M-quantile models with application to poverty mapping

Over the last decade there has been growing demand for estimates of population characteristics at small area level. Unfortunately, cost constraints in the design of sample surveys lead to small sample sizes within these areas and as a result direct estimation, using only the survey data, is inappropriate since it yields estimates with unacceptable levels of precision. Small area models are designed to tackle the small sample size problem. The most popular class of models for small area estimation is random effects models that include random area effects to account for between area variations. However, such models also depend on strong distributional assumptions, require a formal specification of the random part of the model and do not easily allow for outlier robust inference. An alternative approach to small area estimation that is based on the use of M-quantile models was recently proposed by Chambers and Tzavidis (Biometrika 93(2):255–268, 2006) and Tzavidis and Chambers (Robust prediction of small area means and distributions. Working paper, 2007). Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumption and automatically provide outlier robust inference. In this paper we illustrate for the first time how M-quantile models can be practically employed for deriving small area estimates of poverty and inequality. The methodology we propose improves the traditional poverty mapping methods in the following ways: (a) it enables the estimation of the distribution function of the study variable within the small area of interest both under an M-quantile and a random effects model, (b) it provides analytical, instead of empirical, estimation of the mean squared error of the M-quantile small area mean estimates and (c) it employs a robust to outliers estimation method. The methodology is applied to data from the 2002 Living Standards Measurement Survey (LSMS) in Albania for estimating (a) district level estimates of the incidence of poverty in Albania, (b) district level inequality measures and (c) the distribution function of household per-capita consumption expenditure in each district. Small area estimates of poverty and inequality show that the poorest Albanian districts are in the mountainous regions (north and north east) with the wealthiest districts, which are also linked with high levels of inequality, in the coastal (south west) and southern part of country. We discuss the practical advantages of our methodology and note the consistency of our results with results from previous studies. We further demonstrate the usefulness of the M-quantile estimation framework through design-based simulations based on two realistic survey data sets containing small area information and show that the M-quantile approach may be preferable when the aim is to estimate the small area distribution function.     

Bayesian prediction for small areas using sur models

Sample surveys are usually designed and analyzed to produce estimates for larger areas and/or populations. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To circumvent such difficulties, borrowing strength from related small areas via modeling becomes essential. In line with this, we propose a hierarchical multivariate Bayes prediction method for small area estimation based on the seemingly unrelated regressions (SUR) model. The performance of the proposed method was evaluated through simulation studies.     

Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.     

Semi‐parametric small‐area estimation by combining time‐series and cross‐sectional data methods

In survey sampling, policymaking regarding the allocation of resources to subgroups (called small areas) or the determination of subgroups with specific properties in a population should be based on reliable estimates. Information, however, is often collected at a different scale than that of these subgroups; hence, the estimation can only be obtained on finer scale data. Parametric mixed models are commonly used in small‐area estimation. The relationship between predictors and response, however, may not be linear in some real situations. Recently, small‐area estimation using a generalised linear mixed model (GLMM) with a penalised spline (P‐spline) regression model, for the fixed part of the model, has been proposed to analyse cross‐sectional responses, both normal and non‐normal. However, there are many situations in which the responses in small areas are serially dependent over time. Such a situation is exemplified by a data set on the annual number of visits to physicians by patients seeking treatment for asthma, in different areas of Manitoba, Canada. In cases where covariates that can possibly predict physician visits by asthma patients (e.g. age and genetic and environmental factors) may not have a linear relationship with the response, new models for analysing such data sets are required. In the current work, using both time‐series and cross‐sectional data methods, we propose P‐spline regression models for small‐area estimation under GLMMs. Our proposed model covers both normal and non‐normal responses. In particular, the empirical best predictors of small‐area parameters and their corresponding prediction intervals are studied with the maximum likelihood estimation approach being used to estimate the model parameters. The performance of the proposed approach is evaluated using some simulations and also by analysing two real data sets (precipitation and asthma).     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

A Bayesian predictive inference for small area means incorporating covariates and sampling weights

The main goal in small area estimation is to use models to ‘borrow strength’ from the ensemble because the direct estimates of small area parameters are generally unreliable. However, model-based estimates from the small areas do not usually match the value of the single estimate for the large area. Benchmarking is done by applying a constraint, internally or externally, to ensure that the ‘total’ of the small areas matches the ‘grand total’. This is particularly useful because it is difficult to check model assumptions owing to the sparseness of the data. We use a Bayesian nested error regression model, which incorporates unit-level covariates and sampling weights, to develop a method to internally benchmark the finite population means of small areas. We use two examples to illustrate our method. We also perform a simulation study to further assess the properties of our method.     

A framework for progressively improving small area population estimates

Summary.  The paper presents a framework for small area population estimation that enables users to select a method that is fit for the purpose. The adjustments to input data that are needed before use are outlined, with emphasis on developing consistent time series of inputs. We show how geographical harmonization of small areas, which is crucial to comparisons over time, can be achieved. For two study regions, the East of England and Yorkshire and the Humber, the differences in output and consequences of adopting different methods are illustrated. The paper concludes with a discussion of how data, on stream since 1998, might be included in future small area estimates.     

