hat{\bar{\theta}}_i, \widehat{\text{v}}\left( \hat{\theta}_i^{\text{S1}} \right) \right), \]

which accommodates some of the uncertainty of the stage 1 model; a
sampling model,

\[ \hat{\bar{\theta}}_i \sim N\left( \hat{\theta}_i, \gamma_i^{\text{S1}} \right), \]

which accommodates the sampling variance; and a
linking model,

\[ \begin{eqnarray} \hat{\theta}_i & = & \mathbf{Z}_{i} \boldsymbol{\lambda} + v_i \\ v_i & \sim & N(0, \sigma_v^2), \nonumber \end{eqnarray} \]

The parameter of interest (i.e. proportion) is the posterior distribution of \(\text{logit}^{-1}( \hat{\theta}_i)\).

1. “in our case, y may be whether the sampled individual smoked or not, where a one means that the person is a smoker.”

1. The posterior draws from stage 1 are fed as input to the stage 2 model.

Simulation study

Individual level

• LOG: Pseudo-likelihood logistic model
  • Similar to stage 1 of the TSLN approach

Area level

• BETA: Beta model
• ELN: Empirical-logistic normal model
  • Similar to stage 2 of the TSLN approach

ChatGPT4 nailed this!

Results

Results

“Our TSLN approach provides superior estimates”

Key findings 📢

• TSLN approach provides superior proportion estimates in the sparse setting
  • Similar bias but much smaller variance.
  • Consistently smaller MSEs and credible intervals.
  • Interval coverage more stable.

“As you can see in our paper, the interval coverage provided by our TSLN approach is more stable.”

Objective 1

Small area estimation

❕ 1.1 Method: Develop a Bayesian SAE method for proportions, which specifically tackles data sparsity.

❕ 1.2 Application: Generate small area level prevalence estimates for several cancer risk factors across Australia.

Objective 1

Small area estimation

✅ 1.1 Method:

❕ 1.2 Application: Generate small area level prevalence estimates for several cancer risk factors across Australia.

Chapter 4/Paper 2

Mapping the prevalence of cancer risk factors at the small area level in Australia

Paper published

James Hogg, Jessica Cameron, Susanna Cramb, Peter Baade, Kerrie Mengersen. Mapping the prevalence of cancer risk factors at the small area level in Australia. International Journal of Health Geographics 22, 37 (2023). https://doi.org/10.1186/s12942-023-00352-5

Previously (SHA)

🛑 Population health areas (n = 1,165)

🛑 Missing estimates for very remote areas

Previously (SHA)

🛑 Population health areas (n = 1,165)

🛑 Missing estimates for very remote areas

Now (this work)

✅ Statistical area level 2 (n = 2,221)

✅ Estimates for very remote areas

Previously (SHA)

Now (this work)

Contribution 🎉

The first Australia-wide prevalence estimates for eight cancer risk factors at the statistical area level 2 (SA2) level.

Thesis aim

Develop and apply Bayesian small area methods to health data in Australia

Sparsity issues

👉 Improve spatial resolution

👉 Support informed decision making

Survey data

2017-18 National Health Survey

• Conducted by the Australian Bureau of Statistics (ABS)
• Sample size 17,248 persons 15 years and older
• Median SA2 sample size: 8
• 1,694 (76%) of SA2s had survey data

Auxiliary data

• 2016 Australian census data — collapsed to six principal components for analysis (Chidumwa et al., 2021)
• Average Estimated Resident Population between 2017 and 2018
• Index of Relative Socio-Economic Disadvantage (IRSD) from Socio-Economic Indexes for Areas by the ABS (ABS, 2016)
• Remoteness (ARIA+)
• Social Health Atlas estimates (PHIDU, 2018) at Primary Health Network (n = 31) and Population Health Area level (n = 1,165)

Model

• TSLN approach (Chapter 3 of this thesis)
• Single TSLN specification for all risk factors

Validation

How to check our estimates?

Validation

How to check our estimates?

• Internal: Fully Bayesian benchmarking (Zhang & Bryant, 2020)
  • State benchmark (n = 7)
  • Major cities-by-state benchmark (n = 12)

“Benchmarking validates non-sampled areas”

Validation

How to check our estimates?

