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In this vignette, we demonstrate the use of basic area level models for small area estimation, as originally implemented in SUMMER and adapted for use in survey. The models implemented in survey include common small area estimation models, as outlined below. A in-depth review of our approach to model-based small area estimation is provided by Wakefield, Okonek, and Pedersen (2020).
First, we load the necessary packages and data. The required package INLA is not available via a standard repository, so we include code for installation if it is not found. The survey package will be used to generate direct estimates, while dplyr and tidyr will be used for data manipulation.
library(survey)
if (!isTRUE(requireNamespace("INLA", quietly = TRUE))) {
install.packages("INLA", repos=c(getOption("repos"),
INLA="https://inla.r-inla-download.org/R/stable"), dep=TRUE)
}In their vignette for the sae package, Molina and Marhuenda (2015) generate an artificial dataset on income and other related variables to illustrate the use of area level models. In this example, the objective is to estimate prevalence of poverty in Spanish counties.
data("incomedata", package="sae")
data("sizeprov", package="sae")
data("sizeprovedu", package="sae")
povertyline <- 0.6 * median(incomedata$income) # 6557.143
incomedata$in_poverty <- as.integer(incomedata$income < povertyline)The incomedata data frame contains information on 17199 observations of individuals in 52 Spanish provinces. Income values and sampling weights are provided for each individual along with covariate information including age group and education level. Molina and Marhuenda (2015) define the poverty line and calculate an indicator variable (which we name in_poverty) with value 1 if the corresponding income value is below the poverty line and 0 otherwise.
saeBefore considering model-based methods for small area estimation, we compute direct weighted estimators for the desired small area means. The Horvitz-Thompson estimator of domain means is given by
(1/Ni)∑j ∈ Siwjyj
where Ni is the population size of domain i, Si is the set of sampled observations in domain i, wj is the sampling weight for unit j, and yj is the observation for unit j, for all j ∈ Si. The sae::direct function also estimates standard deviation and coefficient of variation for each domain. Note that Ni is assumed known and is provided in the data frame sizeprov. The domains of interest are identified via the provlab variable.
We can use the survey::svyby function to compute the Horvitz-Thompson estimates:
incomedata$pop <- sum(sizeprov$Nd[match(incomedata$provlab, sizeprov$provlab)])
design <- survey::svydesign(ids = ~1, weights = ~weight,
data = incomedata, fpc = ~pop)
# estimate area totals
svy.DIR <- survey::svyby(~in_poverty, ~provlab, design, svytotal)
# calculate corresponding area mean estimates
svy.DIR$prov_pop <- sizeprov$Nd[match(svy.DIR$provlab, sizeprov$provlab)]
svy.DIR$Domain <-svy.DIR$provlab
svy.DIR$Direct = svy.DIR$in_poverty/svy.DIR$prov_pop
svy.DIR$SD= svy.DIR$se/svy.DIR$prov_popThe basic area level model, also called the Fay-Herriot model, treats direct estimates of small area quantities as response data and explicitly models differences between areas using covariate information and random effects (Fay and Herriot 1979). The Fay-Herriot model can be viewed as a two-stage model: in the first stage, a sampling model represents the sampling variability of a direct estimator and in the second stage, a linking model describes the between area differences in small area quantities.
Sampling model:
Let θ̂iDIR be a direct estimator of an area level mean or total θi. The sampling model treats θ̂iDIR as a noisy observation of the true finite population quantity θi:
θ̂iDIR = θi + ϵi; ϵi ∼ind N(0,Vi), i = 1, …, M
where Vi is the known sampling variance of the direct estimator θ̂iDIR.
Linking model:
θi = xiTβ + ui, ui ∼ind N(0,σu2) i = 1, …, M,
where σu2 (between-area residual variance) is estimated. In this basic Fay-Herriot model, the area-specific random effects ui are assumed to be independent and identically distributed (IID) between areas.
