Nicky Best, Chris Jackson, Sylvia Richardson

1
Studying place effects on health by synthesising
individual and area-level outcomes using a new
class of multilevel models
Nicky Best, Chris Jackson, Sylvia Richardson
Department of Epidemiology and Public
Health Imperial College, London
http//www.bias-project.org.uk
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Outline

• Introduction and motivating example
• Models for analysing individual and contextual
  effects
• Standard multilevel model
• Ecological regression
• Hierarchical related regression
• Concluding remarks

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A Introduction and motivating example
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BIAS project Overall goals

• To develop a set of statistical frameworks for
  combining data from multiple sources
• To improve our capacity to handle biases inherent
  in the analysis of observational data.
• Key statistical tools Bayesian hierarchical
  models and ideas from graphical models form the
  basic building blocks for these developments

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Example Socioeconomic predictors of health

• Question
• Characterising individual level socio-demographic
  predictors of limiting long term illness (LLTI)
  and heart disease
• Is there evidence of contextual effects?
• Design
• Data synthesis using
• Individual-level survey data Health Survey for
  England.
• Area-level administrative data Census small-area
  statistics and Hospital Episode Statistics
• Methodological issues
• Sparse individual data per area (0-9 subjects per
  area) so difficult to estimate contextual effects
• Cant separate individual and contextual effects
  using only aggregate data (ecological bias)
• Improve power and reduce bias by combining data

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B Models for analysing individual and contextual
effects
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Multilevel model for individual data
b
g
s2
ai
xij
yij
Zi
person j
area i
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Multilevel model for individual data
yij Bernoulli(pij), person j, area i
b
g
s2
logit pij ai b xij g Zi
ai
xij
yij
Zi
person j
area i
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Multilevel model for individual data
yij Bernoulli(pij), person j, area i
b
g
s2
logit pij ai b xij g Zi
ai
ai Normal(0, s2)
xij
yij
Zi
person j
area i
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Multilevel model for individual data
yij Bernoulli(pij), person j, area i
b
g
s2
logit pij ai b xij g Zi
ai
ai Normal(0, s2)
xij
yij
Zi
Weak priors on s2, b, g
person j
area i
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Multilevel model for individual data
yij Bernoulli(pij), person j, area i
b
g
s2
logit pij ai b xij g Zi
ai
ai Normal(0, s2)
xij
yij
Zi

• Weak priors on s2, b, g
• b individual-level effects
• g contextual effects
• ai unexplained area effects

person j
area i
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Data sources

• INDIVIDUAL DATA
• Health Survey for England
• Self-reported limiting long term illness
• Self reported hospitalisation for heart disease
• age and sex
• ethnicity
• social class
• car access
• income
• etc.

• AREA (WARD) DATA
• Census small area statistics
• Carstairs deprivation index

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Results from analysis of individual survey data
Heart Disease (n5226)
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Results from analysis of individual survey data
Limiting Long Term Illness (n1155)
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Comments

• CI wide and not significant for most effects
• Some evidence of contextual effect of area
  deprivation for both heart disease and LLTI
• Adjusting for individual risk factors
  (compositional effects) appears to explain
  contextual effect for heart disease
• Unclear whether contextual effect remains for
  LLTI after adjustment for individual factors
• Survey data lack power to provide reliable
  answers about contextual effects
• What can we learn from aggregate data?

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Area-level data

• AREA (WARD) DATA
• Census small area statistics
• Carstairs deprivation index
• population count by age and sex
• proportion reporting LLTI
• proportion non-white
• proportion in social class IV/V
• proportion with no car access
• PayCheck (CACI)
• mean variance of household income
• Hospital Episode Statistics
• number of admissions for heart disease

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Ecological inference

• This is the group level association. Not
  necessarily equal to individual-level association
• i.e. b ? b ? ecological bias

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Standard ecological regression model
c
b
s2
ai
Zi
Yi
Ni
area i
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Standard ecological regression model
c
b
s2
logit qi ai bXi cZi
ai
Zi
Yi
Ni
area i
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Standard ecological regression model
Yi Binomial(qi, Ni), area i
c
b
s2
logit qi ai bXi cZi
ai
ai Normal(0, s2)
Zi
Yi
Ni
area i
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Standard ecological regression model
Yi Binomial(qi, Ni), area i
c
b
s2
logit qi ai bXi cZi
ai
ai Normal(0, s2)
Zi
Yi
Priors on s2, b, c
Ni
area i
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Comparison of individual and ecological
regressions Heart Disease
Individual
Area deprivation
Ecological
No car
Social class IV/V
Non white
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Comparison of individual and ecological
regressions Limiting Long Term Illness
Individual
Area deprivation
Ecological
Female
Non white
Doubled income
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Ecological bias

• Bias in ecological studies can be caused by
• Confounding
• confounders can be area-level (between-area) or
  individual-level (within-area).
• Solution try to account for confounders in model
  
• Non-linear exposure-response relationship,
  combined with within-area variability of exposure
  
• No bias if exposure is constant in area
  (contextual effect)
• Bias increases as within-area variability
  increases
• unless models are refined to account for this
  hidden variability

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Improving ecological inference

• Alleviate bias associated with within-area
  exposure variability.
• Obtain information on within-area distribution
  fi(x) of exposures, e.g. from individual-level
  exposure data.

• Use this to form well-specified model for
  ecological data by integrating (averaging) the
  underlying individual-level model.

