saeHB.Spatial.Beta

We designed this package to provide several functions for area-level small area estimation under Spatial Simultaneous Autoregressive (SAR) and Leroux Conditional Autoregressive (CAR) models, accommodating survey design effect (DEFF) adjustments, using hierarchical Bayesian (HB) method with Beta distribution for variables of interest. Some datasets simulated by a data generation are also provided. The rjags package is employed to obtain parameter estimates using Markov Chain Monte Carlo (MCMC) algorithms. Model-based estimators involve the HB estimators which include the mean estimation, the estimated model coefficients, the random effect, and the random effect variance. For the reference, see Rao and Molina (2015), Liu (2009), Liu et al. (2014), Kubacki and Jedrzejczak (2016), Leroux et al. (2000), Chung and Datta (2020), Anselin (1988), and Anselin and Morrison (2019).

Author

Boby Iwan, Cucu Sumarni

Maintainer

Boby Iwan bobyiwanboby2122@gmail.com

Functions

betadeff_sar() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Spatial SAR Model with a Beta distribution, incorporating survey design effect (DEFF) adjustments.
beta_sar() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Spatial SAR Model with a Beta distribution without DEFF adjustments, by estimating the unknown precision parameter.
betadeff_lerouxcar() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Spatial Leroux CAR Model with a Beta distribution, incorporating survey design effect (DEFF) adjustments.
beta_lerouxcar() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Spatial Leroux CAR Model with a Beta distribution without DEFF adjustments, by estimating the unknown precision parameter.
betadeff_nonspatial() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Non-Spatial Model with a Beta distribution and Independent and Identically Distributed (IID) random effects, incorporating DEFF adjustments.
beta_nonspatial() Estimates small area proportions using a Hierarchical Bayes (HB) method under a Non-Spatial Model with a Beta distribution and IID random effects without DEFF adjustments, by estimating the unknown precision parameter.
build_w() A utility function to construct spatial weights matrices (contiguity, distance, or kernel) required for spatial modeling.
moran_test() A diagnostic function to perform Moran’s I test for spatial autocorrelation.

Installation

System Requirement: Since this package relies on rjags for MCMC computations, you must first install the JAGS (Just Another Gibbs Sampler) software on your computer before installing this package.

You can install the development version of saeHB.Spatial.Beta from GitHub with:

# install.packages("devtools")
devtools::install_github("BobyIwan/saeHB.Spatial.Beta")

Or, to include the vignette, use the following command:

devtools::install_github("BobyIwan/saeHB.Spatial.Beta", build_vignettes = TRUE)

Example

This is a basic example of using the betadeff_sar() function to make an estimate based on synthetic data in this package:

library(saeHB.Spatial.Beta)

# Load dataset and proximity matrix
data(databeta)
data(weight_mat)

# Fitting the Spatial SAR model
model_sar_deff <- betadeff_sar(
  formula = y ~ x1 + x2,
  deff = "deff",
  n_i = "n_i",
  proxmat = weight_mat,
  data = databeta
)

Extract the mean estimation for the areas:

head(model_sar_deff$est)
#>        Estimate   Est.Error  l-95% CI  u-95% CI
#> mu[1] 0.6904202 0.122276168 0.4491634 0.9066046
#> mu[2] 0.7358053 0.075425351 0.5852298 0.8758472
#> mu[3] 0.8315318 0.048186333 0.7266684 0.9207047
#> mu[4] 0.9098379 0.052079819 0.7889493 0.9734647
#> mu[5] 0.9147718 0.034302527 0.8348752 0.9695360
#> mu[6] 0.9933859 0.003904432 0.9845815 0.9986099

Extract the estimated model coefficients:

model_sar_deff$coefficient
#>          Estimate  Est.Error  l-95% CI u-95% CI     Rhat       ESS
#> beta[0] 2.1552971 0.18871070 1.7942579 2.469571 2.351008  63.83022
#> beta[1] 0.9846968 0.16170654 0.6613031 1.298958 1.822902  43.60133
#> beta[2] 0.9015832 0.09514377 0.7211005 1.099114 1.044164  51.33361
#> rho     0.7882047 0.10719963 0.5517969 0.959489 1.215377 536.84195

