Logistic Regression Type Small Area Estimations Based on Relative Error

There is growing need for reliable survey-based small area estimates of crime and confidence in police work to design and evaluate place-based policing strategies. Crime and confidence in policing are geographically aggregated and police resources can be targeted to areas with the most problems. High levels of spatial autocorrelation in these variables allow for using spatial random effects to improve small area estimation models and estimates’ reliability. This article introduces the Spatial Empirical Best Linear Unbiased Predictor (SEBLUP), which borrows strength from neighboring areas, to place-based policing. It assesses the SEBLUP under different scenarios of number of areas and levels of spatial autocorrelation and provides an application to confidence in policing in London. The SEBLUP should be applied for place-based policing strategies when the variable’s spatial autocorrelation is medium/high, and the number of areas is large. Confidence in policing is higher in Central and West London and lower in Eastern neighborhoods.

This paper develops allocation methods for stratified sample surveys in which small area estimation is a priority. We assume stratified sampling with small areas as the strata. Similar to Longford (2006), we seek efficient allocation that minimizes a linear combination of the mean squared errors of composite small area estimators and of an estimator of the overall mean. Unlike Longford, we define mean-squared error in a model-assisted framework, allowing a more natural interpretation of results using an intra-class correlation parameter. This allocation has an analytical form for a special case, and has the unappealing property that some strata may be allocated no sample. We derive a Taylor approximation to the stratum sample sizes for small area estimation using composite estimation giving priority to both small area and national estimation.

Large-scale sample surveys are not designed to produce reliable estimates for small areas. Here, small area estimation methods can be applied to estimate population parameters of target variables to detailed geographic scales. Small area estimation for noncontinuous variables is a topic of great interest in the social sciences where such variables can be found. Generalized linear mixed models are widely adopted in the literature. Interestingly, the small area estimation literature shows that multivariate small area estimators, where correlations among outcome variables are taken into account, produce more efficient estimates than do the traditional univariate techniques. In this article, the author evaluate a multivariate small area estimator on the basis of a joint mixed model in which a small area proportion and mean of a continuous variable are estimated simultaneously. Using this method, the author “borrows strength” across response variables. The author carried out a design-based simulation study to evaluate the approach where the indicators object of study are the income and a monetary poverty (binary) indicator. The author found that the multivariate approach produces more efficient small area estimates than does the univariate modeling approach. The method can be extended to a large variety of indicators on the basis of social surveys.

Shrinkage estimation with singular priors and an application to small area estimation

Data of Social Survey and Economic National is a relatively small sample of data, so that data is called small area. Estimation of parameter in small area can be done in two ways, there are direct estimation and indirect estimation. Direct estimation is unbias estimation but give a high variance because from small sample of data. The technique that use to increase efectivity of sample size is indirect estimation or called Small Area Estimation (SAE). SAE is done by adding auxiliary variable. on estimating parameter. Assumed that auxiliary variable has a linear correlation with the direct estimation. If that assumption is incomplete, use an nonparametric approaching. This research is using Kernel Gaussian approaching to build a correlation between direct estimation which expenditure per capita and auxiliary variable which population density. Evaluation of estimation result is done by comparing the value of direct estimation variance with the value of indirect estimation variance using Kernel Gaussian approaching. The result of parameter estimation which approached by SAE is the best estimation, because it produce the small value of variance that is 5,31275, while the value of direct estimator variance is 6,380522. Keywords : Direct Estimation, Small Area Estimation (SAE), Kernel Gaussian

Modelling forest volume with small area estimation of forest inventory using GEDI footprints as auxiliary information

