1 Spatial assessment of deprivation and mortality risk in Nova Scotia: Comparison between Bayesian and non-Bayesian approaches Prepared for 2008 CPHA Conference, June 1-4, 2008 Mikiko Terashima Dr. Pantelis Andreou Dr. Judith Guernsey Department of Community Health & Epidemiology Dalhousie University

2 Objectives To present basic ideas of Bayesian approaches in disease mapping  Perspectives of a health researcher with minimum statistical background To demonstrate an example application of Bayesian approaches to spatial analysis of health in comparison with non-Bayesian approaches

4 Key differences Non-Bayesian  Parameters are assumed fixed (constant)  Use likelihood to calculate parameters  “Confidence interval”: Based on hypothetical situations where we were to sample e.g.100 times, 95 times out of them will contain the constant (true) parameter value  Standard Error (SE)  Chi-square, AIC, -2Log likelihood etc. for Goodness- of-fit Bayesian  Parameters are assumed random  Use likelihood and prior information to calculate posterior distribution, and estimate parameters from it (Estimated parameters [e.g. mean, median] are used like constant parameter values in non- Bayesian models)  “Credible interval”: Based on a number of simulations, we know the distribution of the parameter— posterior distribution—and 95% of them fall between the interval A and B.  Markov Chain Standard Error (MCSE)  DIC for goodness-of-fit (like AIC but does not require Maximum Likelihood of parameters)

5 Why use Bayesian approaches in spatial analysis of health/disease mapping? It can incorporate spatial random effects in the analysis of spatial variation in health, disease or health/disease risks—Bayesian way is more ‘natural’ and straightforward.  The belief of parameters being random fits assumption that geographical area rates are somewhat different. Because it “borrows strength” from prior knowledge about factors not in the data, the model is closer to reality than inferring things from only data –reduce noise from maps. It allows us to estimate values of areas with missing data by sampling the same way as other parameters.

6 Challenges “Black box” for non-statisticians “PC vs. Apple computers” New and constantly evolving Application of prior information requires statisticians’ insights and background knowledge about the variables.

7 Example application: Spatial analysis of deprivation and mortality in Nova Scotia communities

8 Scenario (Deprivation and morality in Nova Scotia) A number of deaths were observed in 278 communities of Nova Scotia over the period (obs). Counts of events are likely Poisson distributed. Obs~Po(  ) We want to see the association between deprivation at the community level and observed counts of death. The community level deprivation was measured by two sets of variables: material (Mat) and psychosocial (Psy). Spatial autocorrelation might be playing a role (assumed normal). We want to know if other (unknown) factors are affecting the rates. We know the expected counts for each community (exp). We want to map community level risks.

9 Poisson regression model of counts Offset

10 1. Log(  i )=offset i (+ β 0i ) + β 1i x Mat + β 2i x Psy 2. Log(  i )=offset i (+ β 0i ) + β 1i x Mat + β 2i x Psy β 0i ~ flat() β 1i ~ N(0, 1.0E-5) β 2i ~ N(0, 1.0E-5) v 0i ~ N(0, 1/σ 2 v ) u 0i (neigh)  CAR tau.u ~ gamma(0.5, ) tau.v ~ gamma(0.5, ) β 0i = β 0 + u 0i, u 0i ~ N(0, σ 0 ) β 1i = β 1 + u 1i, u 1i ~ N(0, σ 1 ) β 2i = β 2 + u 1i, u 1i ~ N(0, σ 2 ) e i ~ N(0, σ e ) Models Conditional Autoregressive model Simulated from a similar model template provided within WinBUGS software (unstructured random + neighbourhood effect) [+ e i ] tau.v + v 0i + u 0i (neigh)

11 Conditional AutoRegressive model (CAR)-Normal Where Number of neighbours Average of surrounding (adjacent) areas (neighbours) Weights are typically 1 (e.g. Besag, York & Mollie, 1991) Spatial autocorrelation effect for neighbourhood i A common model to deal with spatial autocorrelation in disease mapping  usually results in smoothing Spatial autocorrelation effect for neighbourhood i depends on the number of neighbours and average of surrounding neighbours

WinBUGS code model{ for (i in 1 : N) { # Likelihood O[i] ~ dpois(mu[i]) log(mu[i]) <- log(Mrexp[i]) + beta0 + beta1* Mat[i] + beta2 * Psy[i] + b[i] + h[i] RR[i] <- exp(beta0 + beta1* Mat[i] +beta2 * Psy[i] + u[i] + v[i]) Mrexp[i] ~ dgamma (1, 0.1) Mat[i] ~ dnorm (0.0, 1.0) Psy[i] ~ dnorm (0.0, 1.0) # Exchangeable prior on unstructured random effects v[i] ~ dnorm(0, tau.v) } # CAR prior distribution for spatial random effects: u[1:N] ~ car.normal(adj[], weights[], num[], tau.u) for(k in 1:sumNumNeigh) { weights[k] <- 1 } # Other priors: beta0 ~ dflat() beta1 ~ dnorm(0.0, 1.0E-5) beta2 ~ dnorm(0.0, 1.0E-5) tau.u ~ dgamma(0.5, ) sigma.u <- sqrt(1 / tau.u) tau.v ~ dgamma(0.5, ) sigma.v <- sqrt(1 / tau.v) } CAR Filling missing values Other parameter priors Unstructured random effects Poisson regression model RR for each community

13 node mean sd MC error2.5%median97.5%startsample beta beta beta sigma.u sigma.v Parameter outputs Model 2 output with WinBUGS beta0/β 0 :Intercept Beta1/β 1 : Material deprivation Beta2/β 2 : Psychosocial deprivation sigma.u: Variance due to spatial autocorrelation sigma.v: Variance due to unstructured random effects Log (  ) = offset x Mat x Psy Model 1 output with MLwiN β0β0 β1β1 β2β2 DIC= DIC=

SMR map (Model 1) versus RR map (Model 2 ) Extremely high and low rates were reduced Rate at each community shrunk to the mean (1.0) Communities with missing data on the SMR map now have predicted values

15 Three benefits of Bayesian approaches for spatial health analyses are: 1) random effects due to area variations can be easily incorporated; 2) smoothing effects; and 3) missing values can be inferred using prior knowledge (no holes in the map).  Working closely with knowledgeable statisticians, health researchers can use Bayesian approaches for these benefits (e.g. spatially analyzing and mapping health and health risks). Conclusion

16 Acknowledgement The student is funded by Killam Pre-doctoral Scholarship and CIHR Strategic Training Program in Public Health and the Agricultural Rural Ecosystem (PHARE)

17 Thank you!