Below, we provide a quantile-quantile plot comparing the direct estimates to a Gaussian distribution. Here the observed quantiles align well with those from a Gaussian distribution, which lends some support to the basic IID model.
par(pty = "s")
mu.DIR <- mean(svy.DIR$Direct)
sd.DIR <- sd(svy.DIR$Direct)
qqnorm((svy.DIR$Direct - mu.DIR) / sd.DIR, main = "")
abline(0, 1, col = "red")
The svysmoothArea function adopts a Bayesian approach to inference using models such as the basic area level model, carrying out computation via the INLA package. The svysmoothArea function computes direct estimates and then produces smoothed estimates using a Bayesian Fay-Herriot model. The main arguments of interest are:
formula: Describing the response variable and any area-level covariatesdomain A one-sided formula with the variable containing domain labels on the right. The domain labels variable should be contained in the dataset used to generate the design.design: A survey.design object containing survey data and specifying the survey design.In addition, other commonly used optional arguments include:
adj.mat: Optional adjacency matrix if a spatial smoothing model is desired.transform: If "logit" is specified, a logit transform will be applied to the direct estimates and an appropriate transformation will be applied to the estimated sampling variances before smoothing.direct.est: The direct estimates may be specified directly and smoothing will be applied directly to these user-provided estimates.X.domain: Data frame of area level covariates.domain.size: Data frame of domain sizes used for computing direct estimates if domain sizes are known.Other optional arguments can be specified to change the priors and are described further in the documentation.
For the artificial poverty rate example, we fit the Fay-Herriot model and obtain the following smoothed estimates.
# specify known domain sizes
domain.size <- sizeprov[, c("provlab", "Nd")]
colnames(domain.size)[2] <- "size"
# fit model and obtain svysae object
svy.FH <- svysmoothArea(formula = in_poverty~1,
domain = ~provlab,
design = design,
domain.size = domain.size,
return.samples = T)
svy.FH.table <- data.frame(
Domain = svy.DIR$Domain,
Median = svy.FH$iid.model.est$median,
SE = sqrt(svy.FH$iid.model.est$var)
)
head(svy.FH.table)
## Domain Median SE
## 1 Alava 0.23394447 0.03989036
## 2 Albacete 0.15516988 0.02843149
## 3 Alicante 0.20488000 0.02071508
## 4 Almeria 0.23808371 0.03515380
## 5 Avila 0.07854126 0.02392518
## 6 Badajoz 0.20805100 0.02200545Using the plot() method, we can obtain a comparison of the direct estimates (with corresponding frequentist 95% confidence intervals) and smoothed estimates (with Bayesian 95% credible intervals).
# plot comparison of direct and smoothed estimates
plot(svy.FH)
The SUMMER package includes functions to generate additional diagnostic plots based on samples from the model posterior. The SUMMER::compareEstimates() function generates a heatmap of the posterior pairwise probabilities of one area’s mean exceeding another’s mean.
The svysmoothArea function also allows the use of a model with spatially correlated area effects, where the default implementation assumes a BYM2 model for u (Riebler et al. 2016).
In particular, under the BYM2 model, we assume u is composed of an unstructured term u1′ and a spatially structured term u2*′:
Here, u1′ ∼ N(0,I) is a vector of iid Gaussian random area effects and assume an intrinsic conditional autoregressive (ICAR) Gaussian prior for u2*′. Under this parameterization, σu2 represents the marginal variance of u and ϕ represents the proportion of variation assigned to the spatial term.
