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### HEALTH APPLICATIONS OF BAYES TECHNIQUES

### What is population health?

### Application Theme 5

### Spatial Common Factor Models

Population Health Perspectives

Peter CongdonResearch Professor of Quantitative Geography & Health Statistics, QMUL

- e-mail: p.congdon@qmul.ac.uk
- http://www.geog.qmul.ac.uk/staff/congdonp.html

http://webspace.qmul.ac.uk/pcongdon/

Major tasks in definition and analysis

Population Health: Ecological (Contextual) Risk Factors

- To describe/analyze health variation over
- areas or area categories (poverty status, area socioeconomic classifications, “deprivation quintiles”)
- by area SES scales (deprivation gradients), or other area characteristics (social “fragmentation”, social capital)
- according to area environmental exposures (e.g. pollution levels or categories)

Population health: individual level risk factors

- To describe health variation over
- demographic categories (age, race, gender, family type)
- individual socioeconomic variables (income, education)
- health behaviours (smoker or not, obese or not)
- Assess how individual and contextual factors (aka upstream & downstream factors) interact in their impacts on health

Role of Statistical Analysis

- Assess which potential sources of variation in health are significant (or not)
- Summarise health variations parametrically
- Provide stable estimates

Policy Implications of Population Health Analysis: Tasks

- Assess how health need (need for healthcare) is distributed over areas or social groups
- to guide distribution of scarce health resources and effective targeting of healthcare interventions
- May involve “health need indices” based on characteristics of areas or area populations

AREA STUDIES.

- A major focus of my talk will be on models for spatial variations in health, and predictors of those variations (“ecological studies”)
- These models typically use area counts (deaths, incidence or prevalence totals) from official registration systems
- Statistical models often seek to assess the implications of area health variation (e.g. locating areas with excess risk, ranking areas according to health risk, measuring inequality, smoothing ragged observed area rates).

Area Studies (continued)

- Relevance of “ecological studies” (despite “ecological fallacy”) to broader upstream/downstream debate: what are contextual effects, what causes them, how should they be modelled, etc
- Crude rates for rare events unreliable stabilized/robust/smoothed area health outcome rates essential to accurate description of population health.
- Smoothing may draw on spatial structure of known or unknown risk factors

MULTILEVEL & SURVEY STUDIES.

- I will also consider multilevel perspectives
- Assess how individual and contextual factors interact in their impacts on health, e.g. area variables may act as effect modifiers for individual risk factors.
- Health surveys (e.g. Health Survey for England, Behavioral Risk Factor Surveillance System) are the most suitable for analysing the effects on health of age, ethnicity, and individual SES, and their possible modification by geographic variables.
- But administrative or census data also can be analyzed profitably by ML methods

Outline

RELEVANT METHODS

- Some distinctive aspects of methodologies for modelling population health

APPLICATION THEMES

- 1 Assessing varying health risks in areas
- 2 Spatially varying predictor effects
- 3 Age and area: life table methods
- 4 Spatial aspects of health care use
- 5 Multilevel modelling
- 6 Prevalence Modelling
- 7 Common Spatial Factor Models

Relevant Methods: general linear model regression

- General linear models (e.g. with count or binary response) more frequently used than linear models
- For health survey data often need binary regression, with logit or log link, and maybe accounting for differential survey weights
- For area counts of health events (e.g. deaths, prevalence) typically need Poisson or binomial or over-dispersed versions of these densities

Relevant methods: pooling strength

- Often use random effects to pool strength (or borrow strength) over areas or other relevant dimensions (e.g. age)
- Essentially refers to adding stability/precision to estimates by referring to overall population density
- Maybe pool strength over variables too multivariate random effect and common factor models

Relevant methods: spatially oriented applications

- In area health applications, both outcomes (e.g. mortality or prevalence) and ecological risk factors (e.g. area deprivation, area smoking rates) are typically spatially structured. Also applies to “unknown” risk factors
- So in statistical models, spatially correlated random effects often involved (in Bayesian terminology “spatial priors”)
- Modelling aims to account for spatial structure
- Inter alia, a good model will ensure regression residuals are free of spatial correlation

Relevant methods: hierarchical nesting and interactions across levels

- In multilevel modelling effects of individual risk factors may vary according to area contexts
- For example, ethnic relativities in diabetes prevalence may not be constant over areas
- So use random effects (maybe spatially structured) to model spatial variation in impacts of individual level risk factors

Relevant methods: benefits of MCMC and Bayesian techniques

- Bayesian approach using MCMC sampling assists in monitoring “derived parameters” or outputs, providing full densities, and in testing hypotheses about derived parameters.
- Classical estimation typically provides confidence intervals under assumed asymptotic normality for model parameters only, with delta method for derived parameters
- Bayesian approach arguably more flexible for models with multiple or nested random effects, or where there is partially missing data

An example of a derived model output that is not the model response. Model response (Poisson) are deaths by area & age. Derived model output is life expectancy

EASTERN REGION OF ENGLAND, MALE LIFE EXPECTANCY.