• Yi Binomial(qi , Ni) qi ? pij(x) fi(x)
  dx
• qi is average group-level risk
• pij(x) is individual-level risk given covariates
  x
• fi(x) is distribution of exposure x within area
  i (or joint distribution of multiple exposures)

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Improving ecological inference

• Suppose we have single binary covariate x

• Individual-level model
• log pij a b xij (log link
  assumed for simplicity)
• ? pij ea if person j is unexposed
  (xij0)
• pij eab if person j is exposed
  (xij1)

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Standard ecological regression model
Yi Binomial(qi, Ni), area i
c
b
s2
logit qi ai bXi cZi
ai
ai Normal(0, s2)
Zi
Yi
Priors on s2, b, c
Ni
area i
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Integrated ecological regression model
Yi Binomial(qi, Ni), area i
g
b
s2
qi ? pij(xij,Zi,ai, b,g)fi(x)dx
ai
ai Normal(0, s2)
Zi
Yi
Priors on s2, b, g
Ni
area i
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Combining individual and aggregate data
Multilevel model for individual data
Integrated ecological model
b
g
g
s2
b
s2
ai
ai
Zi
xij
yij
Yi
Zi
person j
Ni
area i
area i
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Combining individual and aggregate data
b
s2
g
Hierarchical Related Regression (HRR) model
Joint likelihood for yij and Yi depending on
shared parameters b, g, s2
ai
xij
yij
Yi
Zi
person j
Ni
area i
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Extending HRR model to multiple covariates
b
g
s2
ai
xij
yij
Zi
Yi
person j
Ni
area i
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Extending HRR model to multiple covariates
b
g
s2
ai
xij1
yij
Zi
Yi
xijQ
person j
Ni
area i
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Extending HRR model to multiple covariates
b
g
s2
district d
person k
xdk1
xdkQ
ai
fi
xij1
yij
Zi
Yi
xijQ
person j
Ni
area i
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Extending HRR model to multiple covariates

• Suppose x1xQ are all binary variables
• R 2Q possible combinations
• fi fi1,, fiR where fir is probability that
  individual in area i has covariate combination r
  (r1,,R)
• We estimate fi using Q-way cross-tabulation of
  covariates in district d(i) from Sample of
  Anonymised Records (SAR)..
• with constraint that marginal probabilities for
  each covariate match observed ward proportions
  from Census
• Assumes within-district correlations are
  representative of within-ward correlations for
  all wards in a district

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Combined data

• INDIVIDUAL DATA
• Health Survey for England
• health outcomes and covariates
• ward code available under special license

• AREA (WARD) DATA
• Census small area statistics
• PayCheck (CACI)
• Hospital Episode Statistics
• aggregate health outcomes
• aggregate covariates (marginal)

• Sample of Anoymised Records (SAR)
• 2 sample of individual data from Census
• district code available
• provides estimate of within-area distribution of
  covariates
• ? assume same distribution for all wards within a
  district

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Comparison of results from different regression
models Heart Disease
Individual
Area deprivation
Standard ecological
Integrated ecological
No car
HRR
Social class IV/V
Non white
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Comparison of results from different regression
models Limiting Long Term Illness
Individual
Area deprivation
Standard ecological
Integrated ecological
Female
HRR
Non white
Doubled income
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Unexplained area variability in risk

• Random effects account for unexplained
  differences in risk between areas, after
  accounting for observed covariates
• Large variance s2 ? large unexplained differences
• Median odds ratio (Larsen Merlo 2005) is a
  simple transformation of s2 to scale of odds
  ratio
• MOR exp( v2s F-1(0.75) )
• MOR median of the residual odds ratios over all
  pairs of areas
• Directly comparable to odds ratio for an observed
  covariate

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Unexplained area variability in risk of Heart
Disease
Area deprivation
Individual
HRR
No car
Social class IV/V
Non white
MOR
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Unexplained area variability in risk of LLTI
Area deprivation
Individual
HRR
Female
Non white
Doubled income
MOR
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Comments

• Integrated ecological model yields odds ratios
  that are consistent with individual level
  estimates from survey
• Large gains in precision achieved by using
  aggregate data
• Significant contextual effect of area deprivation
  for LLTI but not heart disease
• For LLTI, unexplained area variation is small
  compared to that explained by deprivation
  (MOR1.2, deprivation OR2.6)
• For heart disease, there is more unexplained area
  variation (MOR1.5)

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Comments

• Little difference between estimates based on
  aggregate data alone and combined individual
  aggregate data
• Individual sample size very small (0.1 of
  population represented by aggregate data)
• In other applications with larger individual
  sample sizes and/or less informative aggregate
  data, combined HRR model yields greater
  improvements (see simulation study)
• Care needed to check consistency between data
  sources

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Simulation Study
log RR of disease for exposed
whites
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Are aggregate and individual data consistent?
Health Survey for England aggregated over
areas 1991 Census
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Are aggregate and individual data consistent?

• LLTI
• Health Survey for England 23
• Census 13
• Similar discrepancies noted by other authors
• May reflect differences between interview and
  self-completed surveys
• Remedy include fixed offset in regression model
  for Census data

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C Concluding remarks
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• Aggregate data can be used for individual level
  inference if appropriate integrated model is used
• requires large exposure contrasts between areas
• requires information on within-area distribution
  of covariates
• Combining samples of individual data with
  administrative data can yield improved inference
• improves ability to investigate contextual
  effects
• increase statistical power compared to analysis
  of survey data alone
• requires geographical identifiers for individual
  data
• Important to check compatibility of different
  data sources when combining data
• Important to explore sensitivity to different
  model assumptions and data sources

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• Jackson C, Best N and Richardson S. (2008)
  Studying place effects on health by synthesising
  individual and area-level outcomes. Social
  Science and Medicine, to appear.
• Papers available from
• www.bias-project.org.uk
• Thank you for your attention!