Extract the random effect for the areas:

model_sar_deff$randeff
#>          Estimate Est.Error    l-95% CI    u-95% CI
#> v[1]  -1.55295590 0.5415769 -2.46067324 -0.30183277
#> v[2]  -0.40668766 0.4868080 -1.27137753  0.68989860
#> v[3]  -0.56451854 0.3761887 -1.29802589  0.20921295
#> v[4]   1.25377103 0.6708254  0.02248222  2.62497249
#> v[5]   1.57656883 0.4135432  0.83389540  2.37519730
#> v[6]   1.07018762 0.6214890  0.08702998  2.30824241
#> v[7]  -2.25189587 0.3772757 -2.96683152 -1.43691527
#> v[8]  -2.22335204 0.4368014 -2.92910385 -1.40066119
#> v[9]  -0.34010596 0.4563517 -1.26326303  0.45044624
#> v[10]  0.64987774 0.6773952 -0.64682674  2.08807649
#> v[11]  2.04639123 0.6617645  0.97155382  3.78632657
#> v[12]  1.09010398 0.6111019 -0.11469939  2.24601657
#> v[13] -1.05056022 0.6390102 -2.11056767  0.54101461
#> v[14] -0.97210465 0.5683333 -2.50486357 -0.06298717
#> v[15] -0.29789316 0.6093787 -1.30150329  0.92165587
#> v[16]  1.03759858 0.5797622  0.17884612  2.48395933
#> v[17]  1.63925819 0.5189512  0.78898753  2.78170881
#> v[18]  0.69487104 0.9392959 -1.42223674  2.17441560
#> v[19] -0.84008891 0.5460784 -1.92505921  0.17527037
#> v[20] -0.26573043 0.5216119 -1.40333441  0.60022711
#> v[21]  0.51006130 0.5192477 -0.50860557  1.41488293
#> v[22]  0.05989928 0.2736546 -0.48650270  0.61845032
#> v[23]  1.45811532 0.6405493  0.54155041  2.95391925
#> v[24]  1.41442004 0.7454328  0.09578068  3.01440523
#> v[25] -0.77148674 0.7099261 -2.04790565  0.60050748
#> v[26]  0.47685012 0.7054332 -0.95129397  2.12708828
#> v[27]  0.26415565 0.7737525 -1.08873258  1.94640679
#> v[28]  1.04560152 0.4962706  0.14640639  2.03711501
#> v[29]  0.80995469 0.7839949 -0.28068586  2.44469111
#> v[30]  1.52865418 0.8541867  0.20885108  3.05165411
#> v[31]  0.40121995 0.6305752 -0.64229385  1.68319967
#> v[32]  0.15909805 0.5044961 -0.71972688  1.12836390
#> v[33]  0.14512840 0.5387659 -1.25674861  1.07377001
#> v[34]  0.87835185 0.6600124 -0.18512457  2.35154680
#> v[35]  2.64810355 0.8006772  1.33574917  4.21903674
#> v[36]  2.02274589 0.9177265  0.64723607  3.78524746

Extract the random effect variance for the areas:

