Comparing Bayesian Spatial Models with Spatial Smoothing Methods

1. Comparing Bayesian spatial models in the presence of spatial smoothing Earl Duncan Newcastle University 4 Sept 2019

2. Intro Methods 1 2 3 4 Results Conclusions What is a spatial model? • A model that attempts to account for the spatial nature of data – autocorrelated and/or clustered. • Tobler’s first law of geography: “…near things are more related than distant things.” • Ignoring it is like ignoring the order of longitudinal data! • Spatial models account for the spatial dependencies in different ways via spatial smoothing. • Bayesian spatial models make this task easy: • Spatial data is inherently hierarchical; so are Bayesian models. • Different spatial structures can be specified through the prior. Earl Duncan Newcastle University, Sept 20191

3. Intro Methods 1 2 3 4 Results Conclusions What is spatial smoothing? • Smoothing “flattens” a surface. • Two aspects: magnitude (extreme values) and local variation Earl Duncan Newcastle University, Sept 20192

4. Intro Methods 1 2 3 4 Results Conclusions What is spatial smoothing? • Consider the time-series equivalent Trend? Trend? Trend? Seasonality Earl Duncan Newcastle University, Sept 20193

5. Intro Methods 1 2 3 4 Results Conclusions Why smooth? • Reduces statistical uncertainty. • Provides more insight. • The smooth surface represents a “layer” of the underlying data-generating process (similar to time-series decomposition) • Improves data confidentiality • This can be very important in epidemiological studies Earl Duncan Newcastle University, Sept 20194

6. Intro Methods 1 2 3 4 Results Conclusions The problem: • Some amount of smoothing is beneficial. But HOW MUCH? • How can we quantify the degree of smoothing? • How can we compare spatial models? • Can we just rely on traditional goodness-of-fit (GoF) measures? • Typically we want to satisfy 3 objective functions: • GoF • Parsimony (simplicity over complexity) • Spatial smoothing DIC, WAIC Earl Duncan Newcastle University, Sept 20195

7. Intro Methods 1 2 3 4 Results Conclusions Plan: • Literature search. • Any attempts to quantify smoothing? • Can time-series approaches be applied? • Compare methods using real data. • It can be tricky to make simulated data realistic. • Simulated data can unwittingly bias results. • Publically available data is better for reproducibility and provides existing analyses. Earl Duncan Newcastle University, Sept 20196

8. Intro Methods 1 2 3 4 Results Conclusions Data: • Scottish lip cancer data set • First published in Clayton and Kaldor (1987) • Analysed by Breslow and Clayton (1993), Spiegelhalter et al. (2002), Duncan, White, and Mengersen (2017), and others… Earl Duncan Newcastle University, Sept 20197

9. Intro Methods 1 2 3 4 Results Conclusions Data: • North Carolina SIDS • First presented by Atkinson (1978) • Analysed by Cressie and Read (1989) and Banerjee, Carlin, and Gelfand (2014) amongst others… Earl Duncan Newcastle University, Sept 20198

10. Intro Methods 1 2 3 4 Results Conclusions Data: • New York leukemia incidence • First published by Turnball et al. (1990) • Analysed by Waller et al. (1992), Waller et al. (1994), Waller and Gotway (2004), and others… Earl Duncan Newcastle University, Sept 20199

11. Intro Methods 1 2 3 4 Results Conclusions Models: • Leroux et al. (2000) model • 1 spatial random effect parameter • conditional autoregressive (CAR) prior • Can be implemented in CARBayes • Inverse-gamma hyperprior on the variance • Also tested half-Normal hyperprior (using BUGS) • Besag, York and Mollié (1991) model • 2 spatial random effects • Inverse-gamma and half-Normal hyperpriors used Earl Duncan Newcastle University, Sept 201910

12. Intro Methods 1 2 3 4 Results Conclusions Leroux model: Earl Duncan Newcastle University, Sept 201911

13. Intro Methods 1 2 3 4 Results Conclusions BYM model: Earl Duncan Newcastle University, Sept 201912

14. Intro Methods 1 2 3 4 Results Conclusions Spatial weights matrix: • Determines the spatial proximity between random effects • (Controls the spatial dependencies) • Most commonly defined as a first-order, binary, adjacency weights matrix, e.g. Earl Duncan Newcastle University, Sept 201913

