Detecting and Modeling Spatial Disease Clustering. A Bayesian Approach

Transcription

1 Detecting and Modeling Spatial Disease Clustering: A Bayesian Approach Ronald E. Gangnon Department of Biostatistics and Medical Informatics and Murray K. Clayton Department of Statistics University of Wisconsin Madison March 31, March

2 Organization 1. New York Leukemia Data 2. Literature Review/Problem Description 3. Proposed Solution 4. Simulation Results 5. Analysis of New York Leukemia Data 6. Conclusions March

3 New York Leukemia Data Onondaga Madison Cayuga Cortland Chenango Tompkins Broome Tioga March

4 Goals of Analysis ffl Statistical: - Find clusters, i.e., areas with high or low disease rates. - Estimate cell-specific disease rates. ffl Epidemiologic: - Screening. - Surveillance. - Hypothesis generation. March

5 Basic Statistical Model ffl i = 1; 2; :::; N cells. ffl O i = number of cases in cell i. ffl n i = population at risk in cell i. ffl r i = E( O i n i ) = true rate of disease in cell i. ffl O i ο Poisson(r i n i ). March

6 Previous Approaches ffl Distance-based test statistics, e.g., Whittemore et al. (1987). ffl Pre-specified location, e.g., Waller et al. (1992). ffl Multiple cluster locations/sizes. - Openshaw et al. (1988) GAM All nominally significant circles. - Besag and Newell (1991) Circles of fixed case radius. - Turnbull et al. Circles of fixed population radius. - Kulldorff and Nagarwalla (1995) Arbitrary collections of potential clusters. March

7 A Simple Model for Clustering ffl Region divided into background & k clusters. ffl Background: large region with a common rate. ffl Clusters: small regions with higher or lower rates. ffl c = (c1;c2; :::; c n ) = vector of cluster membership indicators. ffl j = rate of disease in cluster j. ffl Structure for cell-specific rates: r i = ci. ffl Conditional on c, gamma priors for rates =) Gamma posteriors for rates. Negative binomial marginal likelihood. March

8 Prior for Clustering Models ffl Define size and shape using perimeter and area. R1 = p A=ß, R 2 = P=(2ß) Size = R2, Shape = R1=R2 ffl Markov connected component field (MCCF) (Møller and Waagepetersen, 1997) Score each cluster based on size and shape. Denote this score by S j. Calculate prior for model as P (c) = expf kx j=1 S j g ffl Higher weight on small, circular clusters. March

9 Posterior for Clustering Models ffl Modified Occam s window (Madigan and Raftery, 1994) ffl Basic idea: Most models have no support in data. ffl Only consider good models. P (cjo) W max c P (cjo): ffl Randomized search for plausible models. ffl Start with saturated model and repeatedly merge 2 components. ffl Select merger with probability roughly proportional to associated posterior. March

10 Details 1. Select initial background cell at random. Informed selection problematic. 2. Reduce the size of the problem. Restart the search from intermediate models. 3. Stopping the search. If probability of finding new model with next search is small. 4. Mixture of point mass on null (M0) + MCCF (M1) as prior. Estimate P (OjM1) using P (OjM1) = E cjo 1 P (Ojc) 1 March

11 Simulations ffl Spatial layout grid of 5 5 squares. Population ο1 million (2,500 per cell) ffl Prior for rates: Background: Gamma(1; 1000) Cluster(s): Gamma(2; 1000) ffl Prior for clusters: Prior Probability (Log10) Relative to Null Model Prior Probability (Log10) Relative to Null Model Cluster Size (km) Cluster Shape ffl Prior for null model: P (M0) = 0:9. March

12 Simulation #1 Null Model Observed Disease Rates - Median RMSE Estimated Disease Rates - Median RMSE Estimated Disease Rates - Mean True Disease Rates March

13 Simulation #2 2 2 Cluster Observed Disease Rates - Median RMSE Estimated Disease Rates - Median RMSE Estimated Disease Rates - Mean True Disease Rates March

14 Simulation #3 4 5 Cluster Observed Disease Rates - Median RMSE Estimated Disease Rates - Median RMSE Estimated Disease Rates - Mean True Disease Rates March

15 Simulation # Cluster (RR 2) Observed Disease Rates - Median RMSE Estimated Disease Rates - Median RMSE Estimated Disease Rates - Mean True Disease Rates March

16 Simulation # Cluster (RR 4) Observed Disease Rates - Median RMSE Estimated Disease Rates - Median RMSE Estimated Disease Rates - Mean True Disease Rates March

17 Simulation Results ffl Can estimate a null model correctly. ffl Can detect a priori likely clusters. Small, circular, positive. ffl Increased detection with larger populations, cluster sizes, cluster risks and overall disease rate. ffl Some ability to detect a priori unlikely clusters. Large and linear or negative. ffl Asymptotics provide accurate assessment of our detection ability. March

18 Analysis of New York Data ffl Prior for rates: Background: Gamma(0:739; 1339) Cluster(s): Gamma(1:478; 1339) ffl Prior for clusters: Prior Probability (Log10) Relative to Null Model Prior Probability (Log10) Relative to Null Model Cluster Size (km) Cluster Shape ffl Prior for null model: P (H0) = 0:99. March

19 NewYorkData Estimated Rates March

20 NewYorkData Probability Cell Belongs to Cluster March

21 Conclusions ffl Bayesian approach to analyzing spatial pattern of disease. ffl Estimate (not just test for) clustered rates. ffl Incorporates uncertainty in rates, cluster membership, etc. ffl Simulation results encouraging. March