Graphical models for combining multiple sources of information in

observational studies

Nicky BestSylvia Richardson

Chris JacksonVirgilio GomezSara Geneletti

ESRC National Centre for Research Methods – BIAS node

Outline

• Overview of graphical modelling• Case study 1: Water disinfection byproducts and

adverse birth outcomes – Modelling multiple sources of bias in observational

studies

• Bayesian computation and software• Case study 2: Socioeconomic factors and heart

disease (Chris Jackson)– Combining individual and aggregate level data– Application to Census, Health Survey for England, HES

Graphical modelling

Modelling

Inference

Mathematics

Algorithms

1. Mathematics

Modelling

Inference

Mathematics

Algorithms

• Key idea: conditional independence• X and W are conditionally independent given Z if, knowing

Z, discovering W tells you nothing more about XP(X | W, Z) = P(X | Z)

Example: Mendelian inheritance• Y, Z = genotype of parents • W, X = genotypes of 2 children• If we know the genotypes of the parents, then the

children’s genotypes are conditionally independent

P(X | W, Y, Z) = P(X | Y, Z)

Y

W

Z

X

Joint distributions and graphical models

Graphical models can be used to:

• represent structure of a joint probability distribution…..

• …..by encoding conditional independencies

Factorization thm:

Jt distribution P(V) = P(v | parents[v])

Y

W

Z

XP(X|Y, Z)P(W|Y, Z)

P(Z)P(Y)

P(W,X,Y,Z) = P(W|Y,Z) P(X|Y,Z) P(Y) P(Z)

Where does the graph come from?

• Genetics– pedigree (family tree)

• Physical, biological, social systems– supposed causal effects (e.g. regression models)

• Conditional independence provides basis for splitting large system into smaller components

Y

W

Z

X

A B

D

C

• Conditional independence provides basis for splitting large system into smaller components

Y

W

Z

WD

C

Y Z

X

Y

A B

2. Modelling

Modelling

Inference

Mathematics

Algorithms

Building complex models

Key idea• understand complex system• through global model• built from small pieces

– comprehensible– each with only a few variables– modular

Example: Case study 1

• Epidemiological study of low birth weight 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– SAHSU are carrying out study in Great Britain using

routine data, to investigate risk of low birth weight associated with exposure to different THM levels

Data sources

• National postcoded births register• 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.)

Model for combining data sources

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Regression sub-model (MCS)

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Regression model for MCS data relating risk of low

birth weight (yim) to mother’s THM exposure

and other confounders (cim)

Regression sub-model (MCS)

[c]

[T]

yim

cim

THMim[mother]

Regression model for MCS data relating risk of low

birth weight (yim) to mother’s THM exposure

and other confounders (cim)

Logistic regression

yim ~ Bernoulli(pim)

logit pim = b[c] cim + b[T] THMim

i indexes small area

m indexes mother

[mother]

cik = potential confounders,e.g. deprivation, smoking, ethnicity

Regression sub-model (national data)

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Regression model for national data relating risk of

low birth weight (yik) to mother’s THM exposure

and other confounders (cik)

Regression sub-model (national data)

[c]

[T]

yik

cik

THMik[mother]

Regression model for national data relating risk of

low birth weight (yik) to mother’s THM exposure

and other confounders (cik)

Logistic regression

yik ~ Bernoulli(pik)

logit pik = b[c] cik + b[T] THMik

i indexes small areak indexes mother

[mother]

Missing confounders sub-model

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Missing data model to estimate confounders (cik)

for mothers in national data, using information on within area distribution of

confounders in MCS

Missing confounders sub-model

cik

i

cim

Missing data model to estimate confounders (cik)

for mothers in national data, using information on within area distribution of

confounders in MCS

cim ~ Bernoulli(i) (MCS mothers)

cik ~ Bernoulli(i) (Predictions for

mothers in national data)

THM measurement error sub-model

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Model to estimate true tap water THM concentration

from raw data

THM measurement error sub-model

2

THMzt[true]

THMztj[raw]

Model to estimate true tap water THM concentration

from raw data

THMztj ~ Normal(THMzt, 2)

z = water zone; t = season; j = sample

(Actual model used was a more complex mixture of Normal distributions)

[raw] [true]

THM personal exposure sub-model

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Model to predict personal exposure using estimated tap water THM level and

literature on distribution of factors affecting individual

uptake of THM

THM personal exposure sub-model

THMik[mother]

THMzt[true]

THMim[mother]

Model to predict personal exposure using estimated tap water THM level and

literature on distribution of factors affecting individual

uptake of THM

THM = ∑k THMzt x quantity (1k) x uptake factor (2k)

where k indexes different water use activities, e.g. drinking, showering, bathing

[mother] [true]

3. Inference

Modelling

Inference

Mathematics

Algorithms

Bayesian

… or non Bayesian

• 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

Recall: Joint distribution P(V) = P(v | parents[v])

• Condition on parts of graph that are observed (data) • Calculate posterior probabilities of remaining nodes

using Bayes theorem• Automatically propagates all sources of uncertainty

Bayesian Full Probability Modelling

[c]

[T]

yik

2

yim

cik

i

cim

THMik[mother]

THMzt[true]

THMztj[raw]

THMim[mother]

Data

Unknowns

4. Algorithms

Modelling

Inference

Mathematics

Algorithms

• MCMC algorithms are able to exploit graphical structure for efficient inference

• Bayesian graphical models implemented in WinBUGS