Bayesian spatial models for estimating means of sampled and non-sampled small areas
Section 1. Introduction

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Sample surveys provide useful data in estimating various characteristics of a population of interest. Surveys are generally designed so that design-based estimators have adequate accuracy. However, when it comes to estimating a sub-population characteristic, a design-based direct estimate, based solely on data from that sub-population alone, is usually inaccurate as the accessible sample size is small and sometimes nonexistent. Sub-populations that lack a reasonable sample size to produce reliable direct estimates are known as small areas. Also, limited resources often preclude many sub-populations from selecting in the sample, creating non-sampled small areas. For example, the American Community Survey (ACS) is conducted to produce reliable statistics for the U.S. counties. However, the ACS usually samples about one-third of the counties resulting in many non-sampled small areas.

To enhance the accuracy of direct estimates of small areas, a model-based approach has been widely used to facilitate borrowing information from direct estimates of other domains and other auxiliary data. In many applications, supplementary information from other surveys and administrative data provide useful covariates. A model-based estimate of an area is produced by suitably shrinking its direct estimate (if available) to a synthetic regression estimate based on auxiliary variables. The improvement in prediction greatly depends on to what extent the sub-population means of the characteristic are related to the auxiliary variables. If a small area has no direct estimate, the traditional independent random-effects model of Fay and Herriot (1979) estimates estimates the mean by a synthetic regression estimate alone.

Fay and Herriot (1979) proposed a useful model for developing estimates of small area means based on direct survey estimates (if available) and computed synthetic regression estimates from auxiliary variables. This model, which is essentially a mixed linear model, is popularly known as the Fay-Herriot (FH) model in small area estimation. For $i=\mathrm{1,}\dots \mathrm{,}m,$ let $Y_i$ be the direct estimate of the small area characteristic ${\theta }_i$ obtained from a survey. Also let $x_i$ and $\beta$ be the $p$ -component vectors of covariates and corresponding regression coefficients, respectively. Denoting the sampling error of $Y_i$ as $e_i,$ the independent FH model can be written as

$$Y_i={\theta }_i+e_i\mathrm{,}{\theta }_i=x_i^T\beta +v_i\mathrm{,}i=\mathrm{1,}\dots \mathrm{,}m\mathrm{,}(1.1)$$

where $e_i’\text{s}$ and random effects $v_i’\text{s}$ $i=\mathrm{1,}\dots \mathrm{,}m,$ are all independently distributed with $e_i~N\mathrm{<mo\; stretchy="false">(</mo>0,}{D}_i\mathrm{<mo\; stretchy="false">)</mo>},$ and $v_i\stackrel{\text{i}\text{.i}\text{.d}\text{.}}{{\displaystyle ~}}N\mathrm{<mo\; stretchy="false">(</mo>0,}{\sigma }_v^2\mathrm{<mo\; stretchy="false">)</mo>.}$ Sampling variances $D_i,$ $i=\mathrm{1,}\dots \mathrm{,}m$ are taken as known, whereas the regression parameter $\beta$ and model error variance ${\sigma }_v^2,$ called model parameters, are unknown quantities. For non-sampled areas with auxiliary variables, only the second part of (1.1) holds for ${\theta }_i.$

There has been extensive research on the independent FH model and its many variants. While Fay and Herriot (1979) used an empirical Bayes (EB) approach, subsequently, Prasad and Rao (1990), Datta and Lahiri (2000) and Datta, Rao and Smith (2005) used the frequentist approach and derived the second-order mean squared error (MSE) of empirical best linear unbiased predictor (EBLUP) of ${\theta }_i$ and various second-order approximate unbiased estimators of the MSE’s (see Datta and Lahiri, 2000). However, Ghosh (1992) proposed a hierarchical Bayesian (HB) approach for the Fay-Herriot model (see also Datta et al. (2005)). In the Bayesian framework, the FH model in (1.1) can be expressed as the following HB model:

$$Y_i|{\theta }_1\mathrm{,}\dots \mathrm{,}{\theta }_m\mathrm{,}\beta \mathrm{,}{\sigma }_v^2\stackrel{\text{ind}}{{\displaystyle ~}}N\mathrm{<mo\; stretchy="false">(</mo>}{\theta }_i\mathrm{,}{D}_i\mathrm{<mo\; stretchy="false">)</mo>,}i=\mathrm{1,}\dots \mathrm{,}m\mathrm{,}(1.2)$$