• Internal: Fully Bayesian benchmarking (Zhang & Bryant, 2020)
  • State benchmark (n = 7)
  • Major cities-by-state benchmark (n = 12)
• External: Comparing our broad results to other Australian surveys and the Social Health Atlas

Risky alcohol consumption: Individuals exceeding the 2020 National Health and Medical Research Council (NHMRC) guidelines (2020) of up to 10 standard drinks per week and no more than 4 standard drinks on any day.

“I have also provided insets for each of the 8 capital cities”

The results suggest that less disadvantaged areas have higher proportions of risky alcohol consumption, which generally manifests in higher prevalence in major cities. This is supported by other Australian surveys and previous research

Note that the regions in NT are disadvantaged areas, so have been predicted to have lower rates. We’ll see in a second that these areas have high uncertainty though.

1. “We can see pockets of areas that are significantly higher and lower than the national average”

2. “Zooming into Brisbane now, we can see a large patch of areas in the south side where the prevalence of risky alcohol consumption is low.”
3. “Finally note how areas in the Northern Territory are grey here. As we saw before, these estimates are very uncertainty.”

Key findings 📢

• Prevalence of cancer risk factors \(\to\) considerable spatial disparities across Australia
• High spatial resolution estimates \(\to\) detailed understanding of spatial variation

Exciting…

These modelled estimates will be available in the Australian Cancer Atlas 2.0 (https://atlas.cancer.org.au/)

“The official launch is scheduled for the 28th of May!”

Objective 1

Small area estimation

✅ 1.1 Method:

❕ 1.2 Application: Generate small area level prevalence estimates for several cancer risk factors across Australia.

Objective 1

Small area estimation

✅ 1.1 Method:

✅ 1.2 Application:

Chapter 5/Paper 3

Creating area level indices of behaviours impacting cancer in Australia with a Bayesian generalised shared component model

Under review at Health & Place

James Hogg, Susanna Cramb, Jessica Cameron, Peter Baade, Kerrie Mengersen. Creating area level indices of behaviours impacting cancer in Australia with a Bayesian generalised shared component model.

Status: Submitted

Background

• Resolution and accessibility of health data for small areas have improved considerably
• BUT…
• Increasing quantity and complexity of such data can make it more challenging to:
  • Collate
  • Draw meaningful conclusions
  • Formulate effective policy decisions

Indices

Indices

• Measures a multifaceted concept that cannot be captured by a single feature.
• Often can help:
  • Determine priorities for future policies
  • Benchmark or monitor the performance of current policies

Creating indices with models

• Increasingly popular (Hedibert et al., 2012; Hogan & Tchernis, 2004; Nethery et al., 2015; Norwood et al., 2020)
• Shared component model (SCM) (Knorr-Held & Best, 2001)
  • Identify underlying factor shared between features (e.g., Index)
  • Single shared factor
  • No heteroscedastic error

“As mentioned previously, the use of models to create indices is increasingly popular. One such approach is the…”

Generalised shared component model

Generalised shared component model

• Multiple shared factors
• Heteroscedastic error
• Provides indices with uncertainty measures

Area Indices of
Behaviors Impacting
Cancer (AIBIC) product

Contribution 🎉

• A Bayesian model-based method for index creation.
• The first Australia-wide area level cancer risk factor indices.

Thesis aim

Develop and apply Bayesian small area methods to health data in Australia

Sparsity issues

Improve spatial resolution

👉 Support informed decision making

Generalised Shared Component Model (GSCM)

\(N\)-dimensional multivariate normal distribution for the \(k\)th feature

\[ \begin{eqnarray} \mathbf{Y}_{k} & \sim & \text{MVN}_{N}\left( \boldsymbol{\mu}_{k}, \text{diag}\left( \boldsymbol{\sigma}_k \right) \right) \\ \boldsymbol{\mu}_{k} & = & \mathbf{z} \left( \boldsymbol{\Lambda}_k \right)^T + \boldsymbol{\epsilon}_{k} \end{eqnarray} \]

Where:

• \(\mathbf{Y}_{k}\) is \(N\)-dimensional vector of estimates for the \(k\)th feature
• \(\boldsymbol{\sigma}_k\) is \(N\)-dimensional vector of standard deviations for the \(k\)th feature
• \(\boldsymbol{\mu}_{k}\) is \(N\)-dimensional vectors of true values for the \(k\)th feature
• \(\mathbf{z}\) is \(N \times L\) matrix of shared factor scores, with independent columns
  • Leroux CAR priors (Leroux et al., 2000)
• \(\boldsymbol{\epsilon}_{k}\) is \(N\)-dimensional vector of feature-specific residual errors for the \(k\)th feature with associated variance \(\tau_k^2\)
  • Leroux CAR priors
• \(\boldsymbol{\Lambda}_k\) is \(k\)th row of the \(K\) by \(L\) factor loadings matrix

