MILAINE DOMINICI SANFINS Neuropatia auditiva/ dessincronia ...
Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca...
-
date post
22-Dec-2015 -
Category
Documents
-
view
232 -
download
0
Transcript of Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca...
![Page 1: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/1.jpg)
Health Risks of Exposure to Chemical Composition of Fine Particulate Air
Pollution
Francesca DominiciYeonseung Chung
Michelle BellRoger Peng
Department of BiostatisticsSchool of Public Health
Harvard University
![Page 2: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/2.jpg)
PM2.5 PM10
PM10-2.5
Chemical constituents
Size Total mass
SO4=
Si
Ca
Z
Ni
NH4 +
NO3-
Fe
EC
OC
Inorganicfraction of PM
MetalsAl
Groups
Bell Dominici Ebisu Zeger Samet EHP 2007
![Page 3: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/3.jpg)
New Scientific Questionsand Statistical Challenges
What are the mechanisms of PM toxicity?
Size? Chemical components? Sources?
![Page 4: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/4.jpg)
Three questions• Are day-to-day changes in the levels of the PM2.5 chemical
components associated with day-to-day changes in admission rates? (short-term effects of PM2.5 components)
– Multi-site time series studies of PM components
• Are the short-term effects of PM2.5 total mass on admission rates modified by long-term averages of PM2.5 chemical components?– Multi-site time series studies of PM total mass and
second stage regression on PM components
• Are the long-term effects of PM2.5 total mass on mortality modified by long-term averages of PM2.5 components?
– Spatially varying coefficient models
![Page 5: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/5.jpg)
National Data
![Page 6: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/6.jpg)
The National Medicare Cohort Study, 1999-2009 (MCAPS)
• Medicare data include: – Billing claims for everyone over 65
enrolled in Medicare (~48 million people), •date of service•disease (ICD 9)•age, gender, and race•place of residence (zip code)
• Approximately 204 counties linked to the PM2.5 monitoring network
![Page 7: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/7.jpg)
MCAPS study population: 204 counties with
populations larger than 200,000 (11.5 million people)
![Page 8: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/8.jpg)
Daily time series of hospitalization rates and PM2.5 levels in Los Angeles county (1999-2005)
![Page 9: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/9.jpg)
Exposure data: Chemical composition data on PM2.5 from the STN network
1. Constructed a database of time series data for 52 PM2.5 chemical constituents from over 250 STN monitors for 2000 to 2008
2. Identified a subset of PM2.5 components that substantially contribute and/or co-vary with daily PM2.5 concentrations
3. Constructed a database that links by zip code the chemical composition data to human health data
Bell et al EHP 2007
![Page 10: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/10.jpg)
• Only seven of the 52 components contributed 1% or more to total mass for yearly or seasonal averages
1. OCM 2. Sulfate 3. Nitrate 4. EC5. Silicon6. Sodium Ion7. Ammonium
Chemical composition data on PM2.5
![Page 11: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/11.jpg)
PM2.5 chemical components and mortality rates: 1999-2008
![Page 12: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/12.jpg)
Short Term Exposures
![Page 13: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/13.jpg)
Multi-site time series data
1. Semi-Parametric Regression for time series data
2. Hierarchical Models for combining health risks across locations
3. Model Uncertainty in effect estimation
![Page 14: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/14.jpg)
Ytc ~ Poisson(t
c )
logtc logN t
c kc
k
xkt lc s1(tempt
c,1) s2(dewtc,2)
Confounders:•weather variables•seasonality
JASA 2004
s3(temp[1 3],tc ,3) s4 (dew[1 3],t
c ,4 )s5(t,5)
![Page 15: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/15.jpg)
Everson and Morris, JRSSB 2000Dominici Samet Zeger JRSSA 2000R package for TLNISE, released on March 26 2008 by Roger Peng
logtc logN t
c kc
k
xkt lc z t
cc
c |0,1,2 ~ N(0 1(h
c h ), 2)cor( c1 , c2 )exp[ d(c1,c2)]
Smooth part
![Page 16: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/16.jpg)
Assessing the sensitivity of the results to model assumptions
Sensitivity of the exposure effect estimate to:
• the number of degrees of freedom in the smooth functions of time to adjust for seasonality• the degree of flexibility in the adjustment for weather variables• other potential confounders (e.g other pollutants)
![Page 17: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/17.jpg)
PM2.5 and Admissions PM10-2.5 and Admissions
US EPA PM Fact Sheet 2006: To better protect public health EPA issued the Agency most protective suite of national air quality standards for particle pollution ever
Dominici et al JAMA 2006 Peng et al JAMA 2008
![Page 18: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/18.jpg)
National average estimates and 95% posterior intervals for the percent increase in hospital admissions for cardiovascular diseases per 1 IQR increase in each of the seven PM2.5 components, 119 U.S. counties, 2000--2006.
Peng et al submitted
Peng et al 2008, EHP
![Page 19: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/19.jpg)
Do the PM2.5 chemical constituents modify the short-term effects of PM2.5 on mortality and morbidity?