$${\theta }_i|\beta \mathrm{,}{\sigma }_v^2\stackrel{\text{ind}}{{\displaystyle ~}}N\mathrm{<mo\; stretchy="false">(</mo>}{x}_i^T\beta \mathrm{,}{\sigma }_v^2\mathrm{<mo\; stretchy="false">)</mo>,}i=\mathrm{1,}\dots \mathrm{,}m\mathrm{,}(1.3)$$

$$\pi \mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{,}{\sigma }_v^2\mathrm{<mo\; stretchy="false">)</mo>}\propto g\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{,}{\sigma }_v^2\mathrm{<mo\; stretchy="false">)</mo>,}(1.4)$$

where $g(\cdot )$  is a suitably chosen function of $\beta$ and ${\sigma }_v^2,$ which expresses a prior probability density function (pdf) for these parameter. An EB predictor for ${\theta }_i,$ which does not require a prior pdf as in (1.4), was originally developed by Fay and Herriot (1979). While a standard EB approach usually underestimates the measure of uncertainty of the EB estimator of ${\theta }_i,$ the HB approach facilitates quantification of uncertainty due to estimation of unknown model parameters, $\beta$ and ${\sigma }_v^2.$ The uncertainty is fully captured by the posterior distribution of the model parameters.

In model-based estimations, random effects are of great importance in capturing the remaining variability of the ${\theta }_i’\text{s}$ that is not explained by the regression model. In real applications, small areas generally involve features such as population size, ethnicity, age-group, and education level, which might affect the variability of small area effects. Furthermore, when disease prevalence rates are of interest, it is reasonable to assume that random effects of adjacent small areas are correlated in a certain way. In such cases, the FH model given in (1.1), which we refer to as the independent FH random-effects model, oversimplifies and misspecifies the distribution of random effects by assuming a common and independent distribution. Opsomer, Claeskens, Ranalli, Kauermann and Breidt (2008) and Rao, Sinha and Dumitrescu (2014) proposed nonparametric small area estimation models, which capture spatial proximity effect using the P-spline function. However, these approaches require additional computational cost for model inference and uncertainty quantification.

In this work, we propose spatial FH models which effectively account for heteroscedasticity and spatial dependence of the small area effects. We take a fully Bayesian approach by specifying a class of noninformative priors on the model parameters and model spatial dependence of small area random effects by four widely used autocorrelation structures. These include simultaneous autoregressive and three types of conditional autoregressive models. There is an abundance of literature on spatial models under the Bayesian framework. Sun, Tsutakawa and Speckman (1999) studied an HB model with the conditional and intrinsic autoregressive models on the random effects. The same models were considered by Speckman and Sun (2003) in the context of Bayesian spline smoothing. For small area estimation, You and Zhou (2011) modeled small area effects using a conditional autoregressive model. As an extension of the time series FH model (Datta, Lahiri, Maiti and Lu, 1999), Torabi (2012) proposed a spatio-temporal model with intrinsic autoregressive random effects. Porter, Holan, Wikle and Cressie (2014) proposed an extension of the FH model with functional covariates and intrinsic autoregressive random effects. Porter, Wikle and Holan (2015) incorporated the conditional autoregressive random effects on the multivariate FH model.

The existing Bayesian spatial small area estimation models consider a proper prior on ${\sigma }_v^2$ even though the specification of a proper prior will require subject matter expertise. Furthermore, all existing models assume a conditional autoregressive structure on the random effects. The main contributions of this paper are as follows. First, to the best of our knowledge, the proposed models in Section 2 (Section 2.1) include most of the popularly used spatial structures. Second, in Section 2.2, we further extend the spatial models to estimate means of several non-sampled small areas with no direct estimates. The non-sampled area mean ${\theta }_i$ is estimated by borrowing strength from the auxiliary variables of the area and, for spatial models, from the regression residuals of its neighboring areas. Third, for all proposed models, we provide, in Section 2.3, sufficient conditions for posterior propriety for a class of improper noninformative priors on model parameters. Interestingly, the sufficient conditions do not depend on the assumed spatial model, provided that the model yields a positive definite covariance matrix for the random effects. We provide rejection sampling steps for simulating from the posterior of the proposed models in Section 3. The effectiveness of the proposed spatial models is demonstrated in Sections 4 and 5. We apply the spatial models to simulated datasets and real survey data from the Current Population Survey (CPS). We compare various spatial models in Section 5 to estimate four-person family median incomes for the forty-nine contiguous states of the U.S. based on the CPS data and appropriate covariates from the previous Census and administrative data. Our data analysis and simulation studies reveal that proposed spatial models significantly improve prediction accuracy and reduce the measure of uncertainty, posterior standard deviation. We provide concluding remarks in Section 6. All technical details are provided in Appendix.

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