Ref: Congdon , 2009, International Statistical Review

Relevant methods: latent variable techniques

- Many relevant risk or outcome variables for analysing population health can be regarded as latent constructs, not directly observed but proxied by several observed variables
- Examples in area studies: area unemployment or rates of social housing are proxies for area construct “deprivation”
- Examples in survey studies: battery of survey items on neighbourhood perceptions and trust are proxies for individual level construct “social capital”

APPLICATION THEME 1

- ASSESSING VARYING HEALTH RISKS IN AREAS

Maximum Likelihood

- Observed data (e.g. death totals by area i) are y[i], and E[i] are expected event totals.
- Limitations of conventional (fixed effects) maximum likelihood estimates of relative risks (or “standard mortality ratios”) y[i]/E[i] as description of spatial contrasts.
- OR data might be y[i] and populations P[i], MLE (e.g. crude death rate) is y[i]/P[i] (or such rates feed into “age standardised” rate)
- OR data: y[i] (infant deaths) and births B[i]. MLE is y[i]/B[i]

MLE estimation (continued)

- Maximum likelihood approach (underlies conventional demographic techniques) treats each area (or risk category) as a separate isolated entity, taking no account of:
- overall average for the event,
- the location of the area, or risk category in relation to other areas (or risk categories)
- By neglecting the broader context, MLE estimates also potentially unstable

Bayesian Approach

- Under Bayesian random effects, information on the pattern of disease risk across the collectivity of areas (or risk categories) is used to provide an estimate of the underlying relative risk for each area (or risk category)
- Treat each area’s outcome with reference to the ensemble of areas
- The “prior” specifies the chosen overarching density of relative risk (e.g. normal or gamma) and whether or not the density specifies local or global pooling of strength.

Adaptive Spatial Priors

- However, may be unwise to uncritically assume complete spatial dependence - or homogenous spatial correlation.
- So allow for some unstructured variation or for spatial outliers
- Spatial outliers: areas unlike their neighbours, e.g. socially dissimilar (example, suburban “social housing” estates surrounded by owner occupied housing areas)
- Allow extent of spatial dependence to vary across the map
- Congdon, 2008, Statistical Methodology

Policy relevant posterior inferences

- Use of spatial risk modelling for policy inferences
- One may assess for example, the posterior probability that a particular area has an elevated relative risk (compared to the average)
- Assume RR=1 on average. Then simply count the proportion of MCMC iterations where condition RR[i]>1 holds
- More complicated to do this under frequentist approaches
- e.g. Congdon, Health and Place, 1997, article on area contrasts in suicide and attempted suicide in NE London

APPLICATION THEME 2

- EXTENDING THE SMOOTHING PRINCIPLE: Spatial Heterogeneity In Regression Effects

Spatial Models for Regression Effects

- Spatial pooling of strength may be applied not only to disease risks but to effects of area risk factors. Example: how are lung cancer incidence relativities iaffected by area smoking rates xi
- Conventionally assume constant slope on xi over all areas
- However, risk relationship may vary (smoothly) over space varying slopes i
- e.g. Congdon, Health and Place, 1997, article on area contrasts in suicide and attempted suicide in part of NE London (x=deprivation)

Application Theme 3

- EXTENDING THE SMOOTHING PRINCIPLE: Smoothing over areas and ages to derive small area life tables

Modelling area and age effects

- Modelling mortality data yix (and maybe illness data hix too) by both area i and age group x
- As before, neighbouring areas have similar rates under prior incorporating spatial dependency
- But also assume neighbouring ages have similar rates under pooling (random effects) prior
- Technically, often use “state space” or “random walk” priors for age effects

Why assume similar ages related: strong correlation in successive age effects

Health and Mortality

- Congdon 2006, Demographic Research, A model for geographical variation in health and total life expectancy
- Spatial Framework, 33 London Boroughs, ca 230k population on average
- Use illness data (long term ill status from 2001 UK Census) as well as deaths data (bivariate outcome)
- With mortality and illness data can model both total life expectancy and healthy life expectancy - difference between expectancies is expected years lived in disability (“disease burden”)
- Correlation between disease burden & area deprivation

