1. Graphical models for combining multiple data sources
Nicky Best
Sylvia Richardson
Chris Jackson
Imperial College BIAS node
with thanks to Peter Green

2. Outline Overview of graphical modelling
Case study 1: Water disinfection byproducts and adverse birth outcomes
Modelling multiple sources of bias in observational studies
Case study 2: Socioeconomic factors and limiting long term illness
Combining individual and aggregate level data
Simulation study
Application to Census and Health Survey for England

3. Graphical modelling
Mathematics
Modelling
Algorithms
Inference

4. 1. Mathematics Mathematics Modelling Algorithms Inference
Key idea: conditional independence
X and Y are conditionally independent given Z if, knowing Z, discovering Y tells you nothing more about X
P(X | Y, Z) = P(X | Z)

5. Example: Mendelian inheritanceZ X Y
Z = genotype of parents
X, Y = genotypes of 2 children
If we know the genotype of the parents, then the children’s genotypes are conditionally independent

6. Joint distributions and graphical models
Use ideas from graph theory to:
represent structure of a joint probability distribution…..
…..by encoding conditional independencies A C D F B E
Factorization thm:
Jt distribution P(V) =  P(v | parents[v])

7. Where does the graph come from?
Genetics
pedigree (family tree)
Physical, biological, social systems
supposed causal effects
Contingency tables
hypothesis tests on data
Gaussian case
non-zeros in inverse covariance matrix

8. A C D F B E
Conditional independence provides mathematical basis for splitting up large system into smaller components

9. C D A C D F B E E
Conditional independence provides mathematical basis for splitting up large system into smaller components

10. 2. Modelling
Mathematics
Modelling
Algorithms
Inference
Graphical models provide framework for building probabilistic models for empirical data

11. Building complex models
Key idea
understand complex system
through global model
built from small pieces
comprehensible
each with only a few variables
modular

12. Example: Case study 1
Epidemiological study of birth defects and mothers’ exposure to water disinfection byproducts
Background
Chlorine added to tap water supply for disinfection
Reacts with natural organic matter in water to form unwanted byproducts (including trihalomethanes, THMs)
Some evidence of adverse health effects (cancer, birth defects) associated with exposure to high levels of THM
We are carrying out study in Great Britain using routine data, to investigate risk of birth defects associated with exposure to different THM levels

13. Data sources National postcoded births register
National and local congenital anomalies registers
Routinely monitored THM concentrations in tap water samples for each water supply zone within 14 different water company regions
Census data – area level socioeconomic factors
Millenium cohort study (MCS) – individual level outcomes and confounder data on sample of mothers
Literature relating to factors affecting personal exposure (uptake factors, water consumption, etc.)

14. Model for combining data sources
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

15. Model for combining data sources
Regression model for national data relating risk of birth defects (pzk) to mother’s THM exposure and other confounders (czk) f
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

16. Model for combining data sources
Regression model for MCS data relating risk of birth defects (pzi) to mother’s THM exposure and other confounders (czi)
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

17. Model for combining data sources
Missing data model to estimate confounders (czk) for mothers in national data, using information on within area distribution of confounders in MCS f
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

18. Model for combining data sources
Model to estimate true tap water THM concentration from raw data
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

19. Model for combining data sources
Model to predict personal exposure using estimated tap water THM level and literature on distribution of factors affecting individual uptake of THM f
THMzt
[tap] s2
THMztj
[raw]
THMzk
[pers]
THMzi
[pers]
b[T]
pzk
b[c]
pzi
czk
czi
yzk qz
yzi

20. 3. Inference
Mathematics
Modelling
Algorithms
Inference

21. Bayesian

22. … or non Bayesian

23. Bayesian Full Probability Modelling
Graphical approach to building complex models lends itself naturally to Bayesian inferential process
Graph defines joint probability distribution on all the ‘nodes’ in the model
Condition on parts of graph that are observed (data)
Update probabilities of remaining nodes using Bayes theorem
Automatically propagates all sources of uncertainty

24. 4. Algorithms Mathematics Modelling Algorithms Inference
Many algorithms, including MCMC, are able to exploit graphical structure
MCMC: subgroups of variables updated randomly
Ensemble converges to equilibrium (e.g. posterior) dist.

25. Key idea exploited by WinBUGS software
MCMC
Key idea exploited by WinBUGS software
- need only look at neighbours ?
Updating

26. Case study 2 Socioeconomic factors affecting health Background
Interested in individual versus contextual effects of socioeconomic determinants of health
Often investigated using multi-level studies (individuals within areas)
Ecological studies also widely used in epidemiology and social sciences due to availability of small-area data
investigate relationships at level of group, rather than individual
outcome and exposures are available as group-level summaries
usual aim is to transfer inference to individual level

27. Building the model s2 Multilevel model for individual data ai x[c]ik
b[c]
x[b]ik
pik
b[b]
yik

28. Building the model s2 Multilevel model for individual data
yik ~ Bernoulli(pik), person k, area i ai
x[c]ik
b[c]
x[b]ik
pik
b[b]
yik

