1. © Imperial College London Analysis of space time patterns of disease risk Sylvia Richardson Centre for Biostatistics Joint work with Juanjo Abellan and Nicky Best Small Area Health Statistics Unit Department of Epidemiology and Public Health

2. © Imperial College London Outline Context Space time models for disease risk Use of space time models to investigate the stability of patterns of disease –Simulations –Illustration on the analysis of congenital malformations Space time analysis of related disease –Illustration on the analysis of male & female lung cancer Discussion © Imperial College London

3. Benefits of Space Time Analysis for chronic diseases Study the persistence of patterns over time –Interpreted as associated with stable risk factors, environmental effects, distribution of health care access … Highlight unusual patterns in time profiles via the inclusion of space-time interaction terms –Time localised excesses linked to e.g. emerging environmental hazards with short latency –Variability in recording practices  Increased epidemiological interpretability  Potential tool for surveillance

4. © Imperial College London Joint analysis of two related chronic diseases or health outcomes is of interest in several contexts Epidemiology: quantify ‘expected’ variability linked to shared risk factors and tease out specific patterns Health planning: assess the performance of the health system, e.g. for health outcomes linked to screening policies Data quality issues: uncover anomalous patterns linked to a data source shared by several outcomes Benefits of joint analysis of related diseases

5. © Imperial College London Case study: Congenital anomalies in England All cases of congenital anomalies (non chromosomal) recorded in England for the period 1983 – 1998 Data from national post-coded registers (Office for National Statistics) Annual post-coded data on total number of live births, still births and terminations 136,000 congenital anomalies  84.5 per 10 5 birth-years Congenital anomalies are sparse:  Grid of 970 grid squares with variable size, to equalize the number of birth and expected cases per square Variations could be linked to socio-economic or environmental risk factors or heterogeneity in recording practises  Interest in characterising space time patterns © Imperial College London

6. Map of grid squares used in the Congenital Anomalies study © Imperial College London

7. Expected number of congenital anomalies per year in each square (per quintiles) Annual expected cong anomal. Annual number of births Min00 20%2.602994.8 40%5.716576.2 Media n 7.438569.0 60%9.5410994.6 80%14.4816687.4 Max64.0973872.0 © Imperial College London

8. Case study: Male and Female lung cancer Lung cancer, with its low survival rates is the biggest cancer killer in the UK –Over one fifth of all cancer deaths in UK are from lung cancer (25% for male and 18% for female) Major risk factor is smoking. –Smoking time trends different for men/women: uptake of smoking started to decrease in cohorts of men after 1970, while for women the levelling off was later, after 1980 Other risk factors include exposure to workplace agents, radon, air pollution …  Interested in similarity and specificity of patterns between men and women

9. © Imperial College London Data sets Female and Male lung cancer incidence in Yorkshire. Spatial resolution: wards (626): –between 0 and 20 new cases per year with mean around 4 for male –between 0 and 12 per year with mean 1.8 for female Time periods: 1981-85, 1986-90, 1991-95, 1996-99. Expected counts based on sex-age incidence rates for the region and the total period 1981- 1999

10. © Imperial College London Standardised Incidence Ratios (O/E) Lung Cancer Male

11. © Imperial College London Lung Cancer Female

12. © Imperial College London Outline Context Space time models for disease risk Use of space time models to investigate the stability of patterns of disease –Simulations –Illustration on the analysis of congenital malformations Space time analysis of related disease –Illustration on the analysis of male & female lung cancer Discussion © Imperial College London

13. Space time models in epidemiology Space time extensions of Bayesian hierarchical models for disease mapping have been considered by a number of authors, with models differing in their treatment of space time interactions e.g. Knorr-Held and Besag (1998), Waller et al (1997), Bernardinelli et al (1995), Knorr-Held (2000), Richardson, Abellan and Best (2006) © Imperial College London