Below, we provide an example using data from the Behavioral Risk Factor Surveillance System (BRFSS).
data(BRFSS)
data(KingCounty)
BRFSS <- subset(BRFSS, !is.na(BRFSS$diab2))
BRFSS <- subset(BRFSS, !is.na(BRFSS$hracode))
head(BRFSS)
## age pracex educau zipcode sex street1 street2 seqno year
## 1 30 White college grad 98001 male NA NA 2009000041 2009
## 2 26 White college grad 98107 female NA NA 2009000309 2009
## 3 33 Black college grad 98133 male NA NA 2009000404 2009
## 4 25 White some college 98058 male NA NA 2009000531 2009
## 5 23 White some college 98102 male NA NA 2009000675 2009
## 6 19 Asian some college 98106 male NA NA 2009000694 2009
## hispanic mracex strata hracode tract rwt_llcp genhlth2 fmd obese
## 1 non-Hisp White 53019 Auburn-North NA 2107.463 0 0 0
## 2 non-Hisp White 53019 Ballard NA 2197.322 0 1 0
## 3 non-Hisp Black 53019 NW Seattle NA 3086.511 0 0 0
## 4 non-Hisp White 53019 Renton-South NA 3184.740 1 1 1
## 5 non-Hisp White 53019 Capitol Hill/E.lake NA 3184.740 0 0 0
## 6 non-Hisp Asian 53019 North Highline NA 4391.304 0 0 0
## smoker1 diab2 aceindx2 zipout streetx ethn age4 ctmiss
## 1 0 0 NA 98001 0 1 3 1
## 2 0 0 NA 98107 0 1 3 1
## 3 0 0 NA 98133 0 2 3 1
## 4 0 0 NA 98058 0 1 3 1
## 5 0 0 NA 98102 0 1 4 1
## 6 0 0 NA 98106 0 3 4 1
mat <- SUMMER::getAmat(KingCounty, KingCounty$HRA2010v2_)
design <- svydesign(ids = ~1, weights = ~rwt_llcp,
strata = ~strata, data = BRFSS)
direct <- svyby(~diab2, ~hracode, design, svymean)Below, we fit two versions of the spatial area level model in SUMMER. For this example, we apply the area level model to the logit-transformed direct estimates (as specified via the transform = "logit" argument). In other words, we use the following model:
logit(θ̂iDIR) = logit(θi) + ϵi; ϵi ∼ind N(0, V(logit(θi)), i = 1, …, M where V(logit(θi)) ≈ Vi/θi2 is the sampling variance of θ̂iDIR. The linking model is then logit(θi) = xiTβ + ui, ui ∼ind N(0,σu2) i = 1, …, M,
If we change pc.u and pc.alpha from the default value u = 1, α = 0.01 to u = 0.1, α = 0.01, we assign more prior mass on smaller variance of the random effects, inducing more smoothing. Note that in this example, we include no covariates to avoid having to introduce additional data.
svy.brfss <- svysmoothArea(diab2~1, domain= ~hracode,
design = design,
transform = "logit",
adj.mat = mat, level = 0.95,
pc.u = 0.1, pc.alpha = 0.01)The SUMMER::mapPlot() function can be used to visualize the posterior median estimates and uncertainty estimates.
The nested error model, introduced by Battese, Harter, and Fuller (1988), uses auxiliary data at the unit level.
Nested error model: yij = xijTβ + ui + ϵij, ui ∼ind N(0,σu2), ϵij ∼ind N(0,σϵ2)
Here ui are area random effects and ϵij are unit level errors.
This model assumes there is no sample selection bias.
The cornsoybean and cornsoybeanmeans datasets contain info on corn and soy beans production in 12 Iowa counties (Battese, Harter, and Fuller 1988). The objective here is use satellite imagery of the number of pixels assigned to corn and soy to estimate the hectares grown of corn.
SampSegments: sample size.PopnSegments: population size.MeanCornPixPerSeg: county mean of the number of corn pixels (satellite imagery).MeanSoyBeansPixPerSeg county mean of the number of soy beans (satellite imagery) pixels.The variables MeanCornPixPerSeg and MeanSoyBeansPixPerSeg provide the known county means of the auxiliary variables.
We load the sample data:
data("cornsoybean", package="sae")Next, we load the population auxiliary information:
data("cornsoybeanmeans", package="sae")
Xmean <-
data.frame(cornsoybeanmeans[, c("CountyIndex",
"MeanCornPixPerSeg",
"MeanSoyBeansPixPerSeg")])
Popn <-
data.frame(cornsoybeanmeans[, c("CountyIndex",
"PopnSegments")])The smoothUnit function provides the ability to fit unit level models with unit level covariates for Gaussian and binomial response variables. Below we use the is.unit argument to specify a unit level model and then provide the column names of the unit level covariates in X.unit. Finally, the X argument provides the area level means of each covariate for use when generating predictions. Note that we specify a relatively flat prior on the variance of the area-specific random effect (the arguments pc.u = 100, pc.alpha = 0.01 specify a penalized complexity prior such that P(σu>100) = 0.01 where σu is the standard deviation of the area-specific random effects).
cornsoybean$id <- 1:dim(cornsoybean)[1]
Xsummer <- Xmean
colnames(Xsummer) = c("County", "CornPix", "SoyBeansPix")
des0 <- svydesign(ids = ~1, data = cornsoybean)
svy.bhf.unit <- smoothUnit(formula = CornHec ~ CornPix + SoyBeansPix,
family = "gaussian",
domain = ~County,
design = des0, X.pop = Xsummer,
pc.u = 1000, pc.alpha = 0.01, level = 0.95)
summary(svy.bhf.unit)
## Call:
## smoothUnit(formula = CornHec ~ CornPix + SoyBeansPix, domain = ~County,
## design = des0, family = "gaussian", X.pop = Xsummer, pc.u = 1000,
## pc.alpha = 0.01, level = 0.95)
##
## Methods used: direct.est, iid.model.est
##
## direct.est
## domain mean median var lower upper method
## 1 1 165.7600 165.7600 0.000000 165.76000 165.7600 Direct
## 2 2 96.3200 96.3200 0.000000 96.32000 96.3200 Direct
## 3 3 76.0800 76.0800 0.000000 76.08000 76.0800 Direct
## 4 4 150.8900 150.8900 610.238739 102.47299 199.3070 Direct
## 5 5 158.6233 158.6233 7.430247 153.28077 163.9659 Direct
## 6 6 102.5233 102.5233 430.314438 61.86580 143.1809 Direct
## 7 7 112.7733 112.7733 213.115805 84.16083 141.3858 Direct
## 8 8 144.2967 144.2967 665.975547 93.71685 194.8765 Direct
## 9 9 117.5950 117.5950 87.414716 99.27015 135.9198 Direct
## 10 10 109.3820 109.3820 40.331533 96.93483 121.8292 Direct
## 11 11 110.2520 110.2520 24.148695 100.62048 119.8835 Direct
## 12 12 114.8100 114.8100 178.260890 88.64166 140.9783 Direct
##
## iid.model.est
## domain mean median var lower upper method
## 1 1 123.7876 123.2013 86.06266 108.32013 142.2333 Unit level model: IID
## 2 2 124.2452 124.7868 97.86504 104.93144 142.9875 Unit level model: IID
## 3 3 110.3233 111.5757 102.53029 88.20383 126.9989 Unit level model: IID
## 4 4 113.6742 114.1331 71.81986 96.38425 127.9440 Unit level model: IID
## 5 5 140.1470 139.9905 75.00325 123.76649 155.1843 Unit level model: IID
## 6 6 110.0132 109.4203 57.25731 95.71681 124.7636 Unit level model: IID
## 7 7 115.9491 116.0076 57.80478 101.21077 129.8307 Unit level model: IID
## 8 8 122.6541 122.4973 55.39583 108.86131 138.6098 Unit level model: IID
## 9 9 112.6702 112.1757 48.62457 98.12156 127.7731 Unit level model: IID
## 10 10 123.8395 123.8251 36.20162 111.43605 135.8105 Unit level model: IID
## 11 11 111.2287 111.9254 44.52737 96.51444 123.5814 Unit level model: IID
## 12 12 131.3522 131.1526 35.26051 120.12848 142.9651 Unit level model: IIDThese binaries (installable software) and packages are in development.
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