Life Expectancies

- Calculate life expectancies Eix for areas i and ages x using usual life table calculations and “smoothed” age and area specific mortality rates Mix
- Life expectancy at birth Ei0.
- Monitor “derived outcomes” Ei0 in MCMC whereas likelihood for deaths uses Mix (“actual model parameters”)
- Problems with conventional calculations for life expectancies when populations small, rates Mix unstable apply Bayesian random effects smoothing

Goal: model should reflect spatial clustering in “derived outcomes”

- Congdon (2007) A model for spatial variations in life expectancy; mortality in Chinese regions. Int J Health Geographics
- Negative binomial model because of large death counts/overdispersion but allowing for correlated area and age effects

Spatio-temporal models

- Similar ideas apply if the second dimension is time rather than age
- Correlation between adjacent times is expected and should be included in the model
- For example, could have “random walk” in time parameters

Area-age-time model with “derived output”

- Congdon, 2004, J Appl Stat “Modelling Trends and Inequality in Small Area Mortality” has three dimensions: area, age, time (years) in an analysis of area mortality through time
- “Derived outputs” monitored by MCMC are Theil and Gini indices of inequality in life expectancies Eit between areas i =(1,..,n) at year t.
- If Rit=Eit/Et where Et is average, then Theil entropy index in year t is

Ht=i [Ritlog(Rit)]/n

Health care usage

- Mortality, prevalence and incidence variations between areas reflect primarily population health need (e.g. age/ethnicity/SES composition), maybe together with area contextual effects
- But health care use in different areas (e.g. hospitalisation rates) affected not only by population need but by health supply factors & efficacy of different health sectors
- Same applies to flows f[i,j] from areas i to care providers j, e.g. of acute (hospital) care

Varying hospital admission rates

- For example, emergency hospital use in different areas i affected (inter alia) by
- area deprivation, age structure, etc
- efficacy of primary care in handling chronic disease (and preventing “ambulatory sensitive” emergency hospital admissions)
- access to primary care (e.g. primary care physicians per head, adequate “out of hours” cover)
- referral thresholds
- hospital capacity
- proximity of area i to hospitals j allowing for competing/intervening populations in other areas

Gravity models for area to hospital flows

- Gravity models for flows f[i,j] from area i to providers j take account of:
- Population sizes in areas i (maybe weighted for need) and capacity of hospital provider j (e.g. bed mass, staff)
- Efficacy of, and access to, primary care
- Distance decay (home to hospital distances)
- Relative accessibility R[i,j]: capacity B[j] of providers j adjusted for distances d[i,j] relative to other providers
- Can model situation of new sites, or hospital closures (via Bayesian “predictions”) using implied changes in R[i,j]

Multilevel models

Multilevel Models

- Usual paradigm: want to assess effects on health of
- Compositional variables (individual level risk factors)
- Contextual influences (area variations)
- Interplay between composition and context
- Example is additive effect: poor people more likely to smoke, but poor people living in deprived areas more likely to smoke than poor living in less deprived areas.

Preliminaries: intercept and slope variation

- For example suppose yij is binary health status (e.g. whether long term ill) for individuals i in areas j, and xij is individual level measure of socioeconomic status (e.g. years of education).
- Then ij=Probability(yij=1)
- Logit regression to predict ij,
- ij=1/(1+exp(-ij)
- Intercept variation only ij=j+xij
- Intercept and slope variation ij=j+jxij

Central Questions

- Are there “place effects”: does inter-area health variation (intercept variation) remain when individual risk variables are added into the model
- Conversely are effects of individual variables diminished when area effects added
- Explaining “contextual variation” or “place effects” in substantive terms. Which aspects of areas cause contextual effects (e.g. “healthy food access”, deprivation amplification, residential segregation)
- How should (known and unknown sources of) contextual variation be modelled (e.g. spatial prior or not)
- Do impacts of individual risk factors vary by area (interaction between levels). This includes slope variation, as well as more complex forms for categorical predictors (e.g. multivariate conditional autoregressive prior)

Example: place effects on obesity

- Neighbourhood access to healthy environments, positive health choices
- Access to healthy food outlets (e.g. work by Neil Wrigley on “Food Deserts” in British cities). Many GIS studies on access to healthy food
- Access to physical activity facilities also an influence on child & adolescent obesity

