Spatial Statistics for environmental health data Montse Fuentes (NC State)

Transcription

1 Spatial Statistics for environmental health data (NC State) Josh Warren (Yale), Amy Herring (UNC) and Peter Langlois (Texas SHD)

2 Why Congenital Anomalies? Congenital Anomalies: abnormalities, physiological or structural, which present at the time of birth Around 3% of all births result in defect Leading cause of infant mortality (>20% of all infant deaths) Cardiac defects: leading cause of mortality among defect related deaths (a ect around 1% of all births) Cause of most defects is unknown More than $2.5 billion a year in hospital costs alone Sources: March of Dimes Prematurity Campaign, 2010; Martin et al., 2010.; Rynn, Cragan, and Correa, 2008; Martin et al., NC State University 2 / 44

3 Scientific Motivation Birth Defects and Pollution Exposure: Vrijheid et al. (2010) literature review and meta-analysis Meta-analysis: Only few cardiac anomalies and pollutant combinations found to be significant NO 2,SO 2,andPM 10 Atrial septal defect, coarctation of the aorta, tetralogy of Fallot Recommendations: Exposure assessment improvement Other defect groups being analyzed Identify the more susceptible time windows during pregnancy NC State University 3 / 44

4 Texas Vital Records Data Full birth records from all births in Texas where 1 Mother resided at delivery in a region and time period covered by the birth defect registry; At least some demographic information available Geocoding Information: Each birth geocoded to the residence at delivery Texas ( ) study found that 68% of pregnant women did not move between date of conception and date of birth (Nuckols et al. (2004)) Of the women who did move, 49% stayed very close geographically (same water supply source) NC State University 4 / 44

5 Data Sources and Information Infant/Fetus and Pregnancy Information Included: Case/Control Status Sex Birth Weight Pregnancy Outcome Plurality of Pregnancy Gestational Age Estimates Number of Previous Live Births Mother/Father Information Included: Age Birthplace Race/Ethnicity Education NC State University 5 / 44

6 Texas Vital Records Data Figure: Birth Locations in Texas, NC State University 6 / 44

7 Analysis Data Case-control study design Two controls per case Matched on mother s age group, year of birth, and infant s gender Three cardiac defects of interest Atrial septal defects Pulmonary artery and valve defects Ventricular septal defects Groups chosen based on largest observed sample sizes and similar groupings in Gilboa (2005). NC State University 7 / 44

8 Analysis Data The Birth Defects Epidemiology and Surveillance Branch (BDES) in Texas was established in 1993 after cluster of birth defect cases seen in Brownsville, TX. Further investigation showed Hispanic mothers in Southern Texas were at higher risk of their babies developing birth defects (TDSHS, 2011). We focus on the southern part of Texas, TDSHS health service regions 6, 8, and 11, from Regions include Houston, San Antonio, Corpus Christi, and Brownsville Shares border with Mexico 5134 controls, 2567 cases in final dataset Births removed if gestational age < 20 weeks and if birth resulted in an induced termination of the pregnancy or unspecified fetal death, based on previous epidemiologic cardiac defect and air pollution studies (Dolk et al., 2010; Dadvand et al., 2010; Rankin et al.,2009) NC State University 8 / 44

9 Analysis Data (a) Health Regions. (b) Locations of births. NC State University 9 / 44

10 Analysis Data ASD PAVD VSD 5134 Figure: Cardiac defect breakdown of analyzed data. NC State University 10 / 44

11 Pollution Data Sources Air Quality System (AQS): Texas, Ozone- Maximum daily 8-hour average (parts per million (ppm)) PM Daily average (micrograms per cubic meter (ug/m 3 )) Nitrate (NO 3 )- Daily average (ug/m 3 ) Sulfate (SO 4 )- Daily average (ug/m 3 ) Elemental carbon (EC)- Daily average (ug/m 3 ) Organic carbon (OC)- Daily average (ug/m 3 ) Note: EC and OC available from in Texas NC State University 11 / 44

