Transcript Document

Bayesian graphical models for inference from
combinations of data
Nicky Best and Chris Jackson
With Sylvia Richardson
Department of Epidemiology and Public Health
Imperial College, London
[email protected]
http://www.bias-project.org.uk
Example: low birth weight and air pollution
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Does exposure to air pollution during pregnancy
increase the risk of low birth weight?
Example illustrates various biases.
Combine datasets with different strengths:
 Survey data (Millennium Cohort Study)
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Administrative data (national births register)
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Small, great individual detail.
Large, but little individual detail.
Single underlying model assumed to govern both
datasets: elaborate as appropriate to handle biases
Low birth weight
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Important determinant of future health  population
health indicator.
Established risk factors:
 Tobacco smoking during pregnancy.
 Ethnicity (South Asian, issue for UK data)
 Maternal age, weight, height, number of previous
births.
Role of environmental risk factors, such as air
pollution, less clear.
 Various studies around the world suggest a link.
 Exposure to urban air pollution correlated with
socioeconomic factors  ethnicity, tobacco
smoking  confounding
Data sources (1): Millennium Cohort Study
About 15,000 births in the UK between Sep 2000
and August 2001 (we study only England and Wales, singleton births)
 Postcode made available to us under strict security
 Match individuals with annual mean
concentration of certain air pollutants (PM10,
NO2, CO, SO2) (NETCEN)
 Birth weight, and reasonably complete set of
confounder data available
Allows a reasonable analysis, but issues remain:
 Low power to detect small effect  could be
improved by incorporating other data.
 Selection bias…
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Selection of Millennium
Cohort
SELECTION PROBABILITY
High child poverty
0.04
Low child poverty
0.02
High ethnic minority
0.11
High child poverty
0.07
Low child poverty
0.04
High child poverty
0.18
Low child poverty
0.06
High child poverty
0.16
Low child poverty
0.08
ENGLAND
SCOTLAND
ALL UK WARDS
WALES
NORTHERN
IRELAND
Selection bias in the Millennium Cohort
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Survey disproportionately represents population.
 If selection scheme (=child poverty / ethnicity)
related to exposure (=pollution) and outcome (=low
birth weight), then estimate of association biased.
Accounting for selection bias:
 Adjust model for all variables affecting selection, or
 Weight cases by inverse probability of selection
Cluster sampling  within-ward correlations
 for correct standard errors for inference on
population, use a hierarchical (multilevel) model with
groups defined by wards.
Data sources (2): National birth register
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Every birth in the population recorded.
Individual data with postcode ( pollution exposure)
and birth weight available to us under strict security.
Social class and employment status of parents also
available for a 10% sample.
We study only this 10% sample: 50,000 births
between Sep 2000 and Aug 2001.
Larger dataset, no selection bias,
…but no confounder information, especially
ethnicity and smoking.
Data sources (3): Aggregate data
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Ethnic composition of the population
 2001 census
 for census output areas (~500 individuals)
Tobacco expenditure
 consumer surveys (CACI, who produce ACORN
consumer classification data)
 for census output areas.
…linked by postcode to Millennium Cohort and
national register data.
Birth weight
and pollution
(source: MCS)
Birth weight
and ethnicity
(source: MCS)
Birth weight
and smoking
(source: MCS)
Pollution and
confounders
(source: MCS)
Models for formally analysing combined data
Want estimate of the association between low
birth weight and pollution, using all data,
accounting for:
 Selection bias in MCS
 Adjust models for all predictors of selection
 Or weight by inverse probability of selection
 Missing confounders in register
 Bayesian graphical model…
Graphical model representation
ETHi
POLLi
POLLj
ETHj
MODEL
LBWi
baby i in register
LBWi:
LBWj
baby j in MCS
low birth weight
POLLi:
pollution exposure (plus other confounders observed
in both datasets)
ETHi:
known
ethnicity and smoking. Only observed in the MCS.
Same MODEL assumed to govern both datasets.
unknown
Adding in the imputation model
AGGi
ETHi
MODEL
(imputation)
POLLi
AGGj
POLLj
ETHj
MODEL
(LBW)
LBWi
baby i in register
AGGi:
LBWj
baby j in MCS
aggregate ethnicity/smoking data for area of residence of baby i
MODEL for imputation
of ETHi in terms of aggregate data and other variables.