15. Intro Methods 1 2 3 4 Results Conclusions Posterior summary – the risk surface: • The log risk-surface can be decomposed into: • The spatial random effect(s) (SRE): • Structured (SSRE), and • Unstructured (USRE), (BYM model only) • The covariate effect (including the intercept): • Note that the SSRE is zero centred (sum-to-zero constraint) • The risk (aka SIR) surface is then simply the exponentiated log-risk surface. Given that the smoothing applies to the SSRE only, do you expect the (log)risk-surface to be smooth? Earl Duncan Newcastle University, Sept 201914

16. Intro Methods 1 2 3 4 Results Conclusions Posterior summary – the risk surface: • The SIR isn’t necessarily smooth due to covariate effects. • Define the covariate-adjusted SIR (CASIR) • Simply • By analogy, we can also define the covariate-adjusted raw SIR, CARSIR: • “raw” (observed) SIR is simply for BYM Earl Duncan Newcastle University, Sept 201915

17. Intro Methods 1 2 3 4 Results Conclusions Posterior summary – the risk surface: • CASIR forms the basis of one approach to quantifying smoothing (see slide 20 and 21) • Why use CASIR over ? • It makes it comparable to the SIR (both on ratio scale) • We can compute an approximate range of permissible values: • CARSIR, and • The mean value of the neighbouring CASIR values, i.e. • (Taking logs to compute a range for is not reliable since may be zero.) Earl Duncan Newcastle University, Sept 201916

18. Intro Methods 1 2 3 4 Results Conclusions Risk surface decomposition (Scottish lip cancer): Over-smoothed? Under-smoothed? (Results for the Leroux, inverse-gamma model) Earl Duncan Newcastle University, Sept 201917

19. Intro Methods 1 2 3 4 Results Conclusions Risk surface decomposition (North Carolina SIDS data): (Results for the Leroux, inverse-gamma model) Earl Duncan Newcastle University, Sept 201918

20. Intro Methods1 2 3 4 Results Conclusions Position of CASIRs relative to permissible range: • If no smoothing occurs, CASIR = CARSIR (because SIR = raw SIR) • If the maximum amount of smoothing is applied to area , then • The degree of smoothing exhibited by a given area can be quantified by considering how far the CASIR estimate has “moved” from the CARSIR towards the mean of the neighbouring CASIR values • 0 = no movement. • 1 = CASIR is equal to the neighbouring values. Earl Duncan Newcastle University, Sept 201919

21. Intro Methods1 2 3 4 Results Conclusions Position of CASIRs relative to permissible range Earl Duncan Newcastle University, Sept 201920

22. Intro Methods1 2 3 4 Results Conclusions Distribution of CASIR position: Earl Duncan Newcastle University, Sept 201921

23. Intro Methods1 2 3 4 Results Conclusions Quantitative metric: • Overall measure of degree of smoothing is the proportion of areas that lie within a user-specified range • Allows user to specify desired level of smoothing. E.g. • the proportion of areas within the 90% quantile range • The proportion of areas within the range (greater preference towards models that exhibit more smoothing) Earl Duncan Newcastle University, Sept 201922

24. Intro Methods 1 2 3 4 Results Conclusions Kurtosis Preservation: • Rongand Bailis (2017). • Address over-smoothing by using a simple moving average smoothing function such that: • the moving average window size minimises the “roughness” • with the constraint that the kurtosis of the smoothed time-series kurtosis of the original unsmoothed time-series • The idea is that the smoothed time series retains rare large-scale deviations while smoothing out more frequent modestly sized deviations. • Can we use this idea: • In a spatial context? • For quantifying rather than implementing smoothing? Earl Duncan Newcastle University, Sept 201923

25. Intro Methods 1 2 3 4 Results Conclusions Kurtosis Preservation: • Spatial dependencies are quite different to longitudinal dependencies. • Consequently, defining “roughness” isn’t as straightforward • (Rong and Bailis(2017)define it as the standard deviation of the first-order difference series). • Could consider first-order differences between spatial neighbours. Earl Duncan Newcastle University, Sept 201924

26. Intro Methods 1 2 3 4 Results Conclusions Kurtosis Preservation: Earl Duncan Newcastle University, Sept 201925

27. Intro Methods 1 2 3 4 Results Conclusions Kurtosis Preservation: Drawback: appropriate degree of smoothing is assessed against model with minimum roughness. Several models are required to assess smoothing. Earl Duncan Newcastle University, Sept 201926