% increase in CVD-PM2.5 risk per IQR increase in the fraction of PM2.5 total mass
for each component. Statistically significant associations are shown in bold 101 US counties1999-2005
Bell et al AJRCCM 2009
logtc logN t
c cPM2.5tc z t
cc
c |0,1,2 ~ N(0 1k
k
x k,2)
![Page 20: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/20.jpg)
Long term exposures
![Page 21: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/21.jpg)
Average PM2.5 levels for the period 2000 to 2006 for 518 monitors in the East US
![Page 22: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/22.jpg)
Bayesian Spatially Varying Coefficient Models for estimating spatially varying long term effects of PM2.5 (Stage I)
Mortality counts in zip codes “close” to monitor “i”
average PM2.5
)(~ ijij PoissonY
iijijijiiijij xxxxaaN **10 ,)log()log(
•“i” is the monitor•“j” is the month• “xij” is the average PM2.5 over the 12 previous months
![Page 23: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/23.jpg)
Bayesian Spatially Varying Coefficient Models for estimating spatially varying long term effects of PM2.5 (Stage II)
Investigating whether PM2.5 chemical components explain the spatial variability in mortality risks
Long-term average of log relative proportion of kth component
,01
*00 i
p
k
ikki za
,11
*01 i
p
k
ikki za
€
0 ~ MVN(0,τ 0−1ρ (si,s j;φ0))
€
1 ~ MVN(0,τ 1−1ρ (si,s j;φ1))
iikik zzz *
||)||exp();,( 00 jiji ssss Spatial Coordinate of ith location
![Page 24: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/24.jpg)
Missing data challenge
• The monitoring network that provides the chemical composition (STN) data is sparser than and does not exactly match with the PM2.5 monitoring stations.
• For 241 monitors we have both PM2.5 and composition data
• For 277 monitors we have PM2.5 but composition is missing
• For 10 monitors we have composition data but PM2.5 is missing.
sparser than 518
251
241
Composition dataavailable
All spatial units in our analysis
![Page 25: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/25.jpg)
Analysis options
Option 1. Using only 241 locations where the chemical composition data are available
Option 2. Using all 518 locations with an imputation procedure for missing composition data incorporated in the model
![Page 26: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/26.jpg)
Before the composition is incorporated
-5.39(-5.41,-5.38)
0.0126(0.0088,0.0165)
95.5(78.5,114.7)
4357.8(3141.2 ,5673.8)
17.4(12.8,19.9)
14.1(6.5,19.7)
After the composition is incorporated
-5.41(-5.39,-5.38)
0.0125(0.0088,0.0161)
100.0(82.1,12.1)
4538.6(3319.5805.7)
17.5(12.9,19.8)
14.3(6.5,19.7)
Option 1 : using 241 locations
0 1 0 10 1
1ˆiaPosterior median for slope:0ˆiaPosterior median for slope:
Table 1. Posterior median for each parameter with 95% credible intervals
![Page 27: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/27.jpg)
• We propose a prior for the missing composition data and incorporate an imputation procedure in the MCMC iterations.
1. We denote the component levels for 3 different locations as
2. We assume the component levels for observed + extra locations come from a multivariate Gaussian spatial process as
3. We obtain posterior estimates for using a spBayes R package (Finley et al., 2010).
ExtrapnnpExtra
ObspnnpObs
MisspnnpMiss
nzzzzZ
nzzzzZ
nzzzzZ
ExtraExtra
ObsObs
MissMiss
,)',,,,,,(
,)',,,,,,(
,)',,,,,,(
1111
1111
1111
: # of missing locations
: # of observed locations
: # of extra locations
518
251
241
ZMiss
ZObs
ZExtra
277
10
Option 2 : using 518 locationsConstructing a prior for ZMiss
},{),,1
1(~ ZZZ
Zn
Zn
Extra
ObsOE
Extra
ObsMVNZ
ZZ
},{ ZZ
![Page 28: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/28.jpg)
• We propose a prior for the missing composition data and incorporate an imputation procedure in the MCMC iterations.
4. Using , we specify a multivariate Gaussian process for the component levels for missing+observed locations.
5. We derive the conditional distribution for ZMiss given Zobs
6. Because the component levels cannot be negative, we use a truncated version of the above multivariate Gaussian process as a prior for ZMiss
Option 2 : using 518 locationsConstructing a prior for ZMiss
}ˆ,ˆ{ˆZZ
)ˆˆ
ˆˆ,
ˆ1
ˆ1(~
OOTMO
MOMM
Zn
Zn
Obs
MissMO
Obs
MissMVNZ
ZZ
)ˆˆˆˆ,)ˆ1(ˆˆ]ˆ1([~| 11 TMOOOMOMMZnobsOOMOZnObsMiss ObsMiss
ZMVNZZ
![Page 29: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/29.jpg)
• The hierarchical structure of our full Bayes model is
Option 2 : using 518 locations
))()()()(,,,|(
))()(ˆ,|(
),,,,,,|(
11001100
21
21
b
ZZ
bZZXY
ObsMiss
ObsMiss
Prior for ZMiss
Likelihood
Prior for fixed effects
Prior for spatially correlated random effects
![Page 30: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/30.jpg)
OCobs OCobs+pred
SO4obs SO4obs+pred
ECobs ECobs+Pred
Siobs Siobs+pred
NO3obs NO3obs+pred Sodobs Sodobs+pred
![Page 31: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/31.jpg)
(Using 241 locations)
(Using 518 locations)
Dot is posterior median and line indicates 95% credible interval.
