Graphical models for combining multiple sources of information in observational studies Nicky Best Sylvia Richardson Chris Jackson Virgilio Gomez Sara 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 X P(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 X P(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 AB D C

Y W Z W D C Y Z X Y AB

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] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [mother] 

Regression sub-model (MCS)  [c]  [T] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [mother]  Regression model for MCS data relating risk of low birth weight (y im ) to mother’s THM exposure and other confounders (c im )

Regression sub-model (MCS)  [c]  [T] y im c im THM im [mother] Regression model for MCS data relating risk of low birth weight (y im ) to mother’s THM exposure and other confounders (c im ) Logistic regression y im ~ Bernoulli(p im ) logit p im = b [c] c im + b [T] THM im i indexes small area m indexes mother [mother] c ik = potential confounders, e.g. deprivation, smoking, ethnicity

Regression sub-model (national data)  [c]  [T] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [mother]  Regression model for national data relating risk of low birth weight (y ik ) to mother’s THM exposure and other confounders (c ik )

Regression sub-model (national data)  [c]  [T] y ik c ik THM ik [mother] Regression model for national data relating risk of low birth weight (y ik ) to mother’s THM exposure and other confounders (c ik ) Logistic regression y ik ~ Bernoulli(p ik ) logit p ik = b [c] c ik + b [T] THM ik i indexes small area k indexes mother [mother]

Missing confounders sub-model  [c]  [T] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [mother]  Missing data model to estimate confounders (c ik ) for mothers in national data, using information on within area distribution of confounders in MCS

Missing confounders sub-model c ik ii c im Missing data model to estimate confounders (c ik ) for mothers in national data, using information on within area distribution of confounders in MCS c im ~ Bernoulli(  i ) (MCS mothers) c ik ~ Bernoulli(  i ) (Predictions for mothers in national data)

THM measurement error sub-model  [c]  [T] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [mother]  Model to estimate true tap water THM concentration from raw data

THM measurement error sub-model 22 THM zt [true] THM ztj [raw] Model to estimate true tap water THM concentration from raw data THM ztj ~ Normal(THM zt,  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] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [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 THM ik [mother] THM zt [true] THM im [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 THM zt 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] y ik 22 y im c ik ii c im THM ik [mother] THM zt [true] THM ztj [raw] THM im [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