28. Intro Methods 1 2 3 4 Results Conclusions Variogram: where • is the number of areas which are no more distant than the lag from area • is the variable of interest () • denotes all areas and which satisfy • is the distance between areas and • Great Circle distance • Voronoi tessellation-based distances (“the minimal number of boundaries that have to be crossed to move from one to the other”) – see Knorr-Held and Raβer (2000, pp. 14) Earl Duncan Newcastle University, Sept 201927

29. Intro Methods 1 2 3 4 Results Conclusions Variogram: Earl Duncan Newcastle University, Sept 201928

30. Intro Methods 1 2 3 4 Results Conclusions Other approaches: • Cohen’s kappa: • Criticism: • Designed for categorical data • Hard to choose “epidemiologically meaningful” cut-offs • Interpretation is difficult (no universal guidelines) Order of observations is ignored (see next slide)Newcastle University, Sept 201929 Earl Duncan Newcastle University, Sept 201929

31. Intro Methods 1 2 3 4 Results Conclusions • If there was a high degree of agreement, even 100%, could a large kappa value really imply that case A is under-smoothed? • Similarly, do small kappa values imply that case B is over-smoothed when the surface doesn’t even appear smoothed? • (This problem is related to the notion of a “baseline agreement”.) Earl Duncan Newcastle University, Sept 201930

32. Intro Methods 1 2 3 4 Results Conclusions Classifying the models: Earl Duncan Newcastle University, Sept 201931

33. Intro Methods 1 2 3 4 Results Conclusions Classifying the models: • Scotland Lip cancer, Leroux model, inverse-gamma prior:​ DIC WAIC Earl Duncan Newcastle University, Sept 201932

34. Intro Methods 1 2 3 4 Results Conclusions Classifying the models: • Scotland Lip cancer, Leroux model, half-Normal prior:​ DIC WAIC Earl Duncan Newcastle University, Sept 201933

35. Intro Methods 1 2 3 4 Results Conclusions Classifying the models: • NC SIDS, Leroux model, half-Normal prior:​ WAIC DIC Earl Duncan Newcastle University, Sept 201934

36. Intro Methods 1 2 3 4 Results Conclusions Other metrics: Earl Duncan Newcastle University, Sept 201935

37. Intro Methods 1 2 3 4 Results Conclusions Conclusions: • ‘CASIR position’ approach seems reliable​ • More testing required​ • Provides better comparison of spatial models when some level of smoothing is required?​ • Criteria can be adjusted to suit specific research problem/data characteristics • Variogram approach also promising​ • Kurtosis preservation approach may need more work​ • Kappa not recommended… Earl Duncan Newcastle University, Sept 201936

38. Intro Methods 1 2 3 4 Results Conclusions Key references: Earnest, A. et al. 2007. Evaluating the effect of neighbourhood weight matrices on smoothing properties of Conditional Autoregressive (CAR) models. International Journal of Health Geographics, 6 (1), p.54. DOI: 10.1186/1476-072x-6-54.​ Clayton, D., and J. Kaldor. 1987. Empirical Bayes estimates of age-standardized relative risks for use in disease mapping. Biometrics43 (3): 671–81. doi:10.2307/2532003.​ Atkinson, D. 1978. Epidemiology of Sudden Infant Death in North Carolina: Do Cases Tend to Cluster? PHSB Studies, No. 16. Raleigh, North Carolina: N. C. Department of Human Resources, Division of Health Services, Public Health Statistics Branch.​ Besag, J., J. York, and A. Mollié. 1991. Bayesian image restoration with application in spatial statistics. Annals of the Institute of Statistical Mathematics43 (1): 1–20. DOI: 10.1007/BF00116466.​ Rong, K., and P. Bailis. 2017. ASAP: prioritizing attention via time series smoothing. Proceedings of the VLDB Endowment10 (11): 1358–69. DOI: 10.14778/3137628.3137645.​ Leroux, B. G., X. Lei, and N. Breslow.  2000.  Estimation of disease rates in small areas: a new mixed model for spatial dependence.  In “Statistical models in epidemiology, the environment and clinical trials”, edited by M. E. Halloran and D. Berry, pp. 179-191. The IMA Volumes in Mathematics and its Applications, vol 116.  New York: Springer.  DOI: 10.1007/978-1-4612-1284-3_4.​ Earl Duncan Newcastle University, Sept 201937

39. Intro Methods 1 2 3 4 Results Conclusions Acknowledgements: Thank You! Contact: • earl.duncan@qut.edu.au • https://bragqut.wordpress.com/people/earl-duncan/ Earl Duncan Newcastle University, Sept 201938