29. Building the model s2 Multilevel model for individual data
yik ~ Bernoulli(pik), person k, area i ai
log pik = ai + b[c] x[c]ik + b[b] x[b]ik
x[c]ik
b[c]
x[b]ik
pik
b[b]
yik

30. Building the model s2 Multilevel model for individual data
yik ~ Bernoulli(pik), person k, area i ai
log pik = ai + b[c] x[c]ik + b[b] x[b]ik
x[c]ik
b[c]
ai ~ Normal(0, s2)
x[b]ik
pik
b[b]
yik

31. Building the model s2 Multilevel model for individual data
yik ~ Bernoulli(pik), person k, area i ai
log pik = ai + b[c] x[c]ik + b[b] x[b]ik
x[c]ik
b[c]
ai ~ Normal(0, s2)
x[b]ik
pik
b[b]
Prior distributions on s2, b[c], b[b]
yik

32. Building the model Ecological model s2 ai V[c]i b[c] X[c]i b[b] qi
X[b]i Yi Ni

33. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
V[c]i
b[c]
X[c]i
b[b] qi
X[b]i Yi Ni

34. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
qi =  pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c] ai
V[c]i
b[c]
X[c]i
b[b] qi
X[b]i Yi Ni

35. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
qi =  pik(x[b], x[c]) fi(x[b], x[c]) dx[c]dx[c] ai
V[c]i
Assuming x[b], x[c] independent, with
X[b]i = proportion exposed to ‘b’ in area
i and fi(x[c]) = Normal(X[c]i, V[c]i), then
qi = q0i(1-X[b]i) + q1iX[b]i
where
q0i = marginal prob of disease for unexposed
= exp(ai + b[c]X[c]I + b2[c]V[c]i/2)
b[c]
X[c]i
b[b] qi
X[b]i Yi Ni

36. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
qi =  pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c] ai
V[c]i
Assuming x[b], x[c] independent, with
X[b]i = proportion exposed to ‘b’ in area
i and fi(x[c]) = Normal(X[c]i, V[c]i), then
qi = q0i(1-X[b]i) + q1iX[b]i
where
q1i = marginal prob of disease for exposed
= exp(ai + b[b] + b[c]X[c]I + b2[c]V[c]i/2)
b[c]
X[c]i
b[b] qi
X[b]i Yi Ni

37. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
qi =  pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c] ai
V[c]i
ai ~ Normal(0, s2)
b[c]
X[c]i
b[b] qi
X[b]i Yi Ni

38. Building the model Ecological model Yi ~ Binomial(qi, Ni), area i s2
qi =  pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c] ai
V[c]i
ai ~ Normal(0, s2)
b[c]
X[c]i
Prior distributions on s2, b[b], b[c]
b[b] qi
X[b]i Yi Ni

39. Combining individual and aggregate data
Individual level survey data often lack power to inform about contextual and/or individual-level effects
Even when correct (integrated) model used, ecological data often contain little information about some or all effects of interest
Can we improve inference by combining both types of model / data?

40. Combining individual and aggregate datas2 s2
Multilevel model for individual data
Ecological model ai ai
V[c]i
x[c]ik
b[c]
b[c]
X[c]i
x[b]ik
pik
b[b]
b[b] qi
X[b]i
yik Yi Ni

41. Combining individual and aggregate datas2
Hierarchical Related Regression (HRR) model ai
V[c]i
x[c]ik
b[c]
X[c]i
x[b]ik
pik
b[b] qi
X[b]i
yik Yi Ni

42. Simulation Study

43. Simulation Study

44. Simulation Study

45. Comments
Inference from aggregate data can be unbiased provided exposure contrasts between areas are high (and appropriate integrated model used)
Combining aggregate data with small samples of individual data can reduce bias when exposure contrasts are low
Combining individual and aggregate data can reduce MSE of estimated compared to individual data alone
Individual data cannot help if individual-level model is misspecified

46. Application to LLTI Health outcome Exposures Data sources
Limiting Long Term Illness (LLTI) in men aged yrs living in London
Exposures
ethnicity (white/non-white), income, area deprivation
Data sources
Aggregate: 1991 Census aggregated to ward level
Individual: Health Survey for England (with ward identifier)
1-9 observations per ward (median 1.6)

47. Ward level data
Deprivation
% non white
Mean income
Prevalence of LLTI

48. Results Model Non-white Log income Deprivation Between-area variance
Individual
-0.36
(-0.98, 0.23)
-0.55
(-0.80, -0.32)
-0.022
(-0.032, 0.074)
0.18
(0.052, 0.64)
Ecological
0.50
(0.27, 0.72)
-0.72
(-0.93, -0.51)
0.063
(0.054, 0.073)
0.19
(0.17, 0.21)
Combined
0.48
(0.23, 0.72)
-0.70
(-0.91, -0.50)
0.064
(0.054, 0.074)
(0.17, 0.22)
Combined (correlation modelled)
(0.24, 0.73)
-0.71
(-0.91, -0.51)

49. Thank you for your attention!
Concluding Remarks
Graphical models are powerful and flexible tool for building realistic statistical models for complex problems
Applicable in many domains
Allow exploiting of subject matter knowledge
Allow formal combining of multiple data sources
Built on rigorous mathematics
Principled inferential methods
Thank you for your attention!