12 PM 2.5 Speciated Data Monitors Figure: Active Monitors, NC State University 12 / 44

13 Limitations of the traditional monitoring data Sparse spatial coverage Missing values Question? How to fill the space-time gaps for the monitoring data? What we need? More complete coverage and better understanding of the underlying physical system. Possible solutions Numerical model output data. NC State University 13 / 44

14 CMAQ (Community Multiscale Air Quality) Model CMAQ model Air quality simulations based on di erential equations that represent the underlying physical processes. Primary inputs: meteorological information, emission rates, initial and boundary conditions. Benefits from CMAQ models Air quality simulations for the entire US. High spatial and temporal resolution without missing values; NC State University 14 / 44

15 CMAQ (Community Multiscale Air Quality) Data 12km x 12km grid over Texas, Included pollutants (daily values): Ozone- Maximum daily 8-hour average (ppb) PM Daily average (ug/m 3 ) Nitrate (NO 3 )- Daily average (ug/m 3 ) Sulfate (SO 4 )- Daily average (ug/m 3 ) Elemental carbon (EC)- Daily average (ug/m 3 ) Organic carbon (OC)- Daily average (ug/m 3 ) NC State University 15 / 44

16 Previous Work: Congenital Anomalies Exposure to pollution during pregnancy typically handled through averages over suspected windows of importance; fit separately using multiple models for di erent exposure windows Separate models for di erent pollutant/defect combinations; Multivariate outcomes rarely considered The impact of PM 2.5 or speciated fine PM on birth defects has not been studied NC State University 16 / 44

17 Statistical Challenges Incorporate spatial analysis of environmental health data, typically not considered in classical birth outcome epidemiological studies Create statistical models to identify the specific critical windows during the pregnancy when high exposures to pollutants more negatively a ect the birth defects Model multivariate birth defects to include additional information rarely utilized in previous studies Incorporate nonstationary and non-gaussian spatial-temporal behavior to increase flexibility to model complex relationships between pollution exposure and birth defects Carry out appropriate inference to better characterize the uncertainty associated with the parameter estimates NC State University 17 / 44

18 New Methodology Cardiac Anomaly Model: Y ij is a binary outcome, taking the value 1 when pregnancy i ends on birth defect j. p ij is the probability of that outcome. We use a probit link. Y i =(Y i1,...,y ij ) T, i =1,...,N; where Y ij p ij ind Bern p ij 1 (p ij) =x T i j + z T i j z vector of multi-pollutants, j pollutant and defect specific, spatially and temporally-varying coe cients Represent the e ect of the concentration of air pollutants during the pregnancy and across space on the probability of developing anomaly j for woman i. NC State University 18 / 44

19 Stick-Breaking Process Prior Review To account for uncertainty in the parametric form of a distribution, G, we use a Bayesian nonparametric model and put a prior on the unknown distribution G. The stick-breaking prior for G is the possibly infinite mixture G = d P M k=1 p k k,wherep i are the mixture probabilities and k is the Dirac distribution with point mass at k. The mixture probabilities break the stick into M pieces, so the sum of the pieces is one, P M k=1 p k =1 p 1 = V 1 and the subsequent probabilities are p k = V k Qj<k (1 V j), P where 1 j 1 k=1 p k = Q j 1 k=1 (1 V k) is the probability not accounted by the first j 1 components. NC State University 19 / 44

20 Stick-Breaking Process Prior Review V i Beta (a, b) control the distribution of the probability mixture i G 0,whereG 0 is a known density Dirichlet Process prior: DP ( G 0 ) Ferguson, 1973 M = 1, V i iid Beta (1, ) SBP can be extended to spatial setting by incorporating spatial information in the p k or the k Generally finite M used in practice SBP is a.s. discrete, problematic for assumed continuous data. Instead Y i = i + i,where i is a continuous random error component and i has the SBP prior. NC State University 20 / 44

21 University of Michigan, February Stick breaking 1. Draw θ 1 from H 0 2. Draw V 1 from Beta(1,ν) 3. p 1 = V 1 4. Draw θ 2 from H 0 5. Draw V 2 from Beta(1,ν) p 2 = V 2(1 V 1)

