Strategies for Multiply Imputed Survey Data and Modeling in the Context of Small Area Estimation

To target resources and policies where they are most needed, it is essential that policy-makers are provided with reliable socio-demographic indicators on sub-groups. These sub-groups can be defined by regional divisions or by demographic characteristics and are referred to as areas or domains. Information on these domains is usually obtained through surveys, often planned at a higher level, such as the national level. As sample sizes at disaggregated levels may become small or unavailable, estimates based on survey data alone may no longer be considered reliable or may not be available. Increasing the sample size is time consuming and costly. Small area estimation (SAE) methods aim to solve this problem and achieve higher precision. SAE methods enrich information from survey data with data from additional sources and "borrow" strength from other domains . This is done by modeling and linking the survey data with administrative or register data and by using area-specific structures. Auxiliary data are traditionally population data available at the micro or aggregate level that can be used to estimate unit-level models or area-level models. Due to strict privacy regulations, it is often difficult to obtain these data at the micro level. Therefore, models based on aggregated auxiliary information, such as the Fay-Herriot model and its extensions, are of great interest for obtaining SAE estimators. Despite the problem of small sample sizes at the disaggregated level, surveys often suffer from high non-response. One possible solution to item non-response is multiple imputation (MI), which replaces missing values with multiple plausible values. The missing values and their replacement introduce additional uncertainty into the estimate. Part I focuses on the Fay-Herriot model, where the resulting estimator is a combination of a design-unbiased estimator based only on the survey data (hereafter called the direct estimator) and a synthetic regression component. Solutions are presented to account for the uncertainty introduced by missing values in the SAE estimator using Rubin's rules. Since financial assets and wealth are sensitive topics, surveys on this type of data suffer particularly from item non-response. Chapter 1 focuses on estimating private wealth at the regionally disaggregated level in Germany. Data from the 2010 Household Finance and Consumption Survey (HFCS) are used for this application. In addition to the non-response problem, income and wealth data are often right-skewed, requiring a transformation to fully satisfy the normality assumptions of the model. Therefore, Chapter 1 presents a modified Fay-Herriot approach that incorporates the uncertainty of missing values into the log-transformed direct estimator of a mean. Chapter 2 complements Chapter 1 by presenting a framework that extends the general class of transformed Fay-Herriot models to account for the additional uncertainty due to MI by including it in the direct component and simultaneously in the regression component of the Fay-Herriot estimator. In addition, the uncertainty due to missing values is also included in the mean squared error estimator, which serves as the uncertainty measure. The estimation of a mean, the use of the log transformation for skewed data, and the arcsine transformation for proportions as target indicators are considered. The proposed framework is evaluated for the three cases in a model-based simulation study. To illustrate the methodology, 2017 data from the HFCS for European Union countries are used to estimate the average value of bonds at the national level. The approaches presented in Chapters 1 and 2 contribute to the literature by providing solutions for estimating SAE models in the presence of multiply imputed survey data. In particular, Chapter 2 presents a general approach that can be extended to other indicators. To obtain the best possible SAE estimator in terms of accuracy and precision, it is important to find the optimal model for the relationship between the target variable and the auxiliary data. The notion of "optimal" can be multifaceted. One way to look at optimality is to find the best transformation of the target variable to fully satisfy model assumptions or to account for nonlinearity. Another perspective is to identify the most important covariates and their relationship to each other and to the target variable. Part II of this dissertation therefore brings together research on optimal transformations and model selection in the context of SAE. Chapter 3 considers both problems simultaneously for linear mixed models (LMM) and proposes a model selection approach for LMM with data-driven transformations. In particular, the conditional Akaike information criterion is adapted by introducing the Jacobian into the criterion to allow comparison of models at different scales. The methodology is evaluated in a simulation experiment comparing different transformations with different underlying true models. Since SAE models are LMMs, this methodology is applied to the unit-level small-area method, the empirical best predictor (EBP), in an application with Mexican survey and census data (ENIGH - National Survey of Household Income and Expenditure) and shows improvements in efficiency when the optimal (linear mixed) model and the transformation parameters are found simultaneously. Chapter 3 bridges the gap between model selection and optimal transformations to satisfy normality assumptions in unit-level SAE models in particular and LMMs in general. Chapter 4 explores the problem of model selection from a different perspective and for area-level data. To model interactions between auxiliary variables and nonlinear relationships between them and the dependent variable, machine learning methods can be a versatile tool. For unit-level SAE models, mixed-effects random forests (MERFs) provide a flexible solution to account for interactions and nonlinear relationships, ensure robustness to outliers, and perform implicit model selection. In Chapter 4, the idea of MERFs is transferred to area-level models and the linear regression synthetic part of the Fay-Herriot model is replaced by a random forest to benefit from the above properties and to provide an alternative modeling approach. Chapter 4 therefore contributes to the literature by proposing a first way to combine area-level SAE models with random forests for mean estimation to allow for interactions, nonlinear relationships, and implicit variable selection. Another advantage of random forest is its non-extrapolation property, i.e. the range of predictions is limited by the lowest and highest observed values. This could help to avoid transformations at the area-level when estimating indicators defined in a fixed range. The standard Fay-Herriot model was originally developed to estimate a mean, and transformations are required when the indicator of interest is, for example, a share or a Gini coefficient. This usually requires the development of appropriate back-transformations and MSE estimators. 5 presents a Fay-Herriot model for estimating logit-transformed Gini coefficients with a bias-corrected back-transformation and a bootstrap MSE estimator. A model-based simulation is performed to show the validity of the methodology, and regionally disaggregated data from Germany are used to illustrate the proposed approach. 5 contributes to the existing literature by providing, from a frequentist perspective, an alternative to the Bayesian area-level model for estimating Gini coefficients using a logit transformation