Identifiability is enforced by:

• Shared factors are homoscedastic (variances \(=1\)) (Nethery et al., 2015; Wang & Wall, 2003)
• Hierarchical structural constraint (Aguilar & West, 2000)
  • Factor loading matrix, \(\boldsymbol{\Lambda}\), is constrained to be lower triangular with strictly positive diagonal entries (Nethery et al., 2015)

Special cases:

• Traditional shared component model
• Bayesian spatial factor models (Mezzetti, 2012; Nethery et al., 2015)
• Generic Bayesian factor model

Input data

• Small area estimates of five features (cancer risk factor OR unhealthy behaviors) from Chapter 4
  • Current smokers, risky alcohol consumption, overweight/obese, inadequate diet and inadequate physical
• Available as point estimates and standard deviations of proportions (inverse logit transformed)

Input data

Reverse scale: Reverse the order would put problem for different factors. Would make interpretation more difficult.

Input data

• Small area estimates of five features (cancer risk factor OR unhealthy behaviors) from Chapter 4
  • Current smokers, risky alcohol consumption, overweight/obese, inadequate diet and inadequate physical
• Available as point estimates and standard deviations of proportions (inverse logit transformed)
• Details
  • Number of features \(K = 5\)
  • Number of small areas \(N = 2221\)
  • Number of shared factors \(L = 2\)

Results

Posterior medians (and 95% highest posterior density intervals) of the factor loadings.

The four indices

Factor 1: \(z_{n1}\)

Factor 2: \(z_{n2}\)

Health Behavior Index (HBI): \(w_1 z_{n1} + w_2 z_{n2}\)

Population Adjusted Health Behaviors Index (PAHBI): \(P_n \left( w_1 z_{n1} + w_2 z_{n2} \right)\)

Interpretation

Point estimates:

• Higher values of the index \(\approx\) higher prevalence of unhealthy behaviours

Uncertainty:

• Probabilistic statements about the indices (e.g., exceedance probabilities)

Laurieton - Bonny Hills (NSW)

Modelled percentages -

10 numbers!

Inadequate physical activity: 84% (±2.8%) ¹

Risky alcohol consumption: 39% (±4.1%)

Inadequate diet: 50% (±3.9%)

Overweight/obese: 78% (±2.9%)

Current smoking: 22% (±3.7%)

Health Behaviors Index (HBI) -

2 numbers!

Percentile: 98

Probability that percentile above 80th: 0.99

Population Adjusted HBI

Percentile: 100

Probability that percentile above 80th: 1.00

1. Estimate (±standard error)

Key findings 📢

• The AIBIC reveals a pronounced spatial autocorrelation in unhealthy behaviors
  • Patterns reflect remoteness and socioeconomic categories
  • Faster to interpret than individual prevalence estimates
  • Optimal for broad policies

Indeed, indices inherently involve a loss of information, rendering them less suitable for decisions relating to a single feature (e.g., current smoking only). In such cases, a direct examination of the specific feature is always preferable

Exciting…

The indices (AIBIC) are on Github

• Point estimates, intervals, and posterior probabilities
• Percentiles and ranks computed both nationally and for each of the eight states and territories of Australia.

Objective 2

Index creation

❕ 2.1 Method: Develop a Bayesian model-based approach to index creation.

❕ 2.2 Application: Using cancer risk factor data, create the first area-level cancer risk factor indices for Australia.

Objective 2

Index creation

✅ 2.1 Method:

✅ 2.2 Application:

Chapter 6/Paper 4

Improving the spatial and temporal resolution of burden of disease measures with Bayesian models

Under review at Spatial and Spatio-temporal Epidemiology

James Hogg, K. Staples, A. Davis, S. Cramb, C. Patterson, L. Kirkland, M. Gourley, J. Xiao, W. Sun. Improving the spatial and temporal resolution of burden of disease measures with Bayesian models.