22 Previous Extensions of Stick-Breaking Prior Gelfand et al., 2005: Spatial Dirichlet Process Prior 1X G ( ) (.) = k (.), k =( (s 1 ),..., (s n )) T k=1 p k iid k MVN (0, s ) Duan, Guindani, and Gelfand, 2007: Generalized Spatial Dirichlet Process Prior 1X 1X G ( ) (.) =... p i(s1 ),...,i(s n) (s 1 ) (.)... i(s1 ) (s n) i(sn) (.) i(s 1 )=1 i(s n)=1 Reich and Fuentes, 2007: Spatial Stick-Breaking Prior MX G ( (s)) (.) = p k (s) k (.) k=1 NC State University 21 / 44

23 Advances in Interdisciplinary Statistics, UNCG October 14, 2007 Figure 2: Example to illustrate the spatial stick-breaking prior. In this example, the spatial domain is the one-dimensional interval (0,1) and the model has Gaussian kernels with knots ψ = (0.5, 0.0, 1.0, 0.2, 0.8), bandwidths ϵ =(0.1, 0.2, 0.2, 0.2, 0.2), and V =(0.9, 0.7, 0.7, 0.9, 0.9). Panel (a) shows the masses p i (s) and Panel (b) shows the correlation between Ṽ (s) and Ṽ (s ) (this is a nonstationary covariance). probabilities p1(s) p2(s) p3(s) p4(s) p5(s) p6(s) s s s 23

24 New Methodology Cardiac Anomaly Model: Y ij is a binary outcome, taking the value 1 when pregnancy i ends on birth defect j. p ij is the probability of that outcome. We use a probit link. Y i =(Y i1,...,y ij ) T, i =1,...,N; where 1 (p ij) =x T i j + Y ij p ij QX ind D+l X q=1 d=1+l Bern p ij z q (t i (d), s i ) j (B (s i ), d, q) j (B (s i ), d, q) : pollutant and defect specific, spatially and temporally-varying coe cients Represent the e ect of the concentration of air pollutant q at pregnancy week d (corresponding to calendar week t i (d)) at location s i within region B (s i ) on the probability of developing anomaly j for woman i. NC State University 22 / 44

25 New Methodology E ects grouped across defects and pollutants such that (B (s i ), d) =( 1 (B (s i ), d, 1),..., 1 (B (s i ), d, Q),..., J (B (s i ), d, Q)) T G ( (B(si ),d)) (.) = (B (s i ), d) G ind G ( (B(si ),d)) G KSBP MX p k (B (s i ), d) (B(si ),d) k (.) k=1 Y p k (B (s i ), d) =w k (B (s i ), d) V k (1 w j (B (s i ), d) V j ), j<k V i iid Beta (a, b), and a, b > 0. NC State University 23 / 44

26 New Methodology iid k MVN (0, ), k =1,...,M, where T k = (s 1, 1) T k,..., (s L, D)T k = s t s : spatial correlation matrix t : temporal correlation matrix : unstructured covariance matrix describing cross-correlations between anomaly groups and pollutants Di erent kernel functions available for the w k (B (s i ), d) parameters. Di erent flexible covariance structures induced with di erent kernel functions. Conjugancy for full conditionals allows use of Gibbs Sampler. NC State University 24 / 44

27 Finite Mixture Model Representation For finite # of mixture components (M< 1), the model can be expressed as: (B (s i ), d) = (B (s i ), d) g(b(si ),d) g (B (s i ), d) ind Categorical (p 1 (B (s i ), d),...,p M (B (s i ), d)) p k (B (s i ), d) parameters similarly defined as before NC State University 25 / 44

