Robust Small Area Estimation Based on a Survey Weighted MCMC Solution for the Generalized Linear Mixed Model

福尔松等人（1999年）开发了一种调查加权分层贝叶斯（SWHB）估计方法，用于拟合单位级广义线性混合模型，并将其应用于全国家庭药物滥用调查（NHSDA）。 逻辑混合模型的SWHB解决方案对模型错误指定具有稳健性，因为任何大样本区域的小区域估计值（SAE）接近其基于稳健设计的类似物。 它还确保SAE的国家总量在设计上是一致的，因此，大约可以自我校准到基于稳健设计的国家估计值。 使用单位级模型还确保了不同聚集水平的SAE的内部一致性，即使在这些水平使用不同的预测因子。 然而，福尔松等人的解决方案假设，在纳入某些协变量后，调查设计可以被视为无信息;即，假设超级种群模型适用于抽样单位。 为了防止模型误指定的鲁棒性，最好去掉这个假设。 本研究的第一个目标是通过充分考虑调查设计效果来改进SWHB解决方案的不确定性措施。 第二个项目目标是通过确保将汇总的SAE精确校准到设计一致的国家调查估计值，提高增强解决方案的稳健性。 为了实现这些目标，近似高斯似然假设的调查加权固定和随机效应估计函数的输入向量的联合抽样分布。 在此近似高斯似然中，将使用估计函数向量的设计一致性方差-协方差矩阵来充分说明调查设计。 第二个项目的目标是通过采用“校准”马尔可夫链蒙特卡罗（MCMC）算法与大都会黑斯廷斯步骤，准确地基准SAE的强大的设计为基础的国家估计。 将使用分析模型中未包含的具有固定和随机预测因子的模拟数据来比较校准和未校准溶液对模型错误设定的稳健性。 此外，改进的SWHB解决方案将与一个或多个大型调查数据集上的其他解决方案进行对比，例如，NHSDA、NHIS、BRFSS.尽管在国家一级可以获得大量信息，但由于缺乏地方一级的具体信息，与方案规划有关的联邦、州和地方机构面临困难。 通常，需要各州和州以下规划区或县的信息。 原则上，可以扩大提供国家统计数据的调查，以便收集所需的州和州以下数据;但是，政府机构很少有通过直接调查方法收集这一数量数据所需的经济和基础设施资源。 幸运的是，统计方面的新进展和计算能力的提高为昂贵得令人望而却步的直接调查方法提供了一种可行的、负担得起的替代办法，现在可以为小地区提供有效和可靠的估计数。 该项目的目标是通过改进不确定性措施、提供针对模型错误说明的稳健性以及确保不同汇总水平严重不良事件的内部一致性，促进官方统计更广泛地接受基于模型的严重不良事件。 这项研究得到了人口普查局在调查和统计方法资助机会研究下的支持。

Folsom et al. (1999) developed a survey weighted hierarchical Bayes (SWHB) estimation methodology for fitting unit-level generalized linear mixed models and applied it to the National Household Survey on Drug Abuse (NHSDA). The SWHB solution for the logistic mixed model is robust against model misspecification because the small area estimates (SAEs) for any large sample areas are close to their robust design based analogs. It also assures that national aggregates of the SAEs are design consistent and, therefore, approximately self-calibrated to the robust design based national estimates. The use of unit level models also assures internal consistency of SAEs for different levels of aggregation even when different predictors are used at those levels. However, the Folsom et al. solution assumed that the survey design could be treated as noninformative after inclusion of certain covariates; i.e., the superpopulation model was assumed to hold for the sampled units. In the interest of robustness against model misspecifications, it is desirable to remove this assumption. The first goal of this research project is to improve the uncertainty measures of the SWHB solution by taking full account of the survey design effects. The second project goal is to improve the robustness properties of the enhanced solution by assuring exact calibration of the aggregated SAEs to the design consistent national survey estimate. To achieve these goals an approximate Gaussian likelihood is assumed for the joint sampling distribution of the input vector of survey weighted fixed and random effect estimating functions. In this approximate Gaussian likelihood, a design consistent variance-covariance matrix for the vector of estimating functions will be used to fully account for survey design. The second project goal is achieved by employing a 'calibrated' Markov Chain Monte Carlo (MCMC) algorithm with a Metropolitan Hastings step that exactly benchmarks the SAEs to the robust design based national estimates. Simulated data with fixed and random predictors that are not included in the analysis model will be used to compare the robustness of the calibrated and uncalibrated solutions against model misspecification. Also, the improved SWHB solution will be contrasted with other solutions on one or more large survey data sets, e.g., NHSDA, NHIS, BRFSS.In spite of the wealth of information that is available at the national level, Federal, State and local agencies concerned with program planning face difficulties because of the lack of specific information at the local level. Typically, information is desired for States and for substate planning regions or counties. In principle, surveys that provide national statistics could be expanded so that the needed State and sub-state data were collected; however, government agencies seldom have the economic and infrastructure resources needed to collect this volume of data via a direct survey approach. Fortunately, new advances in statistics and increases in computing power offer a viable, affordable alternative to the prohibitively expensive direct survey approach and now permit the production of valid and reliable estimates for small areas. The goal of this project is to promote wider acceptance of model based SAEs for official statistics by improving the uncertainty measures, by providing robustness against model misspecification, and by assuring the internal consistency of SAEs for different aggregation levels. This research is supported by the Bureau of the Census under the Research on Survey and Statistical Methodology Funding Opportunity.