Status: Second round of revisions

Partnership with the Department of Health Western Australia and the Australian Institute of Health and Welfare

Background

• BOD estimates \(\to\) identify diseases causing excess burden (AIHW, 2019; Murray & Acharya, 1997; Naghavi et al., 2017; Schmidt et al., 2020; Steel et al., 2018)
• Disability-adjusted life years (DALY)
  • Linear and non-linear combinations of fatal (e.g., mortality) and non-fatal (e.g., prevalence) burden
  • Fatal component \(=\) ‘years of life lost’ (YLL)
  • Non-fatal component \(=\) ‘years lived with disability’ (YLD)

Full BOD studies

• Aim to calculate DALYs for all causes (typically over 200)
• Prioritise healthcare resources by ranking diseases
• Wide range of data sources (e.g., registry, administrative, survey or epidemiological)
• Conducted by government agencies in Australia

Limitation of these data for small area estimates is small counts or gaps (poor reach). SAE can help in this case.

Gaps

Gaps

• Few studies use Bayesian models (Foreman et al., 2012; GBD 2015 Chronic Respiratory Disease Collaborators, 2017; MacNab, 2007, 2009; Steel et al., 2018)
  • Not applicable to full scope of data
  • No use of small area estimation methods
  • Not practical for many users (e.g., government)
• Australian full BOD studies
  • No use of Bayesian models and no small area estimates

Not practical both in terms of applying the methods and interpretation/reconcilling the output.

Contribution 🎉

A principled and practical set of Bayesian spatiotemporal methods for modelling the wide range of data in full BOD studies.

Thesis aim

Develop and apply Bayesian small area methods to health data in Australia

👉 Sparsity issues

👉 Improve spatial resolution

👉 Support informed decision making

Models

• Intentionally simple \(\to\) accessible and practical

• Novelty
  • Not in the development of new Bayesian models
  • Application of existing models to the field of BOD

Models

Standard age-year-time (SAYT) model

• Model details:
  • Poisson model
  • Population as offset
  • Spatial, temporal and space-time random effects
• Input data:
  • Aggregated counts by area, year, age, and sex
• Output data:
  • Smoothed counts for each area, year, age, and sex
• Applicable to:
  • Mortality or prevalence counts

Models

Weighted multilevel regression and poststratification (MrP)
First use of small area estimation in BOD

• Model details:
  • Logistic model with poststratification step (Park et al., 2004)
  • Pseudo-likelihood (Savitsky & Toth, 2016)
  • Spatial and temporal random effects
• Input data: Individual level data
• Output data: Probabilities for all combinations of area, year, age, and sex
• Applicable to:
  • Survey microdata

“Pseudo likelihood was used in our small area modelling in Chapter 3.”

“By modelling microdata, we can improve the reach of estimates from survey data, which tackles a core data limitation in full BOD studies.”

Case study

• Data was supplied by Department of Health Western Australia (DOHWA)
• Two conditions: Asthma and coronary heart disease (CHD)
• Geographic: Local Government Areas (n = 137) in Western Australia
• Time: 2015 to 2020 inclusive
• Used fully Bayesian methods with the statistical software package, \(\texttt{nimble}\) (Valpine et al., 2017)

Objective

1. Produce modelled estimates of:
   1. Mortality and age-standardised ‘years of life lost’ (ASYLLs)
   2. Prevalence/proportions and age-standardised ‘years lived with disability’ (ASYLDs)
2. Compare to raw results (i.e., no modelling)

Comparison of the distribution of the point estimates from the Bayesian models and basic stratification (i.e., raw).

Comparison of the distribution of the point estimates from the Bayesian models and basic stratification (i.e., raw).

Comparison of the distribution of the point estimates from the Bayesian models and basic stratification (i.e., raw).

Comparison of the distribution of the point estimates from the Bayesian models and basic stratification (i.e., raw).

Comparison of the 822 annual LGA point estimates and their reliability (% of RSEs less than 50%) for modeled and raw BOD measures by condition.

Comparison of the 822 annual LGA point estimates and their reliability (% of RSEs less than 50%) for modeled and raw BOD measures by condition.

Comparison of the 822 annual LGA point estimates and their reliability (% of RSEs less than 50%) for modeled and raw BOD measures by condition.

Comparison of the 822 annual LGA point estimates and their reliability (% of RSEs less than 50%) for modeled and raw BOD measures by condition.

Comparison of the 822 annual LGA point estimates and their reliability (% of RSEs less than 50%) for modeled and raw BOD measures by condition.

