Graphical models for combining multiple sources of information in observational studies

1. 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

2. 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

3. Graphical modelling Modelling Mathematics Algorithms Inference

4. 1. Mathematics • 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) Modelling Mathematics Algorithms Inference

5. Y Z X W 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)

6. Y Z X W Joint distributions and graphical models Graphical models can be used to: • represent structure of a joint probability distribution….. • …..by encoding conditional independencies P(Y) P(Z) P(X|Y, Z) P(W|Y, Z) P(W,X,Y,Z) = P(W|Y,Z) P(X|Y,Z) P(Y) P(Z) 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 (e.g. regression models)

8. A B Y Z X D W C • Conditional independence provides basis for splitting large system into smaller components

9. Conditional independence provides basis for splitting large system into smaller components A B Y Y Z Y Z W X D W C

10. 2. Modelling Modelling Mathematics Algorithms Inference

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 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

13. 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.)

14. Model for combining data sources f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

15. Regression sub-model (MCS) Regression model forMCS data relating risk of low birth weight (yim) to mother’s THM exposure and other confounders (cim) f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

16. Regression sub-model (MCS) Regression model forMCS 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] THMim [mother] b[T] yim b[c] cim cik = potential confounders, e.g. deprivation, smoking, ethnicity

17. Regression sub-model (national data) Regression model fornational data relating risk of low birth weight (yik) to mother’s THM exposure and other confounders (cik) f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

18. Logistic regression • yik ~ Bernoulli(pik) • logit pik = b[c] cik + b[T] THMik • i indexes small area • k indexes mother [mother] Regression sub-model (national data) Regression model fornational data relating risk of low birth weight (yik) to mother’s THM exposure and other confounders (cik) THMik [mother] b[T] yik b[c] cik

19. Missing confounders sub-model Missing data model to estimate confounders (cik) for mothers in national data, using information on within area distribution of confounders in MCS f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

20. Missing confounders sub-model Missing data model to estimate confounders (cik) for mothers in national data, using information on within area distribution of confounders in MCS cim ~ Bernoulli(qi) (MCS mothers) cik ~ Bernoulli(qi) (Predictions for mothers in national data) cik cim qi

21. THM measurement error sub-model Model to estimate true tap water THM concentration from raw data f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

22. THMztj ~ Normal(THMzt,s2) z = water zone; t = season; j = sample (Actual model used was a more complex mixture of Normal distributions) [raw] [true] THM measurement error sub-model Model to estimate true tap water THM concentration from raw data THMzt [true] s2 THMztj [raw]

23. THM personal exposure sub-model Model to predict personal exposure using estimated tap water THM level and literature on distribution of factors affecting individual uptake of THM f THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

24. THM = ∑kTHMztx quantity (f1k) x uptake factor (f2k) where k indexes different water use activities, e.g. drinking, showering, bathing [mother] [true] THM personal exposure sub-model Model to predict personal exposure using estimated tap water THM level and literature on distribution of factors affecting individual uptake of THM f THMzt [true] THMik [mother] THMim [mother]

25. 3. Inference Modelling Mathematics Algorithms Inference

26. Bayesian

27. … or non Bayesian

28. 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 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

29. Data f Unknowns THMzt [true] s2 THMztj [raw] THMik [mother] THMim [mother] b[T] yim yik b[c] cik cim qi

30. MCMC algorithms are able to exploit graphical structure for efficient inference Bayesian graphical models implemented in WinBUGS 4. Algorithms Modelling Mathematics Algorithms Inference