1. Introduction
Benmei Liu, Partha Lahiri and Graham Kalton
Small area estimation methods are often used to estimate the proportions of units with a given characteristic for small areas. For example, small area estimation methods are used: in the Census Bureau's Small Area Income and Poverty Estimates (SAIPE) program to estimate poverty rates for states, counties, and school districts (Citro and Kalton 2000; Maples and Bell 2005); with data from the National Survey on Drug Use and Health (NSDUH) to estimate substance rates for states (Wright, Sathe and Spagnola 2007); and with data from the National Assessment of Adult Literacy (NAAL) to estimate proportions at the lowest level of literacy for states and counties (Mohadjer, Rao, Liu, Krenzke and Van De Kerckhove 2012). In each case, the survey's sample sizes in the small areas are not large enough to support direct estimates of adequate precision. A wide variety of methods have been developed to address such small area estimation problems. See Rao (2003) and Jiang and Lahiri (2006a) for reviews, and Chattopadhyay, Lahiri, Larsen and Reimnitz (1999), Farrell, MacGibbon and Tomberlin (1997), Malec, Sedransk, Moriarity and LeClere (1997) and Malec, Davis and Cao (1999) for methods specifically for estimating small area proportions. The range of methods includes both empirical best prediction (EBP) and hierarchical Bayes (HB) approaches and models developed at both the area and unit levels. We focus on HB area level models in this paper.
When an HB area level model is used to produce estimates of proportions of units with a given characteristic for small areas, it is commonly assumed that the survey-weighted proportion for each sampled small area has a normal samping distribution and that the sampling variance of this proportion is known. However, these assumptions are problematic when the small area sample size is small or when the true proportion is near 0 or 1. Reliance on the central limit theorem for approximate normality of the sampling distribution of a proportion requires reasonably large samples, particularly when the population proportion is very small or very large (e.g., under 0.1 or over 0.9). Moreover, with very small or very large proportions, the sampling variance of a sample proportion is highly sensitive to the actual value of the proportion, thus making it difficult to establish a suitable value for the sampling variance. In an effort to overcome these problems, we propose two alternative models for small area proportions and compare them with two commonly used models. The models are described in Section 3. The four models are compared by means of a Monte Carlo simulation study in which stratified simple random samples are generated from a fixed finite population. The simulation study is described in Section 4 and the results are presented in Section 5. The paper finishes with some concluding remarks in Section 6. First, however, we introduce the notation for a stratified simple random sample design in Section 2.
- Date modified: