Nicky Best and Chris JacksonWith Sylvia RichardsonDepartment of Epidemiology and Public Health
Imperial College, [email protected]
http://www.bias-project.org.uk
Bayesian graphical models for inference from
combinations of data
Example: low birth weight and air pollution
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) Small, great individual detail.
Administrative data (national births register) Large, but little individual detail.
Single underlying model assumed to govern both datasets: elaborate as appropriate to handle biases
Low birth weight
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…
Selection of Millennium Cohort
ALL UK WARDS
ENGLAND
SCOTLAND
WALES
NORTHERN IRELAND
High child poverty
Low child poverty
High child poverty
Low child poverty
High child poverty
Low child poverty
High child poverty
Low child poverty
High ethnic minority
SELECTION PROBABILITYSELECTION PROBABILITY
0.040.04
0.020.02
0.110.11
0.070.07
0.040.04
0.180.18
0.060.06
0.160.16
0.080.08
Selection bias in the Millennium Cohort
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
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
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
LBWLBWii
POLLPOLLii POLLPOLLjj
MODELMODEL
baby i in register baby j in MCS
ETHETHii ETHETHjj
LBWLBWjj
LBWi: low birth weight
POLLi: pollution exposure (plus other confounders observed in both datasets)
ETHi: ethnicity and smoking. Only observed in the MCS.
Same MODEL assumed to govern both datasets.
knownknown
unknownunknown
Adding in the imputation model
LBWLBWii
POLLPOLLii POLLPOLLjjMODELMODEL
(LBW)(LBW)
baby i in register baby j in MCS
ETHETHii ETHETHjj
LBWLBWjj
AGGAGGii AGGAGGjjMODELMODEL
(imputation)(imputation)
AGGi: aggregate ethnicity/smoking data for area of residence of baby i
MODEL MODEL for for imputationimputation of of ETHi in terms of aggregate data and other variables. in terms of aggregate data and other variables. Estimate it from observed Estimate it from observed ETHj in the MCS.
Bayesian model
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 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 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
*
Smoking South Asian
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)
*One unit of pollution concentration = interquartile range of pollution *One unit of pollution concentration = interquartile range of pollution concentration across England and Walesconcentration across England and Wales
Conclusions so far
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
Must allow for selection mechanism of survey when combining data
Work in progress
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
Aggregate (ecological) data 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.
Needs appropriate models, and often individual data survey/cohort data, case-control data.
Combining aggregate and individual data: 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
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):2136-2159.
C. Jackson, S. Richardson, N. Best. Studying place effects on health by synthesising area-level and individual data. Submitted.