Key findings 📢

• Bayesian models \(\to\) increased the reliability, certainty and reportability of estimates
• Bayesian vs. raw:
  • Agreed (mostly!)
  • Discrepancies due to age-standardisation OR paradigm differences ¹

1. Raw estimates: unstable observed data. Bayesian estimates: smoothed version of the underlying risk.

Objective 3

Burden of disease

❕ 3.1 Method: Explore and develop appropriate Bayesian spatiotemporal methods to model the wide range of health data used in full BOD studies.

❕ 3.2 Application: Using two very different diseases, generate and map the first small area level BOD estimates in Australia.

Objective 3

Burden of disease

✅ 3.1 Method:

✅ 3.2 Application:

Thesis summary

We made it. 🎉 Take a breath.

Contributions

• Bayesian two-stage logistic normal (TSLN) approach for proportions (Chapter 3)
• Cancer risk factor estimates for the 2221 small areas across Australia (Chapter 4)
• Generalised shared component model (GSCM) (Chapter 5)
• Area Indices of Behaviors Impacting Cancer (AIBIC) product (Chapter 5)
• Principled and practical set of Bayesian spatiotemporal methods for BOD in Australia (Chapter 6)

Impact

• Novel Bayesian models push boundary of previously achievable \(\to\) spatial and temporal resolution and reach in Australia (Chapters 3 and 6)
• Modelled risk factors \(\to\) Australian Cancer Atlas platform (Chapter 4)
• Epidemiologists from the West Moreton Public Health Unit \(\to\) interest in utilising estimates for reporting (Chapter 4)
• Department of Health Western Australia (DOHWA) applied methods \(\to\) generate new estimates for their Public Health Atlas (Chapter 6)

Impact

• Interest in adopting Bayesian modelling (Chapter 6)
  • The Burden of Disease and Mortality specialist group (Australian Institute of Health and Welfare)
• Efficient implementation of the Leroux prior (2000) in the statistical software package, \(\texttt{stan}\) (Stan Development Team, 2023) (Chapter 5)

Reflections/future directions

• Theory/model expansion
  • Multivariate variations
• Validate small area estimation (Chapter 3 and 4)
  1. Expansion of NHS
  2. Appending state surveys
  3. \(+\) Health behaviours \(\to\) the census
• Validate BOD methods and models (Chapter 6)
  • Disease states assumed constant and discrete
  • Many types of measurement error

Reflections/future directions

• Temporal dimension for the:
  • TSLN approach (Chapters 3 and 4)
  • GSCM (Chapter 5)
• Indices (Chapter 5)
  • Temporal consistency
  • Robustness
  • Aggregation (Modifiable areal unit problem (MAUP) (Fotheringham & Wong, 1991))

Indices

• particularly if the modelled estimates or indices are intended to track the impact of health interventions.
• Temporal consistency requires consistency in the: survey design, questions and scope; region boundaries; and populations across time, which can be difficult to verify.
• temporal consistency must first be backed by a consistent, reproducible and robust methodology
• This entails the selection of methods that are easy to use, validate and understand
• Indices are extremely sensitive to the order, number and selection of features, the nature of the transformations used, and whether errors and spatial dependence are accommodated.
• the pervasive issue of the modifiable areal unit problem presents challenges in releasing indices at a variety of geographical levels

Professional engagements/activities

• 11 conferences (including 2\(\times\) international conferences): 8 talks \(+\) 3 posters
• Internship and statistical consulting for Department of Health Western Australia (DOHWA)
• Casual statistical consulting for Environmental Protection Authority (EPA) Victoria
• Co-chair of the Bayesian Research and Applications Group (BRAG)
• Organised and hosted virtual StanConnect conference
• Worked with BioGro as part of Good Data Institute hackathon
  • Team awarded 1st place
• 2023 United Nations Datathon
  • Team awarded 2nd place globally out of over 200 teams

Acknowledgments

• Supervisors: Kerrie Mengersen, Susanna Cramb, Peter Baade and Jessica Cameron
• Australian Cancer Atlas 2.0 project team
• Government collaborators: DOHWA and AIHW
• PhD stipend and travel funding: Cancer Council Queensland, QUT
• Others:
  • ABS
  • Colleagues (you rock 🎸)
  • My family (particularly my Mum!)
  • My partner
  • Friends