Updating small area population estimates in England and Wales

"Population estimates have important implications for resource allocation within government and commerce, and are often assumed to be without error. Currently, central government provides annual population estimates for all the local and health authority districts in Britain, but estimates are needed for smaller areas, typically for electoral wards and postal sectors. Small area estimates are provided by some local authorities and commercial organizations, using different methods; the accuracy of these estimates is modelled here within a multilevel framework. Certain characteristics of the small area and of the method of estimation are included as explanatory variables. Results show that the method of estimation used is of great importance."     

On estimating census undercount in small areas

"Net undercount rates in the U.S. decennial census have been steadily declining over the last several censuses. Differential undercounts among race groups and geographic areas, however, appear to persist. In the following, we examine and compare several methodologies for providing small area estimates of census coverage by constructing artificial populations. Measures of performance are also introduced to assess the various small area estimates. Synthetic estimation in combination with regression modelling provide the best results over the methods considered. Sampling error effects are also simulated. The results form the basis for determining coverage evaluation survey small area estimates of the 1900 decennial census."     

Small Area Estimation for Zero-Inflated Data

The commonly used method of small area estimation (SAE) under a linear mixed model may not be efficient if data contain substantial proportion of zeros than would be expected under standard model assumptions (hereafter zero-inflated data). The authors discuss the SAE for zero-inflated data under a two-part random effects model that account for excess zeros in the data. Empirical results show that proposed method for SAE works well and produces an efficient set of small area estimates. An application to real survey data from the National Sample Survey Office of India demonstrates the satisfactory performance of the method. The authors describe a parametric bootstrap method to estimate the mean squared error (MSE) of the proposed estimator of small areas. The bootstrap estimates of the MSE are compared to the true MSE in simulation study.     

Assessing different uncertainty measures of EBLUP: a resampling-based approach

The empirical best linear unbiased prediction approach is a popular method for the estimation of small area parameters. However, the estimation of reliable mean squared prediction error (MSPE) of the estimated best linear unbiased predictors (EBLUP) is a complicated process. In this paper we study the use of resampling methods for MSPE estimation of the EBLUP. A cross-sectional and time-series stationary small area model is used to provide estimates in small areas. Under this model, a parametric bootstrap procedure and a weighted jackknife method are introduced. A Monte Carlo simulation study is conducted in order to compare the performance of different resampling-based measures of uncertainty of the EBLUP with the analytical approximation. Our empirical results show that the proposed resampling-based approaches performed better than the analytical approximation in several situations, although in some cases they tend to underestimate the true MSPE of the EBLUP in a higher number of small areas.     

遥感辅助的农作物种植面积小域估计方法研究

遥感影像是大数据的一种,利用遥感对农作物播种面积进行估算常采用回归估计量或校准估计量,通常都需要将地面样本数据与遥感分类信息相结合。但对于大多数回归估计量,对省级总体的农作物面积估算只能满足对省级总体的精度要求而不能分解到更小区域,比如县和乡级。本文利用黑龙江省2011年的地面实测样本数据结合遥感分类结果,构建了单元层次的多响应变量的多元回归形式的小域模型,并将小域效应设定为固定形式。这样基于回归估计方法,既可以估算分县的主要作物播种面积,也可以使得各县播种面积估计结果相加就等于回归模型含义下的省级总体的总量估计。对黑龙江省玉米、水稻、大豆分县小域估计结果的精度评价（变异系数C.V）,平均而言均可以满足县级精度要求。本文的结果表明小域估计方法在解决省级总体对全省和分县的农作物种植面积多级估算问题中具有很好的应用。     

Small Area Estimation Using Estimated Population Level Auxiliary Data

Unit level linear mixed models are often used in small area estimation (SAE), and the empirical best linear unbiased prediction (EBLUP) is widely used for the estimation of small area means under such models. However, EBLUP requires population level auxiliary data, atleast area specific aggregated values. Sometimes population level auxiliary data is either not available or not consistent with the survey data. We describe a SAE method that uses estimated population auxiliary information. Empirical results show that proposed method for SAE produces an efficient set of small area estimates.     

Information theoretic methods in small domain estimation

Small area estimation techniques are becoming increasingly used in survey applications to provide estimates for local areas of interest. The objective of this article is to develop and apply Information Theoretic (IT)-based formulations to estimate small area business and trade statistics. More specifically, we propose a Generalized Maximum Entropy (GME) approach to the problem of small area estimation that exploits auxiliary information relating to other known variables on the population and adjusts for consistency and additivity. The GME formulations, combining information from the sample together with out-of-sample aggregates of the population of interest, can be particularly useful in the context of small area estimation, for both direct and model-based estimators, since they do not require strong distributional assumptions on the disturbances. The performance of the proposed IT formulations is illustrated through real and simulated datasets.     