28 Induced Covariance Structure Conditional on p: cov ( (B (s), d), (B (s 0 ), d 0 ) p) = MX x exp{ s kb (s) B (s 0 )k t d d 0 } p k (B (s), d) p k (B (s 0 ), d 0 ). Conditionally nonstationary Allowing M = 1 and unconditionally: cov ( (B (s), d), (B (s 0 ), d 0 )) = x exp{ s kb (s) B (s 0 )k t d d 0 } apple x ((B (s), d), (B (s 0 ), d 0 )) 2 a + b +1 1 ((B (s), d), (B (s 0 ), d 0 )) a +1 k=1 Unconditionally stationary if function is stationary 1 Increased flexibility to model nonstationary and non-gaussian situations 2 Covariance structure depends on chosen kernel function NC State University 26 / 44

29 Kernel Function Examples Table: =( 1, 2, 3 )=(s 1, s 2, d), s =(s 1, s 2 ) Name w k (s, d) Model for kj (s, d), s 0, d 0 Q Uniform 3j=1 I j kj < kj Q 2 kj 3j=1 j 0! + j j I (j6=1) 1 j I (j6=1) ( Q Uniform 3j=1 I j kj < kj P3 j 0 ) j 2 kj Expo j I (j6=1) exp j=1 j I (j6=1) Q < 2 = < Exponential 3j=1 j kj exp : kj ; kj 2 j I (j6=1) /2 1 exp 23/2 : 2 j Q < 2 = Exponential 3j=1 j kj 2 exp : kj ; kj IG 1.5, j I (j6=1) /2 1 Q 3j=1 j j P 3j=1 j j = ; I (j6=1) A 2 j I (j6=1) NC State University 27 / 44

30 Simulation Study Comparing proposed model to other two competiting models Model 1: Multivariate spatial-temporal KSBP prior (Nonseparable/Nonstationary/NonGaussian) Model 2: Multivariate Gaussian separable/stationary spatial-temporal process prior (M=1) cov (B (s), d), B s 0, d 0 = x exp s B (s) B s 0 t d d 0. Model 3: Multiple probit regression model, assuming independence in space and time cov (B (s), d), B s 0, d 0 = if B (s) =B (s 0 ) and d = d 0 0 otherwise. Analyzed MSE and coverage probabilities of the estimated risk parameters NC State University 28 / 44

31 Results MSE Method 1 Method 2 Method Setting Figure: Setting 1: Independence in space-time, Setting 2: Separable/Stationary in space-time, Setting 3: Nonstationary, Setting 4: Nonstationary/Non-Gaussian, Setting 5: Nonseparable/Nonstationary/Non-Gaussian. S.E. =0.6 NC State University 29 / 44

32 Results Table: MSE estimates and confidence intervals. Method Mean LCL UCL Table: Coverage probability estimates and confidence intervals Method Mean LCL UCL NC State University 30 / 44

33 Simulation Study Conclusions Newly introduced prior has flexibility to produce similar results when the prior covariance structure and distribution of the risk e ects agrees with alternative model Outperforms the other models in terms of MSE in the more complicated (nonstationary/non-gaussian) data settings Never significantly outperformed by the competitor models due to flexibility NC State University 31 / 44

34 Model Application Results DIC values very similar for Models 1 and 2, significantly larger for Model 3 Classic DIC may not be appropriate in mixture model setting (Celeux et al., 2003) Model adequacy checks needed in this situation Significant covariate e ects from Model 1: ASD: Black vs. White (negative), > One Fetus vs. One Fetus (positive), Summer vs. Winter (positive) PAVD: Other vs. White (negative), One vs. No Previous Births (negative), Summer vs. Winter (positive) VSD: Black vs. White (negative), > One Fetus vs. One Fetus (positive), Summer vs. Winter (positive) NC State University 32 / 44

35 Model Adequacy Pearson Residual Discrepancy Measure (Dey and Chen, 2000): Observation level: D ij (y ij ; j, ) = (y ij p ij ( j, )) 2 p ij ( j, )(1 p ij ( j, )), Total (since model adequacy is of concern): D(y;, ) = NX JX D ij (y ij, ; i=1 j=1 j, ), Compare f (D(y obs ;, ) y obs )withf (D(y new ;, ) y obs ) Should be very similar if model fits well NC State University 33 / 44

