The use of sampling weights in Bayesian hierarchical models for small area estimation

Highlights

• We describe a hierarchical model to acknowledge design weights in a small area estimation context.

• We carry out a simulation study and compare our method with a number of alternative hierarchical models and direct estimation.

• The simulation illustrates how incorporating the weights can reduce bias and hierarchical modeling can reduce variance.

• We illustrate the method using data from the Behavioral Risk Factor Surveillance System (BRFSS).

Abstract

Hierarchical modeling has been used extensively for small area estimation. However, design weights that are required to reflect complex surveys are rarely considered in these models. We develop computationally efficient, Bayesian spatial smoothing models that acknowledge the design weights. Computation is carried out using the integrated nested Laplace approximation, which is fast. An extensive simulation study is presented that considers the effects of non-response and non-random selection of individuals, allowing examination of the impact of ignoring the design weights and the benefits of spatial smoothing. The results show that, when compared with standard approaches, mean squared error can be greatly reduced with the proposed methods. Bias reduction occurs through the inclusion of the design weights, with variance reduction being achieved through hierarchical smoothing. We analyze data from the Washington State 2006 Behavioral Risk Factor Surveillance System. The models are easily and quickly fitted within the R environment, using existing packages.

Introduction

In this paper we consider the application of spatial models for small area estimation (SAE). SAE is an important endeavor since many agencies require estimates of health, education and environmental measures in order to plan and allocate resources and target interventions. The data upon which SAE is based are often gathered via complex designs.

Standard model-based approaches to the analysis often ignore the sampling mechanism and are therefore subject to potentially large biases. Adjusting for the sampling scheme by including in the model the design variables upon which sampling was based (and which are associated with the outcome of interest) is often not possible, because the required variables are unavailable, or the required model would be overly complex (Gelman, 2007). In this paper we consider the situation in which it is not possible to model the sampling scheme. Weighted design-based approaches provide a common technique to bias removal but the resultant estimators can be highly variable for areas in which only small sample sizes are collected. Hierarchical models provide a method to reduce the variance, with Fay and Herriot (1979) providing an early example and, notably, acknowledging the sampling scheme. Since this influential paper many hierarchical modeling approaches have been suggested, see Rao (2003) for a comprehensive summary of the literature and Pfeffermann (2013) for a more recent account.

In terms of spatial smoothing techniques, a number of authors allow for spatial correlation between areas, see for example Singh et al., 2005, Pratesi and Salvati, 2008, Pereira and Coelho, 2010. These models are subject to bias, however, since they do not adjust for the sampling scheme. Pseudo-likelihood (Skinner, 1989, Pfeffermann et al., 1998) has been used within a hierarchical modeling framework with the scaling of the weights being a major issue (Potthoff et al., 1992, Longford, 1996, Asparouhov, 2006, Rabe-Hesketh and Skrondal, 2006). Congdon and Lloyd (2010) used such an approach and introduced residual spatial random effects. In this paper we describe a range of models that can acknowledge the sampling scheme and allow spatial smoothing. We describe a new approach based on the concept of “effective sample size” and “effective number of cases”. A related Bayesian model has recently been suggested by Ghitza and Gelman (2013), while a quite different approach, based on a penalized spline model, is described in Zheng and Little, 2003, Zheng and Little, 2005. A key feature of the models we describe is that computation is fast and can be carried out using existing packages within the R computing environment.

The outline of this paper is as follows. We begin with a motivating example that concerns diabetes prevalence in the Behavioral Risk Factor Surveillance System (BRFSS) in Section 2. In Section 3 we describe hierarchical spatial and non-spatial models, which we then compare with various approaches in Section 4, via an extensive simulation study. We return to the BRFSS data in Section 5 and conclude the paper with a discussion in Section 6. The Supplementary Materials contain more technical details and additional supporting information.