References

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Chapter 3/Paper 1

		Sampled areas				Nonsampled areas
		MRRMSE	MARB	CI width	Coverage	MRRMSE	MARB	CI width	Coverage
50-50	BETA	0.36 (2.17)	0.22 (1.96)	0.46	0.80	0.40 (2.29)	0.16 (1.35)	0.61	1.00
	BIN	0.39 (2.38)	0.38 (3.35)	0.18	0.03	0.40 (2.30)	0.38 (3.29)	0.20	0.03
	ELN	0.34 (2.08)	0.21 (1.84)	0.43	0.91	0.40 (2.32)	0.15 (1.32)	0.64	1.00
	LOG	0.25 (1.52)	0.16 (1.41)	0.32	0.90	0.28 (1.61)	0.10 (0.83)	0.47	1.00
	TSLN	0.17 (1.00)	0.11 (1.00)	0.22	0.94	0.17 (1.00)	0.12 (1.00)	0.23	0.95
Rare	BETA	0.47 (1.46)	0.31 (1.39)	0.21	0.66	0.51 (1.57)	0.22 (1.08)	0.34	0.97
	BIN	0.50 (1.54)	0.48 (2.16)	0.12	0.08	0.50 (1.54)	0.47 (2.29)	0.13	0.08
	ELN	0.78 (2.42)	0.55 (2.52)	0.39	0.68	1.03 (3.17)	0.20 (0.96)	0.77	1.00
	LOG	0.50 (1.55)	0.35 (1.60)	0.28	0.82	0.55 (1.68)	0.15 (0.73)	0.43	1.00
	TSLN	0.32 (1.00)	0.22 (1.00)	0.19	0.86	0.32 (1.00)	0.21 (1.00)	0.20	0.88
Common	BETA	0.16 (2.01)	0.09 (1.76)	0.31	0.89	0.18 (2.25)	0.04 (0.87)	0.43	1.00
	BIN	0.25 (3.13)	0.24 (4.80)	0.19	0.05	0.25 (3.04)	0.23 (4.64)	0.21	0.08
	ELN	0.12 (1.52)	0.07 (1.37)	0.26	0.98	0.13 (1.58)	0.06 (1.18)	0.31	1.00
	LOG	0.11 (1.40)	0.06 (1.28)	0.24	0.97	0.12 (1.45)	0.05 (0.99)	0.29	1.00
	TSLN	0.08 (1.00)	0.05 (1.00)	0.16	0.98	0.08 (1.00)	0.05 (1.00)	0.17	0.99

Chapter 4/Paper 2

Cancer Risk Factors

• Smoking
  • Current smoking: Individuals who identify as current smokers, whether they smoke daily, weekly, or less frequently, and have smoked at least 100 cigarettes in their lifetime.
• Alcohol
  • Risky alcohol consumption: Individuals exceeding the 2020 National Health and Medical Research Council (NHMRC) guidelines (National Health and Medical Research Council, 2020) of up to 10 standard drinks per week and no more than 4 standard drinks on any day, assessed via self-reported alcohol consumption over the last three drinking days from the preceding week.
• Diet
  • Inadequate diet: Based on self-reported dietary intake, individuals failing to meet both the fruit (2 serves/day) and vegetable (5 serves/day) requirements as per the 2013 NHMRC Australian Dietary Guidelines (Health & Council, 2013).
• Weight
  • Obese: Individuals with a Body Mass Index (BMI) of 30 or higher.
  • Overweight/obese: Individuals with a BMI of 25 or higher.
  • Risky waist circumference: Individuals with waist circumference measurements of ≥94cm (men) and ≥80cm (women), focusing on adults aged 18 years and older (PHIDU, 2018).
• Physical Activity
  • Inadequate activity (leisure): Based on self-reported leisure physical activity, individuals who do not meet the 2014 Department of Health Physical Activity guidelines (Department of Health, 2014), which include a mix of moderate and vigorous-intensity physical activities, plus muscle-strengthening activities.
  • Inadequate activity (all): Similar to inadequate activity (leisure) but includes both workplace and leisure self-reported physical activities.