This paper demonstrates a process for translating a database of forest measurements to interactive dashboards through which users can access statistically defensible estimates and analyses anywhere in the conterminous US. It taps the extensive Forest Inventory and Analysis (FIA) plot network along with national remotely sensed data layers to produce estimates using widely accepted model-assisted and small area estimation methodologies. It leverages a decade’s worth of statistical and computational research on FIA’s flexible estimation engine, FIESTA, and provides a vehicle through which scientists and analysts can share their own tools and analytical processes. This project illustrates one pathway to moving statistical research into operational inventory processes, and makes many model-assisted and small area estimators accessible to the FIA community. To demonstrate the process, continental United States (CONUS)-wide model-assisted and small area estimates are produced for ecosubsections, counties, and level 5 watersheds (HUC 10) and made publicly available through R Shiny dashboards. Target parameters include biomass, basal area, board foot volume, proportion of forest land, cubic foot volume, and live trees per acre. Estimators demonstrated here include: the simplest direct estimator (Horvitz–Thompson), model-assisted estimators (post-stratified, generalized regression estimator, and modified generalized regression estimators), and small area estimators (empirical best linear unbiased predictors and hierarchical Bayes both at the area- and unit-level). Auxiliary data considered in the model-assisted and small area estimators included maps of tree canopy, tree classification, and climatic variables. Estimates for small domain sets were generated nationally within a few hours. Exploring results across estimators and target variables revealed the progressive gains in precision using (in order of least gain to highest gain) Horvitz–Thompson, post-stratification, modified generalized regression estimators, generalized regression estimators, area-level small area models, and unit-level small area models. Substantive gains are realized by expanding model-assisted estimators beyond post-stratification, allowing FIA to continue to take advantage of design-based inference in many cases. Caution is warranted in the use of unit-level small area models due to model mis-specification. The dataset of estimates available through the dashboards provides the opportunity for others to compare estimators and explore precision expectations over specific domains and geographic regions. The dashboards also provide a forum for future development and analyses.

Survey data often contain measurements for variables that are semicontinuous in nature, i.e. they either take a single fixed value (we assume this is zero) or they have a continuous, often skewed, distribution on the positive real line. Standard methods for small area estimation (SAE) based on the use of linear mixed models can be inefficient for such variables. We discuss SAE techniques for semicontinuous variables under a two part random effects model that allows for the presence of excess zeros as well as the skewed nature of the nonzero values of the response variable. In particular, we first model the excess zeros via a generalized linear mixed model fitted to the probability of a nonzero, i.e. strictly positive, value being observed, and then model the response, given that it is strictly positive, using a linear mixed model fitted on the logarithmic scale. Empirical results suggest that the proposed method leads to efficient small area estimates for semicontinuous data of this type. We also propose a parametric bootstrap method to estimate the MSE of the proposed small area estimator. These bootstrap estimates of the MSE are compared to the true MSE in a simulation study.

Small area estimation under a spatially non-linear model

Child welfare issues such as child poverty pose a challenge for Indonesia. The provinces in Maluku and Papua have the highest rates of child poverty. Data on child poverty at regencies/municipalities level is needed to address this issue through targeted policies. The direct estimations have a Relative Standard Error (RSE) value of more than 25 percent, necessitating the use of an indirect method, Small Area Estimation (SAE). This study aims to compare the results of indirect estimates of the percentage of children aged 0-17 living in poverty at the regencies/municipalities level in Maluku and Papua using SAE Empirical Best Linear Unbiased Prediction (EBLUP) and Hierarchical Bayes Beta (HB Beta) methods. Susenas KOR March 2023 data is used to produce direct estimates, while Podes 2021 data is used to form auxiliary variables. The results indicate that the SAE HB Beta method provides estimates with better RSE compared to SAE EBLUP. All regencies/municipalities in the Maluku and Papua have a fairly good level of accuracy.

Data linkage can be used to combine values of the variable of interest from a national survey with values of auxiliary variables obtained from another source, such as a population register, for use in small area estimation. However, linkage errors can induce bias when fitting regression models; moreover, they can create non-representative outliers in the linked data in addition to the presence of potential representative outliers. In this paper, we adopt a secondary analyst’s point of view, assuming that limited information is available on the linkage process, and develop small area estimators based on linear mixed models and M-quantile models to accommodate linked data containing a mix of both types of outliers. We illustrate the properties of these small area estimators, as well as estimators of their mean squared error, by means of model-based and design-based simulation experiments. We further illustrate the proposed methodology by applying it to linked data from the European Survey on Income and Living Conditions and the Italian integrated archive of economic and demographic micro data in order to obtain estimates of the average equivalised income for labour market areas in central Italy.