14. Notations Y it = Observed # of cases for area i, i = 1,…, N period t, t=1, …, T. n it = number of person at risk in area i, period t = probability of disease in area i, period t First level: binomial likelihood ¼ i t Often a Poisson approximation is used © Imperial College London Y i t » B i nom i a l ( n i t ; ¼ i t ) Y i t » P o i sson ( E i t ½ i t ) E it = expected # of cases, area i, period t = relative risk, area i, period t ½ i t

15. © Imperial College London ¼ i t Overall spatial pattern Overall time trend Space time interactions Second level: modelling of the structure of the random effects or © Imperial College London ½ i t µ l og i t ( ¼ i t ) l og ( ½ i t ) ¶ = ® + ¸ i + » t + º i t Prior structure for the random effects : encodes prior epidemiological knowledge need to borrow strength to effect smoothing has to be adapted to the analysis’s aim

16. © Imperial College London Prior structure for the random effects Overall spatial pattern: account for local dependence due to geographical ‘continuity’ of populations and risk factors Overall time trends: time dependence for long latency chronic disease expected Space time interactions: capture the non predictable part from simple space + time model

17. © Imperial College London Prior structure for the spatial effects and time trend Convolution prior for the spatial effects: –Gaussian Conditional Autoregressive model for random effects, S S i | S -i ~ N (  h S h /n i, s 2 /n i ) for the h ε set {neighbours of square i}, # {set} =n i –Unstructured heterogeneity for H: H i ~ N (0, u 2 ) Random Walk prior for the temporal effects to smooth the time trend: ξ t | ξ t-1, ξ t-2, … ~ N (ξ t-1, v 2 ) The variance parameters :  2, u 2, v 2 are given inverse gamma priors  3rd hierarchical level Model for the interactions ?  Need to be adapted to the context ¸ i = S i + H i l o g i t ( ¼ i t ) = ® + ¸ i + » t + º i t © Imperial College London

18. Model for the interaction terms (1) Prediction context:  Simple exchangeable structure: it ~ N (0, ζ 2 )  More complex dependence structure to obtain better fit (cf Knorr-Held, 2000) –Distribution of it depends on spatial neighbours of i for each t –Distribution of it depends on and for each i Either of these assumptions lead to considerable smoothing of the space time interactions º i t ¡ 1 º i t + 1

19. © Imperial College London Model for the interaction terms (2) Investigating stability of patterns: Aim is to -- Highlight true departures from the overall stable space + time model -- Shrink idiosynchratic (non interpretable) interactions  Mixture model to characterise ‘stable’ and ‘unstable’ risk patterns over time º i t » p N orma l ( 0 ; ¿ 2 1 ) + ( 1 ¡ p ) N orma l ( 0 ; ¿ 2 2 ) : ¿ 2 » N orma l ( 0 ; 100 ) ¢ I ( 0 ; + 1 ) ¿ 1 » N orma l ( 0 ; 0 : 01 ) ¢ I ( 0 ; + 1 ) Component 1 stable Component 2 unstable

20. © Imperial College London Outline Context Space time models for disease risk Use of space time models to investigate the stability of patterns of disease –Simulations –Illustration on the analysis of congenital malformations Space time analysis of related disease –Illustration on the analysis of male & female lung cancer Discussion © Imperial College London

21. Analysis strategy for investigating stability of patterns Estimate a model: space (CAR +Het) + time + interactions (mixture) Use the posterior probabilities of allocation p it into component 2 to classify areas as ‘unstable’ Rule: area i is unstable if at least for one t, 1, … T p it > p cut (threshold probability) For ‘stable’ areas, investigate spatial patterns, e.g. by using the rule Prob( λ i >1) > 0.8. Investigate the profile pattern of ‘unstable’ areas  Need to evaluate the performance of the mixture model and associated classification rule