36 Model Adequacy Observed Model 1 New Observed Model 2 New Observed Model 3 New Pro- (a) Model 1. posed model. (b) Model 2. Separable Stationary Gaussian Model. (c) Model 3. Gaussian indepent in space time. (d) Model 1. Model. Proposed (e) Model 2.Separable Stationary Gaussian Model. (f) Model 3. Gaussian indepent in space time. NC State University 34 / 44

37 Model Adequacy P D(yobs ;, ) NJ p 2NJ > K y obs P D(ynew ;, ) NJ p 2NJ > K y obs Model # K=3 K=4 K=5 K=6 K=3 K=4 K=5 K= P D(yobs ;, ) NJ p 2NJ > K y obs P D(ynew ;, ) NJ p 2NJ > K y obs Model # K=1 K=2 K=3 K=4 K=1 K=2 K=3 K= P D(yobs ;, ) NJ p 2NJ > K y obs P D(ynew ;, ) NJ p 2NJ > K y obs Model # K=15 K=17 K=19 K=21 K=15 K=17 K=19 K= MC errors ranged from to with an average value of NC State University 35 / 44

38 Results Nitrate Nitrate Nitrate Effect Effect Effect Week Week Week (g) Model 1. Proposed Model. (h) Model 2. Separable Stationary Gaussian Model. (i) Model 3. Gaussian indepent in space time. Figure: Brownsville, PAVD, nitrate. NC State University 36 / 44

39 Results EC EC EC Effect Effect Effect Week Week Week (a) Model 1. Proposed Model (b) Model 2. Separable Stationary Gaussian Model. (c) Model 3. Gaussian indepent in space time. Figure: Site 3: San Antonio, VSD, elemental carbon. NC State University 37 / 44

40 Results Nitrate Nitrate Nitrate Effect Effect Effect Week Week Week (a) Model 1. Proposed Model (b) Model 2. Separable Stationary Gaussian Model. (c) Model 3. Gaussian indepent in space time. Figure: Site 3: San Antonio subgroup, ASD, nitrate. NC State University 38 / 44

41 Results Nitrate Nitrate Nitrate Effect Effect Effect Week Week Week (a) Model 1. Proposed Model. (b) Model 2. Separable Stationary Gaussian Model. (c) Model 3. Gaussian indepent in space time. Figure: Site 2: Brownsville subgroup, ASD, nitrate. NC State University 39 / 44

42 Results OC OC OC Effect Effect Effect Week Week Week (a) Model 1. Proposed Model. (b) Model 2. Separable Stationary Gaussian Model. (c) Model 3. Gaussian indepent in space time. Figure: Site 1: Houston subgroup, ASD, organic carbon. NC State University 40 / 44

43 Model Application Conclusions Model adequacy results suggest that Model 1 provides the best fit of the data Model 2 misses a majority of the significant weeks that Model 1 detects The improved fit of the data for Model 1 and the similarity to the empirical results (Model 3) suggests that while Model 2 does have smaller variances, it is not characterizing the distribution properly and should not be used to make inference Model 3 is rejected based on model adequacy checks Main signal seen for weeks 3, 7, and 8 for multiple defects and pollutants across most locations NC State University 41 / 44

44 Conclusion/Discussion Simultaneously characterized the e ect of exposure to multiple pollutants on cardiac defect outcomes in a continuous manner throughout the pregnancy Introduced spatial models to account for the changing risks across space Critical time periods identified during the pregnancy in which increased pollution exposure negatively impacts the resulting birth outcomes Results further build the evidence supporting the link between air pollution and birth outcomes while extending our knowledge for the common congenital anomaly outcomes NC State University 42 / 44

45 Acknowledgments National Science Foundation (Fuentes DMS , DMS ) Environmental Protection Agency (Fuentes, R833863) National Institutes of Health (Fuentes, 5R01ES ) NC State University 43 / 44

46 Thank You NC State University 44 / 44