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### Graphical models for combining multiple data sources

Nicky Best

Sylvia Richardson

Chris Jackson

Imperial College BIAS node

with thanks to Peter Green

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

1. Mathematics

- 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)

Modelling

Mathematics

Algorithms

Inference

X

Y

Example: Mendelian inheritance- 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

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])

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

Conditional independence provides mathematical basis for splitting up large system into smaller components

A

C

D

F

B

E

Conditional independence provides mathematical basis for splitting up large system into smaller components

C

D

A

C

D

F

B

E

E

2. Modelling

- Graphical models provide framework for building probabilistic models for empirical data

Modelling

Mathematics

Algorithms

Inference

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 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

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.)

s2

f

THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzk

yzi

b[T]

pzi

pzk

b[c]

czk

czi

Model for combining data sourcess2

f

THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzi

yzk

b[T]

pzi

pzk

b[c]

czk

czi

Model for combining data sourcesRegression model fornational data relating risk of birth defects (pzk) to mother’s THM exposure and other confounders (czk)

s2

f

THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzi

yzk

b[T]

pzi

pzk

b[c]

czk

czi

Model for combining data sourcesRegression model forMCS data relating risk of birth defects (pzi) to mother’s THM exposure and other confounders (czi)

s2

f

THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzi

yzk

b[T]

pzi

pzk

b[c]

czk

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Model for combining data sourcesMissing data model to estimate confounders (czk) for mothers in national data, using information on within area distribution of confounders in MCS

s2

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THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzi

yzk

b[T]

pzi

pzk

b[c]

czk

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Model for combining data sourcesModel to estimate true tap water THM concentration from raw data

s2

f

THMztj

THMzt

[raw]

[tap]

THMzk

THMzi

[pers]

[pers]

yzi

yzk

b[T]

pzi

pzk

b[c]

czk

czi

Model for combining data sourcesModel to predict personal exposure using estimated tap water THM level and literature on distribution of factors affecting individual uptake of THM

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

4. Algorithms

Modelling

- Many algorithms, including MCMC, are able to exploit graphical structure
- MCMC: subgroups of variables updated randomly
- Ensemble converges to equilibrium (e.g. posterior) dist.

Mathematics

Algorithms

Inference

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

s2

x[c]ik

yik

x[b]ik

pik

Building the modelMultilevel model for individual data

yik ~ Bernoulli(pik), person k, area i

b[c]

b[b]

s2

x[c]ik

yik

x[b]ik

pik

Building the modelMultilevel model for individual data

yik ~ Bernoulli(pik), person k, area i

log pik = ai + b[c] x[c]ik + b[b] x[b]ik

b[c]

b[b]

s2

x[c]ik

yik

x[b]ik

pik

Building the modelMultilevel model for individual data

yik ~ Bernoulli(pik), person k, area i

log pik = ai + b[c] x[c]ik + b[b] x[b]ik

b[c]

ai ~ Normal(0, s2)

b[b]

s2

x[c]ik

yik

x[b]ik

pik

Building the modelMultilevel model for individual data

yik ~ Bernoulli(pik), person k, area i

log pik = ai + b[c] x[c]ik + b[b] x[b]ik

b[c]

ai ~ Normal(0, s2)

b[b]

Prior distributions on s2, b[c], b[b]

s2

X[c]i

V[c]i

X[b]i

Ni

Building the modelEcological model

Yi ~ Binomial(qi,Ni), area i

qi = pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c]

b[c]

b[b]

qi

Yi

s2

X[c]i

V[c]i

X[b]i

Ni

Building the modelEcological model

Yi ~ Binomial(qi,Ni), area i

qi = pik(x[b], x[c]) fi(x[b], x[c]) dx[c]dx[c]

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]

b[b]

qi

Yi

s2

X[c]i

V[c]i

X[b]i

Ni

Building the modelEcological model

Yi ~ Binomial(qi,Ni), area i

qi = pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c]

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]

b[b]

qi

Yi

s2

X[c]i

V[c]i

X[b]i

Ni

Building the modelEcological model

Yi ~ Binomial(qi,Ni), area i

qi = pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c]

ai ~ Normal(0, s2)

b[c]

b[b]

qi

Yi

s2

X[c]i

V[c]i

X[b]i

Ni

Building the modelEcological model

Yi ~ Binomial(qi,Ni), area i

qi = pik(x[b], x[c]) fi(x[b], x[c]) dx[b]dx[c]

ai ~ Normal(0, s2)

b[c]

Prior distributions on s2, b[b], b[c]

b[b]

qi

Yi

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?

ai

ai

s2

X[c]i

V[c]i

x[c]ik

yik

x[b]ik

X[b]i

Ni

pik

Combining individual and aggregate dataMultilevel model for individual data

Ecological model

b[c]

b[c]

b[b]

b[b]

qi

Yi

s2

X[c]i

V[c]i

x[c]ik

yik

X[b]i

Ni

x[b]ik

pik

Combining individual and aggregate dataHierarchical Related Regression (HRR) model

b[c]

b[b]

qi

Yi

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

Application to LLTI

- Health outcome
- Limiting Long Term Illness (LLTI) in men aged 40-59 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)

Prevalence of LLTI

Prevalence of LLTI

Mean income

% non white

Deprivation

Deprivation

Mean income

Mean income

% non white

% non white

Deprivation

Ward level dataConcluding 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!

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