22. © Imperial College London Simulation set up Realistic set up based the congenital anomaly study, using only a subset of 309 grid squares Squares divided into 2 groups: –Modified (20%, 8%, 1%) and Unmodified –Unmodified: use a simple space+time model for the risks –Modified: add an interaction term following 3 scenarios (a)Risk is multiplied by 2 in all time periods, (Reference) (b)Risk is moderately variable in all time periods (Medium v) (c)Risk is highly variable in all time periods (High v) 50 replicated data sets using multinomial sampling © Imperial College London

23. 20% Scenario8% Scenario

24. © Imperial College London Statistical issues If we over-fit, i.e. estimate a space-time model with interactions when the patterns are stable (Reference case), do we loose power to detect pure spatial patterns with respect to a pure spatial model ? : –Gain of interpretability but loss of power ? Is the mixture model identifiable ? What is the performance of the classification rules ? Can we tease out any structured patterns in the interactions? © Imperial College London

25. Comparison of the distribution of the { λ i } and of the posterior probabilities Prob(λ i > 1) between the space time model and a pure spatial model fitted to the aggregate counts over the 16 years Reference case

26. © Imperial College London Variability of space-time interaction terms Compute the empirical standard deviation of the it, SD( it ) for each area (over the 16 years):  SD( it ) characterises the instability over time of the underlying disease risk Posterior distribution of SD( it ) is influenced overall –by the (unknown) proportion of areas with unstable risks over time identified by the mixture –by the size of the interactions for the modified areas © Imperial College London

27. Posterior distribution of SD( it ) in the 3 cases Mixture model 20% Modified areas8% Modified areas1% Modified areas Model captures well the increased variability of modified areas Clear distinction between medium and high variance cases Increasing number of modified areas influences overall fit

28. © Imperial College London Posterior distribution of SD( it ) in the 3 cases Comparison between Mixture model Exchangeable model it ~ N (0, τ 2 ) º i t » p N orma l ( 0 ; ¿ 2 1 ) + ( 1 ¡ p ) N orma l ( 0 ; ¿ 2 2 ) : Mixture modelExchangeable model

29. © Imperial College London Performance of Classification rule Plot of sensitivity versus 1- specificity (ROC curves) Rule: area i is unstable if p it > p cut at least for one t, 1, … T For 90% specificity (10% False positive), p cut ≈ 0.5 20% Modified areas 8% Modified areas

30. © Imperial College London Risk time profiles for the areas classified as unstable Generated patterns 10 samples High variance 10 samples Medium variance

31. © Imperial College London Interpretation of excess risk in unmodified areas Combining the rule Prob( λ i > 1) > 0.8 and the classification rule is effective We found that for the unmodified areas generated with risk > 1.1: –90% have Prob( λ i > 1) > 0.8 –all are classified as stable  spatially stable excess risk better interpreted

32. © Imperial College London Results for congenital anomalies Map of the global spatial pattern Time trend (83 - 98) Classification of areas Time profiles for unstable areas © Imperial College London

33. Spatial main effect 970 grid squares Post median of exp( λ i ) Congenital anomalies England, 83-98 © Imperial College London Evidence of spatial heterogeneity with higher risk in the North, NW and NE and in the Greater London area Deprivation and maternal age are strong determinants of congenital malformations

34. © Imperial College London Time main effect: exp( ξ t ) Congenital anomalies England, 83-98 The downward shift picked up between pre 1990 and post 1990 is due to the “minor anomalies” exclusion policy that was implemented in 1990 and after. © Imperial College London

35. Mixture estimation Using a cut off p cut = 0.5, 125 areas are classified as unstable

36. © Imperial College London Risk time profiles for the areas classified as unstable We performed hierarchical clustering on the 125 areas Four subgroups exhibit smooth-like trends  interaction terms used to adjust to general time trend One small subgroup has a high peak in 97  warrants investigation

37. © Imperial College London Outline Context Space time models for disease risk Use of space time models to investigate the stability of patterns of disease –Simulations –Illustration on the analysis of congenital malformations Space time analysis of related disease –Illustration on the analysis of male & female lung cancer Discussion © Imperial College London