Possible analytic frameworks

- Usual paradigm for multilevel model is individual level observation nested according to higher level index (e.g. area, school); scheme A
- However, to make analysis feasible when there are many individuals, might turn all individual risk factors into category form and group observations according to risk category; scheme B.
- Also, sometimes lower level units may be small areas (“neighbourhoods”), and higher level units might be larger policy areas; scheme C

Example scheme B, (3 levels) Congdon, J Roy Stat Soc C,1998

- Binary birth events y at level 1 (e.g. stillbirth) as well as maternal characteristics x (mothers age, whether lone mother, etc).
- Individual events nested within J=25 districts, in turn nested within K=7 health authorities (HA’s)
- Form risk groups by cross-classifying (a) maternal age (<20, 20-34,> 35), (b) parity (null,1-2, 3+), (c) previous still-birth (y/n) (d) lone mother (y/n).

Multi-level model for risk categories

- Maximum combinations based on these factors is 900=(3 x 3 x 2 x 2 x 25), of which 549 are non-empty.
- Can look at contextual effects for both districts and HA’s in n=549 “collapsed” data points.
- In particular (policy implications): do HA rankings (monitored by MCMC) change before and after controlling for risk factors at both individual and district level

Scheme C: Congdon, Reg Studies, 1995

- Reduction of intra-district health gradients (over small areas) often forms focus of public health targets. But raises methodological issues…
- Study of long term illness (LTI) rates in i=1,..,1332 wards (small areas) within j=1,.,53 districts (London & Eastern England).
- Slope variations: within district slopes jrelating LTI to small area deprivation (binomial data)
- Negative intercept-slope covariation, cov(j, j): stronger deprivation effects occur in districts with lower LTI rates. Slopes in low LTI districts enhanced by very low illness rates in some wards.

Relative Deprivation

- Relates to broader themes of potential relative deprivation impacts on health: districts with lower average illness may be more internally heterogeneous in terms of small area SES
- Not only absolute deprivation or income that matters for health, but income or deprivation relative to the average of reference group.

APPLICATION AREA 6

- GEOGRAPHIC PREVALENCE ESTIMATION (based usually on multilevel modelling)

Diabetes prevalence estimates in US Zip Code Tabulation Areas (ZCTAs)

- Rising US diabetes levels (Mokdad et al, 2001)
- Information regarding small area prevalence important for effective targeting of diabetes prevention and resources
- Prevalence estimates should incorporate
- spatial clustering
- ethnic relativities
- interactions across levels, e.g. between demography & area

Spatial variation & Clustering

Crude diabetes prevalence rate among adults (source; Behavioural Risk Factor Surveillance System)

Differentiation by ethnicity and cross-level interactions

- In 2007 age standardised rate of diagnosed diabetes was highest among Native Americans and Alaska Natives (16.5%), followed by blacks (11.8%) and Hispanics (10.4%), with whites at 6.6 % (CDC, 2008).
- Age gradient for diabetes prevalence varies by race
- Barnett et al (2001) & Casper et al (2000) also report that ethnic disparities vary by area of residence (their work is on CHD mortality)

Congdon/Lloyd, J Data Sci, 2010

- Survey regression model based on BRFSS survey data. Binary multilevel model for doctor diagnosed diabetes. State of residence for survey subjects is at level 2
- Seek diabetes prevalence estimates for 30,000 Zip Code Tabulation Areas in US (http://www.census.gov/geo/ZCTA/zcta.html)
- ZCTA is not provided as a survey variable. Instead use survey regression parameters in conjunction with ZCTA census data (and the ZCTA’s geographic location) to derive ZCTA prevalence estimates

Matching Risk Factors – Survey & ZCTA

- BRFSS survey model includes age, race, sex, and education effects together with contextual modifiers
- Inclusion of individual risk factors (e.g. age, race) in survey regression model presumes that such factors also available in census tabulations for ZCTA populations.
- Any interaction between risk factors in regression model (e.g. age gradients differing by race) similarly requires matching census cross-tabulation
- However, survey model can include geographic context variables (e.g. state or county level random and fixed effects). These are applied according to ZCTA geographic location

Detailed Aspects of BRFSS Diabetes Model for J Data Sci work

- Differential risks for race (whites, black, hispanic, other races) and for education (four categories) (fixed effects)
- Fixed regression effects of state level predictors, (poverty rate & percent rural)
- Random differential risks specific to age-race group (multivariate CAR, dimension 4 by 12)
- Race specific spatially correlated effects for continental states (use multivariate CAR), {sjr, j=1,..,49,r=1,..,4}
- Race specific spatially unstructured effects for all states, (multivariate normal) {ujr, j=1,..,53,r=1,..,4} (includes Hawaii, PR, Alaska, VI)