Semi-Parametric Empirical Best Prediction for small area estimation of unemployment indicators

The Italian National Institute for Statistics regularly provides estimates of unemployment indicators using data from the Labor Force Survey. However, direct estimates of unemployment incidence cannot be released for Local Labor Market Areas. These are unplanned domains defined as clusters of municipalities; many are out-of-sample areas and the majority is characterized by a small sample size, which render direct estimates inadequate. The Empirical Best Predictor represents an appropriate, model-based, alternative. However, for non-Gaussian responses, its computation and the computation of the analytic approximation to its Mean Squared Error require the solution of (possibly) multiple integrals that, generally, have not a closed form. To solve the issue, Monte Carlo methods and parametric bootstrap are common choices, even though the computational burden is a non trivial task. In this paper, we propose a Semi-Parametric Empirical Best Predictor for a (possibly) non-linear mixed effect model by leaving the distribution of the area-specific random effects unspecified and estimating it from the observed data. This approach is known to lead to a discrete mixing distribution which helps avoid unverifiable parametric assumptions and heavy integral approximations. We also derive a second-order, bias-corrected, analytic approximation to the corresponding Mean Squared Error. Finite sample properties of the proposed approach are tested via a large scale simulation study. Furthermore, the proposal is applied to unit-level data from the 2012 Italian Labor Force Survey to estimate unemployment incidence for 611 Local Labor Market Areas using auxiliary information from administrative registers and the 2011 Census

Potential of ALOS2 and NDVI to estimate forest above-ground biomass, and comparison with lidar-derived estimates

Remote sensing supports carbon estimation, allowing the upscaling of field measurements to large extents. Lidar is considered the premier instrument to estimate above ground biomass, but data are expensive and collected on-demand, with limited spatial and temporal coverage. The previous JERS and ALOS SAR satellites data were extensively employed to model forest biomass, with literature suggesting signal saturation at low-moderate biomass values, and an influence of plot size on estimates accuracy. The ALOS2 continuity mission since May 2014 produces data with improved features with respect to the former ALOS, such as increased spatial resolution and reduced revisit time. We used ALOS2 backscatter data, testing also the integration with additional features (SAR textures and NDVI from Landsat 8 data) together with ground truth, to model and map above ground biomass in two mixed forest sites: Tahoe (California) and Asiago (Alps). While texture was useful to improve the model performance, the best model was obtained using joined SAR and NDVI (R2 equal to 0.66). In this model, only a slight saturation was observed, at higher levels than what usually reported in literature for SAR; the trend requires further investigation but the model confirmed the complementarity of optical and SAR datatypes. For comparison purposes, we also generated a biomass map for Asiago using lidar data, and considered a previous lidar-based study for Tahoe; in these areas, the observed R2 were 0.92 for Tahoe and 0.75 for Asiago, respectively. The quantitative comparison of the carbon stocks obtained with the two methods allows discussion of sensor suitability. The range of local variation captured by lidar is higher than those by SAR and NDVI, with the latter showing overestimation. However, this overestimation is very limited for one of the study areas, suggesting that when the purpose is the overall quantification of the stored carbon, especially in areas with high carbon density, satellite data with lower cost and broad coverage can be as effective as lidar

Nonparametric estimation of mean-squared prediction error in nested-error regression models

Nested-error regression models are widely used for analyzing clustered data. For example, they are often applied to two-stage sample surveys, and in biology and econometrics. Prediction is usually the main goal of such analyses, and mean-squared prediction error is the main way in which prediction performance is measured. In this paper we suggest a new approach to estimating mean-squared prediction error. We introduce a matched-moment, double-bootstrap algorithm, enabling the notorious underestimation of the naive mean-squared error estimator to be substantially reduced. Our approach does not require specific assumptions about the distributions of errors. Additionally, it is simple and easy to apply. This is achieved through using Monte Carlo simulation to implicitly develop formulae which, in a more conventional approach, would be derived laboriously by mathematical arguments.Comment: Published at http://dx.doi.org/10.1214/009053606000000579 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org