38. Joint modelling of several diseases Spatial analysis of related diseases has been formulated in the BHM context by Knorr-Held and Best (2001) Extend the formulation of the shared component model to include a time dimension in order to study shared and specific patterns over time We shall formulate our models in the context of male and female lung cancer

39. © Imperial College London Models and Notation O 1it = Observed # of cases for males, ward i, period t, O 2it for females E 1it = Expected # of cases for males, ward i, period t, E 2it for females First level model RRs males females O 1 i t » P o i sson ( ½ 1 i t E 1 i t ) O 2 i t » P o i sson ( ½ 2 i t E 2 i t )

40. © Imperial College London Second level models of the (log) relative risks We discuss first 2 formulations: Model M I: additive space-time structure, for both shared and female-specific terms Model M II: additional space-time interaction term for the shared component NB To avoid over-parametrisation, the formulation is asymmetric, as the ‘shared’ component is constrained to capture the male RR  female specific component has to be interpreted as a differential effect between male/female

41. © Imperial College London Model M I Additive space-time structure l og½ 1 i t = ® 1 + ¸ i ¢ ± + » t ¢ · l og½ 2 i t = ® 2 + ¸ i = ± + » t = · + ¯ i + ° t

42. © Imperial College London Model M II ‘Common’ space-time interaction Inclusion of terms it  investigate common pattern of departure from a simple additive space time structure l og½ 1 i t = ® 1 + ¸ i ¢ ± + » t ¢ · + º i t l og½ 2 i t = ® 2 + ¸ i = ± + » t = · + º i t + ¯ i + ° t Does the share term it captures all the local space-time patterns ?

43. © Imperial College London l og½ 1 it = ® 1 + ¸ i ¢ ± + » t ¢ · + º it + Á 1 it l og½ 2 it = ® 2 + ¸ i = ± + » t = · + º it + ¯ i + ° t + Á 2 it l og½ 1 it = ® 1 + ¸ i ¢ ± + » t ¢ · + Á 1 it l og½ 2 it = ® 2 + ¸ i = ± + » t = · + ¯ i + ° t + Á 2 it Explore benefit of additional male and female specific residual terms in Model I or II Model M II + het Model M I + het

44. © Imperial College London Priors Gaussian Conditional AutoRegressive model for shared and female-specific spatial effects, e.g: i | - i ~ N (  h h /n i,  2 /n i ) for h in set {neighbours of ward i}, # {set} = n i Random Walk prior for the shared and female-specific temporal effects to smooth the time trend Exchangeable model for the interactions: it ~ N (0, ζ 2 ) [Wishart prior for Σ -1 with 2 df to allow for correlation between the male and female residuals] Models fitted using Winbugs (10 000 sweeps after 15000 burn in, κ found not identifiable and set to 0, δ close to 1)

45. © Imperial College London Deviance Information Criteria results E[D(ρ)] D(E[ρ]) pDDIC M I (additive) 5643.1 5185.8457.36100.4 M II (interaction) 5363.14692.5680.66043.7 M I + het 5253.14449.1804.06057.1 M II + het 5293.14540.9752.26045.3 Additional structure beyond additive space + time? Shared space–time interaction is sufficient to capture residual structure  present results of Model II

46. © Imperial College London Results for Model II Smoothed relative risks for male and female Shared and female specific spatial patterns Time trends Posterior probabilities for space-time interaction terms

47. © Imperial College London RR Male 1 st period RR Male 2 nd period RR Male 3 rd period RR Male 4 th period

48. © Imperial College London RR Female 1 st periodRR Female 2 nd period RR Female 4 th period RR Female 3 rd period

49. © Imperial College London Comments Maps of smoothed RRs show: Evidence of spatial heterogeneity with higher RR in urban areas in the SW: Leeds, Bradford, Huddersfield, Sheffield, and towards Hull Opposite time trends in male and female – Decrease over time period for male – Increase over the period for female  Characterisation of these components

50. © Imperial College London Posterior prob that β i > 1 Shared component Specific female component Higher male/female differential in extended semi rural area north of Leeds Clear urban/rural differences in incidence. Linked to smoking patterns? Air pollution?

