Small Area Estimates for the Conservation Effects Assessment Project
Andreea L. Erciulescu (andreeae@iastate.edu) and Emily J. Berg (emilyb@iastate.edu)
Center for Survey Statistics and Methodology, Iowa State University, Department of Statistics, Ames, IA 50011
MOTIVATION
RESULTS
The Conservation Effects Assessment Project (CEAP) is a series of surveys that evaluate erosion rates on agricultural land. One of the regions of interest is the Boone/Raccoon
River Watershed in Iowa, which is subdivided into smaller watersheds called 8-digit hydrologic unit codes. Because sample sizes for 8-digit hydrologic unit codes are relatively
small, model based estimation methods are considered. Exploratory analysis suggests a positive correlation between the estimated means and standard deviations for erosion rates
in small watersheds. Hierarchical Bayes models are developed that relate direct estimators of variances to covariates. Alternative distributional forms and expectation functions
for the direct estimators of the variances are compared.
RUSLE2
CEAP BACKGROUND
NRI (National Resources Inventory)
Des Moines River Watershed
Objectives: Impacts of conservation practices
Periodic survey of status and changing conditions of the
soil, water, and related resources on non-Federal land in
the US (here focus on soil erosion)
Users: Policy makers, farmers
Scope: HUC8 estimation
Auxiliary information
CEAP
1. Sample of NRI cultivated cropland points
1. APEX model: precipitation
2. Collect data: farmer interview and NRCS (National
Resources Conservation Service) field office databases
2. Soil Survey: wind factor
RUSLE2 posterior means and standard deviations
Area
7100001
7100002
7100003
7100004
7100005
7100006
7100007
7100008
7100009
σ̂
√ei
ni
ȳi
0.1175
0.2545
0.2957
0.2118
0.1420
0.2024
0.3842
0.5555
0.4995
0.0111
0.0609
0.0335
0.0220
0.0171
0.0254
0.0566
0.0840
0.0964
θ̂iF H
0.1174
0.2422
0.2907
0.2147
0.1453
0.2052
0.3683
0.5005
0.4924
sdF H
0.0112
0.0561
0.0324
0.0215
0.0169
0.0249
0.0524
0.0783
0.0860
θ̂iF M
0.1170
0.2430
0.2853
0.2353
0.2043
0.2174
0.3644
0.5085
0.4938
sdF M
0.0107
0.0425
0.0404
0.0489
0.0669
0.0396
0.0497
0.0561
0.0684
θ̂iHM 1
0.1174
0.2422
0.2888
0.2155
0.1512
0.2058
0.3662
0.4881
0.4750
sdHM 1
0.0112
0.0532
0.0337
0.0234
0.0292
0.0251
0.0524
0.0777
0.0838
θ̂iHM 2
0.1172
0.2425
0.2899
0.2156
0.1482
0.2067
0.3701
0.4998
0.4919
sdHM 2
0.0113
0.0551
0.0335
0.0231
0.0229
0.0256
0.0519
0.0755
0.0849
θ̂iHM 2
0.1177
0.2532
0.3919
0.3192
0.3091
0.3476
0.5035
0.6884
0.8582
sdHM 2
0.0229
0.0447
0.0633
0.0391
0.0694
0.0476
0.0937
0.1192
0.2053
Wind Erosion
3. APEX model (black box) 7→ erosion estimates
Response variables
16 continuous variables and 3 ordinal scores
EXPLORATORY DATA ANALYSIS
SMALL AREA LINEAR MODEL
Fay-Herriot model (FH)
ŷi = θi + ei ,
0
θi = xi β + ui ,
2
2
where ei ∼ N(0, n−1
σ
),
and
u
∼
N(0,
σ
i
u ).
ei
i
Wind Erosion posterior means and standard deviations
Assume
2
σ̂ei
is the available direct estimator of the variance and
2
di σ̂ei
2
σei
2
| σei
∼ χ2di
Fixed effects model (FM)
2
σei
=
Area
7100001
7100002
7100003
7100004
7100005
7100006
7100007
7100008
7100009
ȳi
0.1190
0.2521
0.4179
0.3043
0.2715
0.3384
0.5391
0.7713
1.1410
σ̂
√ei
ni
0.0235
0.0431
0.0673
0.0374
0.0641
0.0498
0.1131
0.1321
0.3070
θ̂iF H
0.1165
0.2532
0.3933
0.3165
0.3017
0.3461
0.5065
0.6922
0.8601
sdF H
0.0233
0.0397
0.0635
0.0370
0.0605
0.0456
0.0943
0.1196
0.2089
θ̂iF M
0.1135
0.2461
0.4058
0.3350
0.3202
0.3428
0.5314
0.7627
1.0801
sdF M
0.0234
0.0410
0.0517
0.0843
0.1116
0.0626
0.0936
0.1102
0.1662
θ̂iHM 1
0.1148
0.2483
0.3949
0.3148
0.2963
0.3421
0.5120
0.7256
0.9508
sdHM 1
0.0235
0.0434
0.0651
0.0440
0.0750
0.0491
0.0972
0.1218
0.2209
0
2
γ(xi β) ,
DISCUSSION/FUTURE WORK
β ∼ N ormal, γ ∼ N ormal
Potential scale reduction factors
• greater than 1.1 for some of the parameters in FM and HM1
BAYESIAN HIERARCHICAL VARIANCE MODELS
HM1
HM2
1 d
1
2
X, X ∼ χν
=
2
2
σei
σ0i ν
ν
2
0
2
σ0i =
α(xi β)
ν−2
β ∼ N ormal, α ∼ N ormal, ν ∼ U nif orm
Fit
2
2
(log(σei
)|α) ∼ N (log(xi )0 α, σ00
)
2
σ0i
α∼
2
N ormal, σ00
= e
2 /2
log(xi )0 α+σ00
∼ InvGamma
• HM2 indicates better mixing and convergence of the MCMC chains
DIC
• JAGS in R
• 3 chains, 16000 Monte Carlo samples, 1000 burn-in
samples
• di = ni − 1
Varible
FM
FM*
HM1 HM1*
HM2 HM2*
RUSLE2
12.702 52.507 -57.340 -4.990 -68.967 -5.211
Wind Erosion
5.882 57.204 -29.149 -3.022 -35.665 -3.326
2
Model* is the fitted assuming constant variance σei
= σe2
Include additional information
• multivariate models
Acknowledgement
This research was partially supported by USDA NRCS CESU agreement 68-7482-11-534.
• other watersheds