Hierarchical Bayes GLMs for the analysis of spatial data: An application to disease mapping

This paper considers estimation of cancer incidence rates for local areas. The raw estimates usually are based on small sample sizes, and hence are usually unreliable. A hierarchical Bayes generalized linear model approach is taken which connects the local areas, thereby enabling one to ‘borrow strength’. Random effects with pairwise difference priors model the spatial structure in the data. The methods are applied to cancer incidence estimation for census tracts in a certain region of the state of New York.     

Small area estimation of proportions in business surveys

Binary data are often of interest in business surveys, particularly when the aim is to characterize grouping in the businesses making up the survey population. When small area estimates are required for such binary data, use of standard estimation methods based on linear mixed models (LMMs) becomes problematic. We explore two model-based techniques of small area estimation for small area proportions, the empirical best predictor (EBP) under a generalized linear mixed model and the model-based direct estimator (MBDE) under a population-level LMM. Our empirical results show that both the MBDE and the EBP perform well. The EBP is a computationally intensive method, whereas the MBDE is easy to implement. In case of model misspecification, the MBDE also appears to be more robust. The mean-squared error (MSE) estimation of MBDE is simple and straightforward, which is in contrast to the complicated MSE estimation for the EBP.     

Hierarchical Bayes estimation in small area estimation using cross-sectional and time-series data

Bayesian methods have been extensively used in small area estimation. A linear model incorporating autocorrelated random effects and sampling errors was previously proposed in small area estimation using both cross-sectional and time-series data in the Bayesian paradigm. There are, however, many situations that we have time-related counts or proportions in small area estimation; for example, monthly dataset on the number of incidence in small areas. This article considers hierarchical Bayes generalized linear models for a unified analysis of both discrete and continuous data with incorporating cross-sectional and time-series data. The performance of the proposed approach is evaluated through several simulation studies and also by a real dataset.     

Exploring spatial dependence in area-level random effect model for disaggregate-level crop yield estimation

This paper describes an application of small area estimation (SAE) techniques under area-level spatial random effect models when only area (or district or aggregated) level data are available. In particular, the SAE approach is applied to produce district-level model-based estimates of crop yield for paddy in the state of Uttar Pradesh in India using the data on crop-cutting experiments supervised under the Improvement of Crop Statistics scheme and the secondary data from the Population Census. The diagnostic measures are illustrated to examine the model assumptions as well as reliability and validity of the generated model-based small area estimates. The results show a considerable gain in precision in model-based estimates produced applying SAE. Furthermore, the model-based estimates obtained by exploiting spatial information are more efficient than the one obtained by ignoring this information. However, both of these model-based estimates are more efficient than the direct survey estimate. In many districts, there is no survey data and therefore it is not possible to produce direct survey estimates for these districts. The model-based estimates generated using SAE are still reliable for such districts. These estimates produced by using SAE will provide invaluable information to policy-analysts and decision-makers.     

SMALL‐AREA ESTIMATION OF POVERTY: THE AID INDUSTRY STANDARD AND ITS ALTERNATIVES

Small‐area estimation of poverty‐related variables is an increasingly important analytical tool in targeting the delivery of food and other aid in developing countries. We compare two methods for the estimation of small‐area means and proportions, namely empirical Bayes and composite estimation, with what has become the international standard method of Elbers, Lanjouw & Lanjouw (2003) . In addition to differences among the sets of estimates and associated estimated standard errors, we discuss data requirements, design and model selection issues and computational complexity. The Elbers, Lanjouw and Lanjouw (ELL) method is found to produce broadly similar estimates but to have much smaller estimated standard errors than the other methods. The question of whether these standard error estimates are downwardly biased is discussed. Although the question cannot yet be answered in full, as a precautionary measure it is strongly recommended that the ELL model be modified to include a small‐area‐level error component in addition to the cluster‐level and household‐level errors it currently contains. This recommendation is particularly important because the allocation of billions of dollars of aid funding is being determined and monitored via ELL. Under current aid distribution mechanisms, any downward bias in estimates of standard error may lead to allocations that are suboptimal because distinctions are made between estimated poverty levels at the small‐area level that are not significantly different statistically.     

What Level of Statistical Model Should We Use in Small Area Estimation?

If unit‐level data are available, small area estimation (SAE) is usually based on models formulated at the unit level, but they are ultimately used to produce estimates at the area level and thus involve area‐level inferences. This paper investigates the circumstances under which using an area‐level model may be more effective. Linear mixed models (LMMs) fitted using different levels of data are applied in SAE to calculate synthetic estimators and empirical best linear unbiased predictors (EBLUPs). The performance of area‐level models is compared with unit‐level models when both individual and aggregate data are available. A key factor is whether there are substantial contextual effects. Ignoring these effects in unit‐level working models can cause biased estimates of regression parameters. The contextual effects can be automatically accounted for in the area‐level models. Using synthetic and EBLUP techniques, small area estimates based on different levels of LMMs are investigated in this paper by means of a simulation study.