Compositional adjustment for SES structure of ZCTA populations

- Census provides education and poverty breakdown for ZCTAs
- However, poverty status not in BRFSS
- So education gradient in prevalence from BRFSS survey model used to adjust ZCTA prevalence estimates for small area education mix (“compositional” adjustment)
- BRFSS model does shows clear education gradient in prevalence (steeper for females)
- Adjustment ensures ZCTAs with more college graduates have lower prevalence

Contextual adjustments

- Estimates for a particular ZCTA include relevant random state effects, s[j,r] and u[j,r] (applied in prevalence estimates according to state j the particular ZCTA is located in).
- ZCTA estimates also adjust for relevant state poverty/urbanity effect
- Can apply same principle to counties (i.e. 3141 US counties at level 2) but then can’t easily model random area by race interactions. Also survey data sparse for many counties.

Age standardised prevalence since 2000

- Use European standard population weights or US 2000 Census weights to combine over age/race categories
- Provides age standardised prevalence estimate for all persons
- Apply same survey model to different years

Joint diabetes-obesity prevalence model: counties as level 2

- Congdon (Int J Env Res Pub Hea 2010) uses random county effect in prevalence model for joint conditions
- Joint response over six diabetes- weight categories: diabetes (y/n) & obesity/overweight/normal weight
- Using BRFSS again
- In fact county random effect is form of common spatial factor over different categories of the joint response

Application Area 7

Methodological aspects

- Most commonly applied when there are many indicators (e.g. different types of cancer incidence) and just one common factor
- But will give an example of multiple underlying dimensions
- In spatial health applications, the common factor is usually spatially structured (value for particular area depends on those of its neighbours)
- Can have “multiple causes” of spatial common factors as well as “multiple indicators”

Statistical motivations for common factor models

- Pool over diverse outcomes (these are indicators of the underlying common factor) as a way of pooling strength to assess underlying morbidity levels
- Some indicators may be infrequent, some may be subject to partially missing values (e.g. lung cancer county incidence data not comprehensive across all US states)
- So the common factor pools information of health relativities over different observed indicators (and imputes/predicts values where they are missing)

Substantive health care motivations: health need

- Can use common factor models to develop univariate indices of “health need” (need for healthcare based on demographic/social composition of population & also maybe taking account of relativities in healthcare usage) e.g. Congdon, J Geo Syst, 2008
- Conventional methods for needs indices (e.g. Mental Illness Needs Index) are aspatial and usually based on regressing health activity on bundle of socioeconomic indices. Need index then derived from significant coefficients.

More than one common factor to describe area social structure

- Develop multivariate factor models of different aspects of area social structure with relevance to health outcomes
- Example: area deprivation, area social fragmentation, and rurality all relevant to suicide contrasts
- Explanatory ecological model for US suicide contrasts (male & female suicide 2002-2006)

Multiple causes ( MIMIC models)

- Education application. Multiple tests are indicators of underlying ability
- But investigator might also want to know what causes variations in ability.
- Potential causes might include gender, parent SES, type of school, etc

HSR Application

- Variations in avoidable (“ambulatory sensitive”) hospital admissions. Usually unplanned emergencies.
- Such admissions typically for chronic conditions, and in many cases could be avoided with suitable primary/community care

MIMIC model for primary care quality

- Possible “multiple indicators” of underlying “quality of care” factor for GP practices:
- Emergency admission rate by practice
- Ambulatory sensitive admission rate
- Attendance rate (unplanned attendances at “emergency room” or “Accident and Emergency Unit” that usually don’t result in hospital admission)
- Possible “multiple causes”
- GP practice deprivation
- Average proximity of patients (affiliated to each GP Practice) to hospital

Multiple Causes in Spatial Factor Model

- Similarly Congdon 2010 (J Stat Comp Sim) uses mortality and hospitalisation data (as “indicators”) for CHD need index
- Spatially structured need index for 625 London wards (small areas)
- But ward income and unemployment act as “causes”
- Adjust spatial prior (Besag et al CAR or Leroux et al 1999 model) to allow for predictors

Common factor for joint outcomes

- US prevalence model application based on BRFSS with bivariate outcome
- “Multiple indicators” are six categories defined by diabetic status and weight band:
- diabetic and obese (y=1)
- diabetic and overweight (y=2)
- diabetic and normal weight (y=3)
- non-diabetic and obese (y=4)
- non-diabetic and overweight (y=5)
- non-diabetic and normal weight (y=6).

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