51. © Imperial College London 10 wards selected at random Time trends, male in red, female in blue Time trend for male RRs in 10 wards Time trend for female RRs in 10 wards Differential time trends are linked to lagged uptake of smoking in cohorts of men and women

52. © Imperial College London Model II + ‘Common’ interactions ‘High’ values of interaction terms it indicate a lack of fit of the simple additive formulation of model I : space x time  space + time for the shared part Here we did not use a mixture model but a simple exchangeable model More informative than the display of posterior mean estimates for the terms it, display of the posterior probabilities, Prob( it > 1)

53. © Imperial College London Prob( it >1|data) Posterior Prob for interaction 2 nd period 4 th period 1 st period 3 rd period

54. © Imperial College London Model II Interactions seem predominant in the SW corner Number of areas highlighted are compatible with expected number of false positives For long latency disease like lung cancer, epidemiological patterns tend to be stable Explore link between shared structure and contextual covariates  Standardise the expected counts E it with respect to deprivation index and re-estimate Model II

55. © Imperial College London Shared pattern becomes weaker indicating link to deprivation Female differential stays similar

56. © Imperial College London Discussion (1) Bayesian space time analyses allow a richer interpretation of patterns than purely spatial ones Models become more complex, with more choice of prior structure : in particular, the prior structure for the space time interactions (t distribution, semi-parametric mixture, other smoothing priors ….)  Need to think of prior structure in relation to the different aims of the analyses, the time scale and the hypotheses on the health phenomenon under investigation  sensitivity analyses and careful exploration of models is needed © Imperial College London

57. Discussion (2) To gain epidemiologic interpretability, the stability = repeatability over time of spatial patterns found by a pure spatial analysis should be investigated  little loss of power to detect areas with increased risk  decision rules that discriminate between stable and unstable patterns based on mixture model seem promising Need to investigate the effect of a smaller number of expected events (in our simulations, median per area between 4 and 9 in different years) © Imperial College London

58. Discussion (3) How to explore further the pattern showed by the space-time interactions ? Investigate generalisation of the hierarchical clustering to more flexible clustering of time profiles of space-time interactions, e.g. in a Bayesian hierarchical framework, to better detect structured patterns in the time profiles of risks Comparison with other methods for finding space-time clusters (e.g. Scan statistics)

59. © Imperial College London References L. Bernardinelli, D. Clayton, C. Pascutto, C. Montomoli, M. Ghislandi, and M. Songini. Bayesian. Analysis of space-time variation in disease risk. Statistics in Medicine, 14:2433–2443, 1995. L. A. Waller, B. P. Carlin, H. Xia, and A. M. Gelfand. Hierarchical spatio- temporal mapping of disease rates. Journal of the American Statistical Association, 92:607–617, 1997. L. Knorr-Held and J. Besag. Modelling risk from a disease in time and space. Statistics in Medicine, 17:2045–2060, 1998. L. Knorr-Held. Bayesian modelling of inseparable space-time variation in disease risk. Statistics in Medicine, 19:2555–2567, 2000. L. Knorr-Held and N. G. Best. A shared component model for detecting joint and selective clustering of two diseases. Journal of the Royal Statistical Society - A, 164:73–85, 2001. S. Richardson, J. J. Abellan, and N. Best. Bayesian spatio-temporal analysis of joint patterns of male and female lung cancer risks in Yorkshire (UK). Statistical Methods in Medical Research, 15: 385-407, 2006. J. J. Abellan, S. Richardson, and N. Best. Use of space-time models to investigate the stability of patterns of disease. (2007). Submitted for publication.