Estimate it from observed ETHj in the MCS.
Bayesian model
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Estimate both:
 Imputation model for missing ethnicity and smoking
 Outcome model for the association between low
birth weight and pollution.
All beliefs about unknown quantities expressed as
probability distributions.
 Prior beliefs (often ignorance) modified in light of data 
posterior distributions
Joint posterior distribution of all unknowns estimated by
Markov Chain Monte Carlo (MCMC) simulation
(WinBUGS software)
Graphical representation of the model guides the
MCMC simulation.
Variables in the final models:
(1) regression model for low birth weight
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Probability baby i has birth weight under 2.5 kg
modelled in terms of
 Pollution (NO2 and SO2)
 Ethnicity (White / South Asian / Black / other)
 Smoking during pregnancy (yes/no)
 Social class of mother
 Survey selection strata (for MCS data)
Other variables not significant in multiple regression, or
not confounded with pollution (mother’s weight, height,
maternal age, number of previous births, hypertension
during pregnancy,…)
Variables in the final models:
(2) imputation model for missing data
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Probability baby i is in one of eight categories:
 ethnicity 1. White / 2. South Asian / 3. Black / 4. other
 smoking during pregnancy 1. No / 2. Yes
Modelled in terms of small-area variables for baby i:
 Proportion of population of in each of three ethnic
minority categories (South Asian / Black / other)
 Tobacco expenditure
 MCS survey selection strata
…and some individual-level variables for baby i.
 Pollution exposure
 Low birth weight
 Social class, employment status of mother.
Odds ratios (posterior mean, 95% CI)
Data
NO2 *
SO2 *
Register,
ignore
confounding
1.20
(1.13,1.27)
1.03
(1.00,1.07)
MCS
1.04
(0.89,1.21)
1.04
(0.96,1.12)
2.00
(1.71,2.34)
2.76
(2.14,3.56)
MCS, ignore
selection
1.08
(0.94,1.23)
1.04
(0.96,1.12)
2.00
(1.71,2.34)
3.01
(2.42,3.74)
Register +
MCS
0.97
(0.91,1.03)
1.01
(0.97,1.05)
1.94
(1.80,2.10)
2.92
(2.61,3.26)
Register,
adjust for
confounding
0.97
(0.91,1.04)
1.01
(0.97,1.07)
1.94
(1.76,2.12)
2.93
(2.57,3.33)
Smoking
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*One unit of pollution concentration = interquartile range of pollution
concentration across England and Wales
South Asian
-
Conclusions so far
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No evidence for association of pollution
exposure with low birth weight.
Combining the datasets can
 increase statistical power of the survey data
 alleviate bias due to confounding in the
administrative data
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Must allow for selection mechanism of survey
when combining data
Work in progress
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Sensitivity to different choices for the imputation model
 External data (e.g. small-area data) on confounders
not always available
More investigation of selection bias, and different ways
of accounting for it
Quantify relative influence of each dataset
Other biases, expected to be smaller problem
 Missing data in MCS
 Exposure measurement error
Distinguish between preterm birth and low full-term
birth weight.
Combining aggregate and individual data
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Aggregate (ecological) data
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Needs appropriate models, and often individual data
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Administrative data usually aggregated to preserve confidentiality
Make inferences on individual-level risk factors and outcomes using
aggregate data: “Ecological bias” caused by
 within-area variability of risk factors
 confounding caused by limited number of variables.
survey/cohort data, case-control data.
Combining aggregate and individual data:
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can reduce ecological bias and increase power
distinguish contextual effects from individual.
Publications
Our papers, presentations and software available from
http://www.bias-project.org.uk
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C. Jackson, N. Best, S. Richardson. Hierarchical related regression
for combining aggregate and survey data in studies of socio-economic
disease risk factors. under revision, Journal of the Royal Statistical
Society, Series A.
C. Jackson, N. Best, S. Richardson. Improving ecological inference
using individual-level data. Statistics in Medicine (2006) 25(12):21362159.
C. Jackson, S. Richardson, N. Best. Studying place effects on health
by synthesising area-level and individual data. Submitted.