Validating non-sampled SA2s and those in very remote areas

• Warning from the ABS (Australian Bureau of Statistics, 2017)
  `. . . the estimates from the survey, do not (and are not intended to) match estimates of the total Australian estimated resident population (which include persons living in Very Remote areas of Australia and persons in non-private dwellings, such as hotels) obtained from other sources’
  
  • Not included in internal benchmarking
• Warning that direct estimates for the Northern Territory (NT) could be inaccurate
  • Not included in internal benchmarking

Validating non-sampled SA2s and those in very remote areas

• These excluded regions \(\approx\) 1.5% of the population
  • Validated using the external methods
  • Exhibit much greater uncertainty (as expected)
• Future work
  • Utilise smaller surveys taken in these areas as validation datasests

Chapter 5/Paper 3

Efficient LCAR

/**
* Log probability density of the leroux conditional autoregressive (LCAR) model
* @param x vector of random effects
* @param rho spatial dependence parameter
*           MUST be strictly smaller than 1, use:
*           `real<lower=0,upper=0.99> rho;`
* @param sigma standard deviation - often set to 1
* @param C_w Sparse representation of C
* @param C_v Column indices for values in C
* @param C_u Row starting indices for values in C
* @param offD_id_C_w indices for off diagonal terms
* @param D_id_C_w indices for diagonal terms - length M
* @param C_eigenvalues eigenvalues for C
* @param N number of areas
**
@return Log probability density
**
To use: LCAR_lpdf( x | rho, sigma, C_w, C_v, C_u, offD_id_C_w, D_id_C_w, C_eigenvalues, N );
*/
real LCAR_lpdf(
    vector x,               
    real rho,                   
    real sigma,              
    vector C_w , 
    int [] C_v , 
    int [] C_u , 
    int [] offD_id_C_w ,        
    int [] D_id_C_w ,       
    vector C_eigenvalues,       
    int N                   
    ) {                 
        vector[N] ldet_C;
        vector [ num_elements(C_w) ] ImrhoC;
        vector[N] A_S;
        // Multiple off-diagonal elements by rho
        ImrhoC [ offD_id_C_w ] = - rho * C_w[ offD_id_C_w ];
        // Calculate diagonal elements of ImrhoC
        ImrhoC [ D_id_C_w ] = 1 - rho * C_w[ D_id_C_w ];
        A_S = csr_matrix_times_vector( N, N, ImrhoC, C_v, C_u, x );
        ldet_C = log1m( rho * C_eigenvalues );
        return -0.5 * ( 
        N*log( 2 * pi() ) 
        - ( N * log(1/square(sigma)) + sum( ldet_C ) ) 
        + 1/square(sigma) * dot_product(x, A_S) 
        );
}

Chapter 6/Paper 4

Standard age-year-time (SAYT) model

Let:

• \(y_{ita}\) be the raw mortality counts in age group \(a = 1, \dots, A\), area \(i = 1, \dots, M\) and time \(t = 1, \dots, T\)
• \(\xi_{it}\) be the random effect for the \(i\)th area and \(t\)th year
  • \(\theta_i\), the spatial random effect, is modeled using a BYM2 spatial prior (Riebler et al., 2016);
  • \(\gamma_t\), the temporal random effect, is modeled using an intrinsic conditional autoregressive prior (Lee, 2011); and
  • \(\delta_{it}\), the space-time random effect, is modeled using a standard normal distribution (Type I interaction) which assumes independent variation
• \(\alpha\) be an intercept
• \(N_{ita}\) be the population.

Model \[ \begin{align} y_{ita} & \sim \text{Poisson}\left( \mu_{ita} \right) \\ \log\left( \mu_{ita} \right) & = \log\left( N_{ita} \right) + \alpha + \beta_a + \xi_{it} \end{align} \]

Weighted Multilevel Regression and poststratification (MrP)

\[ \begin{eqnarray} z_{itaf} & \sim & \text{Bernoulli}\left( \pi_{itaf} \right)^{\tilde{w}_{itaf}} \label{eq:wmrp} \\ \text{logit}\left( \pi_{itaf} \right) & = & \alpha_{af} + \mathbf{x}_{it} \boldsymbol{\beta} + \theta_i + \gamma_t, \nonumber \\ \alpha_{af} & \sim & \text{N}\left( 0, 1000^2 \right) \nonumber \\ \boldsymbol{\theta} & \sim & \text{BYM2}\left( \mathbf{W}^{\text{S}}, \rho, \kappa, \sigma_\theta^2 \right) \nonumber \\ \boldsymbol{\gamma} & \sim & \text{ICAR}\left( \mathbf{W}^{\text{T}}, \sigma_\gamma^2 \right) \nonumber \end{eqnarray} \]

Prediction \[ YLD^{(d)}_{it} = \sum_{f,a,h} \pi^{(d)}_{itaf} N_{itaf} \psi_h e_h \]