model_sar_deff$refvar
#>           Estimate Est.Error  l-95% CI u-95% CI
#> a.var[1]  4.920737  46.68959 0.9311315 15.25944
#> a.var[2]  4.837588  46.79251 0.8967958 15.18369
#> a.var[3]  4.620072  46.82920 0.8333057 14.65618
#> a.var[4]  4.620072  46.82920 0.8333057 14.65618
#> a.var[5]  4.837588  46.79251 0.8967958 15.18369
#> a.var[6]  4.920737  46.68959 0.9311315 15.25944
#> a.var[7]  4.837588  46.79251 0.8967958 15.18369
#> a.var[8]  4.793385  46.96290 0.8680262 15.27067
#> a.var[9]  4.578069  47.00342 0.8075354 14.69694
#> a.var[10] 4.578069  47.00342 0.8075354 14.69694
#> a.var[11] 4.793385  46.96290 0.8680262 15.27067
#> a.var[12] 4.837588  46.79251 0.8967958 15.18369
#> a.var[13] 4.620072  46.82920 0.8333057 14.65618
#> a.var[14] 4.578069  47.00342 0.8075354 14.69694
#> a.var[15] 4.407630  47.05779 0.7676809 14.18318
#> a.var[16] 4.407630  47.05779 0.7676809 14.18318
#> a.var[17] 4.578069  47.00342 0.8075354 14.69694
#> a.var[18] 4.620072  46.82920 0.8333057 14.65618
#> a.var[19] 4.620072  46.82920 0.8333057 14.65618
#> a.var[20] 4.578069  47.00342 0.8075354 14.69694
#> a.var[21] 4.407630  47.05779 0.7676809 14.18318
#> a.var[22] 4.407630  47.05779 0.7676809 14.18318
#> a.var[23] 4.578069  47.00342 0.8075354 14.69694
#> a.var[24] 4.620072  46.82920 0.8333057 14.65618
#> a.var[25] 4.837588  46.79251 0.8967958 15.18369
#> a.var[26] 4.793385  46.96290 0.8680262 15.27067
#> a.var[27] 4.578069  47.00342 0.8075354 14.69694
#> a.var[28] 4.578069  47.00342 0.8075354 14.69694
#> a.var[29] 4.793385  46.96290 0.8680262 15.27067
#> a.var[30] 4.837588  46.79251 0.8967958 15.18369
#> a.var[31] 4.920737  46.68959 0.9311315 15.25944
#> a.var[32] 4.837588  46.79251 0.8967958 15.18369
#> a.var[33] 4.620072  46.82920 0.8333057 14.65618
#> a.var[34] 4.620072  46.82920 0.8333057 14.65618
#> a.var[35] 4.837588  46.79251 0.8967958 15.18369
#> a.var[36] 4.920737  46.68959 0.9311315 15.25944

References

Rao, J. N. K., & Molina, I. (2015). Small Area Estimation (2nd ed.). New Jersey: John Wiley & Sons, Inc. doi:10.1002/9781118735855.
Liu, B. (2009). Hierarchical Bayes estimation and empirical best prediction of small-area proportions. https://api.drum.lib.umd.edu/server/api/core/bitstreams/cb8e2cbf-441e-4f0f-b4b3-6182f3cf24de/content.
Liu, B., Lahiri, P., & Kalton, G. (2014). Hierarchical Bayes Modeling of Survey-Weighted Small Area Proportions. Statistics Canada. https://www150.statcan.gc.ca/n1/pub/12-001-x/2014001/article/14030-eng.pdf.
Kubacki, J., & Jedrzejczak, A. (2016). Small Area Estimation of Income Under Spatial SAR Model. Statistics in Transition New Series, 17(3), 365–390. doi:10.59170/stattrans-2016-022.
Leroux, B. G., Lei, X., & Breslow, N. (2000). Estimation of Disease Rates in Small Areas: A New Mixed Model for Spatial Dependence. In M. E. Halloran & D. Berry (Eds.), Statistical Models in Epidemiology, the Environment, and Clinical Trials (Vol. 116, pp. 179–191). New York: Springer. doi:10.1007/978-1-4612-1284-3_4.
Chung, H. C., & Datta, G. S. (2020). Bayesian Hierarchical Spatial Models for Small Area Estimation (Research Report Series). Washington, D.C.: U.S. Census Bureau. https://www.census.gov/content/dam/Census/library/working-papers/2020/adrm/RRS2020-07.pdf.
Anselin, L. (1988). Spatial Econometrics: Methods and Models. Dordrecht: Springer Netherlands. doi:10.1007/978-94-015-7799-1.
Anselin, L., & Morrison, S. (2019). Spatial Weights as Distance Functions. https://spatialanalysis.github.io/lab_tutorials/Spatial_Weights_as_Distance_Functions.html.