Effect modification of the long term effects of PM2.5 on mortality by PM2.5 composition
![Page 32: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/32.jpg)
Summary
• We used three study designs to address three related epidemiological questions on the toxicity of PM2.5
• We implemented MCMC algorithms for very large data sets
![Page 33: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/33.jpg)
SummaryWe found that:
– PM10-2.5, (e.g. crustal materials) lead to smaller health risks than PM2.5 (e.g. combustion-related constituents)
– EC and OCM, which are generated typically from vehicle emissions, diesel, and wood burning, lead to the largest risk of emergency hospital admissions for cardiovascular and respiratory diseases compared to the other PM2.5 chemical constituents
Combustion sources Crustal materials
![Page 34: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/34.jpg)
Region Analysis option
Region 1 Option 1 -5.38( -5.39, -5.35)
0.010 (0.005,0.015)
130 (96,168)
4679(2619, 8589)
18.2(13.1, 19.9)
12.8(2.4, 19.5)
Option 2 -5.38(-5.39, -5.37)
0.009(0.005,0.012)
136(109,163)
6358(3943, 10615)
18.3(13.4,19.9)
9.7(2.2, 19.3)
Region 2 Option 1 -5.38( -5.41, -5.36)
0.010 (0.002, 0.020)
92(58,136)
3404(1589, 6890)
17.7(10.7,19.9)
8.5(0.27,19.3)
Option 2 -5.39(-5.40,-5.38)
0.018(0.011, 0.022)
106(84, 128)
2500(831, 6538)
17.5(12.7, 19.8)
7.1(2.4,15.7)
Region 3 Option 1 -5.44( -5.47, -5.41)
0.018(0.009 , 0.026)
93(66,128)
2787(752,6189)
15.8(8.3, 19.7)
3.7(0.27,18.6)
Option 2 -5.44(-5.45, -5.43)
0.016(0.008, 0.021)
120(97, 156)
4374(2203, 7878)
15.2(10.4, 19.4)
9.2(1.7,19.5)
0 1 0 10 1
Table 1. Posterior median for other parameters with 95% credible intervals
Sub-region analysis
![Page 35: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/35.jpg)
• Our model can be written as
Main effects interactions
• The likelihood function is
€
logλ = logN + Xθ1 + Zθ 2
€
(Y | X,Z,θ1,θ 2)
Option 2 : using 518 locations
![Page 36: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/36.jpg)
References
• Chung Y, Dominici F, Bell M Bayesian Spatially Varying Coefficients Models of Long term effects of PM2.5 and PM2.5 composition (in progress)
• Papers in blue have been presented in these slides
![Page 37: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/37.jpg)
• We denote the component levels for 3 different locations as
• We assume that the component levels for observed + extra locations come from a multivariate Gaussian spatial process as
• We obtain posterior estimates for using a spBayes R package (Finley et al., 2010). },{),,
1
1(~ ZZZ
Zn
Zn
Extra
ObsOE
Extra
ObsMVNZ
ZZ
Option 2 : using 518 locationsConstructing a prior for ZMiss
},{ ZZ
518
251
241
ZMiss
ZObs
ZExtra
277
10
![Page 38: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/38.jpg)
• Using , we specify a multivariate Gaussian process for the component levels for missing+observed locations.
• We derive the conditional distribution for ZMiss given Zobs
)ˆˆˆˆ,)ˆ1(ˆˆ]ˆ1([~| 11 TMOOOMOMMZnobsOOMOZnObsMiss ObsMiss
ZMVNZZ
)ˆˆ
ˆˆ,
ˆ1
ˆ1(~
OOTMO
MOMM
Zn
Zn
Obs
MissMO
Obs
MissMVNZ
ZZ
}ˆ,ˆ{ˆZZ
Constructing a prior for ZMiss
![Page 39: Health Risks of Exposure to Chemical Composition of Fine Particulate Air Pollution Francesca Dominici Yeonseung Chung Michelle Bell Roger Peng Department.](https://reader035.fdocuments.net/reader035/viewer/2022062314/56649d7e5503460f94a6115e/html5/thumbnails/39.jpg)
• We place a prior for ZMiss and incorporate an imputation procedure in the MCMC iterations.
€
(Y | X,ZMiss,ZObs,θ1,θ 2)(ZMiss | ZObs,ZExtra,Ω)
The prior can be obtained from a multivariate spatial process defined for ZMiss, Zobs, ZExtra (Next Slide).
Option 2 : using 518 locations
We obtain posterior estimates for using a spBayes R package (Finley et al., 2010).
€