Small Area Modeling for Survey Data with Smoothed Error Covariance Structure via Generalized Design Effects

We consider the problem of specifying the design-based error covariance structure in small area modeling with survey data because of too many unknown design parameters. As a compromise, it is customary to treat the estimated covariance as known although undesirable because of its instability, i.e., relative variance of this estimate could be high, and as a result, it could cause serious underestimation of variance of estimates. To alleviate this problem, one can either model the estimated error covariance structure in addition to the main task of modeling the estimated small area estimates (SAEs), or smooth the estimated covariance by only specifying its mean function. We advocate smoothing over modeling the error covariance because of strong simplifying assumptions needed in modeling the error covariance structure of estimated covariance of SAEs, and because practitioners, in general, prefer to take the least assumptions in modeling path philospphy. In practice, smoothing may work quite well as supported by the recent work of Singh, You, and Mantel (2005) who used the well known property of constant deff (design effect) over suitable subgroups of estimates to smooth the error covariance in modeling direct estimates from the cross-sectional and longitudinal data obtained from the Canadian Labour Force Survey. In this application, however, cross-sectionally the small areas were treated as strata, i.e., cross-sectionally, the error covariance was diagonal. This is rather restrictive because small areas or domains need not be strata. Moreover, using optimal estimating functions (EFs) of Godambe and Thompson (1986) for estimating model parameters, it can be shown that even when small areas are strata, additional summary statistics (in particular, the estimated domain count if it is not constant) for each domain or area should be used besides the direct SAEs, and thus the error covariance becomes necessarily nondiagonal. Use of EFs was also advocated by Singh, Folsom, and Vaish (2002, 2003) to define approximate EF-based Gaussian likelihood (EFGL) for obtaining efficient SAEs under a HB (hierarchical Bayes) framework. In this paper, we propose the idea of using g-deff (generalized design effect), defined earlier by Rao and Scott (1981) in the context of categorical data analysis, to deal with the case of nondiagonal error covariance. Simulation results for SAEs based on a linear mixed model show that the EFGL method with the proposed smoothing works quite well in general, and provides improved coverage of confidence intervals in particular.