Graphical models for combining multiple data sources

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 Modelling Mathematics Algorithms Inference

4. 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

5. Z 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

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

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

10. 2. Modelling • Graphical models provide framework for building probabilistic models for empirical data 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 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. qz s2 f THMztj THMzt [raw] [tap] THMzk THMzi [pers] [pers] yzk yzi b[T] pzi pzk b[c] czk czi Model for combining data sources

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

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

17. qz s2 f THMztj THMzt [raw] [tap] THMzk THMzi [pers] [pers] yzi yzk b[T] pzi pzk b[c] czk czi 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

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

19. qz s2 f THMztj THMzt [raw] [tap] THMzk THMzi [pers] [pers] yzi yzk b[T] pzi pzk b[c] czk czi 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

20. 3. Inference Modelling Mathematics 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 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

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

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

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

29. ai s2 x[c]ik yik x[b]ik pik Building the model Multilevel 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]

30. ai s2 x[c]ik yik x[b]ik pik Building the model Multilevel 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]

31. ai s2 x[c]ik yik x[b]ik pik Building the model Multilevel 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]

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

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

34. ai s2 X[c]i V[c]i X[b]i Ni Building the model Ecological 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

35. ai s2 X[c]i V[c]i X[b]i Ni Building the model Ecological 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

36. ai s2 X[c]i V[c]i X[b]i Ni Building the model Ecological 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

37. ai s2 X[c]i V[c]i X[b]i Ni Building the model Ecological 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

38. ai s2 X[c]i V[c]i X[b]i Ni Building the model Ecological 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

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. s2 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 data Multilevel model for individual data Ecological model b[c] b[c] b[b] b[b] qi Yi

41. ai s2 X[c]i V[c]i x[c]ik yik X[b]i Ni x[b]ik pik Combining individual and aggregate data Hierarchical Related Regression (HRR) model b[c] b[b] qi Yi

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 • 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)

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

48. Results

49. 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!