Key MessageForest attribute maps generated using national forest inventory and remote sensing data can be used to quantify forest attributes (i.e., growing stock volume, basal area) affected by disturbance events (like bark-beetle induced damage) through small area estimation techniques. Our results showed that mature stands were more sensitive to bark-beetle infestation than younger ones.ContextRecurrent drought episodes in Europe weakened trees and promoted biotic disturbances such as bark-beetle outbreaks, heavily affecting spruce (Picea abies (L.) H. Karst.) and fir stands (Abies alba Mill.). While remote sensing data make it possible to assess and monitor the spatial extent of outbreaks, the assessment of the actual impact on forest attributes is still lacking.AimsWe propose a small area estimation approach to assess the amount of forest attributes impacted in outbreak areas during the period 2021–2022 using forest attribute maps.MethodsWe produced forest attribute maps using National Forest Inventory (NFI) plots, and auxiliary data from Light Detection and Ranging (LiDAR) and Sentinel-2, acquired in 2021. The maps were generated using a general k-nearest neighbour model for all forest species (general model) and a model specific to spruce and fir only (specific model). Small estimation domains, defined as bark-beetle outbreak areas, were obtained from an independent source using time series of Sentinel-2 data (2018–2022). Small area estimations were generated using bootstrapped model-predictions with reliability assessment.ResultsIn total, 1137 ha of forest area was affected by bark-beetle outbreaks during 2021–2022, corresponding to 494,000 m3 (general model) and 505,000 m3 (specific model) of growing stock volume, respectively. The associated errors were 2.2% for the general model and 2.6% for the specific model. In the specific model, the estimated mean growing stock volume in outbreak areas for 2021 and 2022 was 441.5 m3 ha⁻1, which was 64.5 m3 ha⁻1 higher than the estimated mean volume of spruce and fir for the whole area (377 m3 ha⁻1). This suggests that mature stands were more sensitive to infestation.ConclusionRegular updates to forest attribute maps, possibly every third year using photogrammetric methods, could improve monitoring of forest attributes impacted by disturbances such as drought and bark-beetle outbreaks and contribute to forest decision making and planning.

In small area estimation direct survey estimates that rely only on area-specific data can exhibit large sampling variability due to small sample sizes at the small area level. Efficient small area estimates can be constructed using explicit linking models that borrow information from related areas. The most popular class of models for this purpose are models that include random area effects. Estimation for these models typically assumes that the random area effects are uncorrelated. In many situations, however, it is reasonable to assume that the effects of neighbouring areas are correlated. Models that extend conventional random effects models to account for spatial correlation between the small areas have been recently proposed in literature. A new semi-parametric approach to small area estimation is based on the use of M-quantile models. Unlike traditional random effects models, M-quantile models do not depend on strong distributional assumptions and are robust to the presence of outliers. In its current form, however, the M-quantile approach to small area estimation does not allow for spatially correlated area effects. The aim of this paper is to extend the M-quantile approach to account for such spatial correlation between small areas.

Small area estimators are often based on linear mixed models under the assumption that relationships among variables are stationary across the area of interest (Fay–Herriot models). This hypothesis is patently violated when the population is divided into heterogeneous latent subgroups. In this paper we propose a local Fay–Herriot model assisted by a Simulated Annealing algorithm to identify the latent subgroups of small areas. The value minimized through the Simulated Annealing algorithm is the sum of the estimated mean squared error (MSE) of the small area estimates. The technique is employed for small area estimates of erosion on agricultural land within the Rathbun Lake Watershed (IA, USA). The results are promising and show that introducing local stationarity in a small area model may lead to useful improvements in the performance of the estimators.

Labor figures for Mexico’s municipalities were estimated during 2018’s first quarter by using Small Area Estimation (SAE) techniques with the incorporation of a spatial component – given there is no recent information source with such a level of geographic disaggregation. To achieve this, combined information from different sources was used to build statistical models in which the Economically Active Population, the Employed Population and the Informal Employed Population were taken as variables object of estimation – this information was taken from the National Survey of Occupation and Employment (ENOE for its acronym in Spanish). Auxiliary variables were selected from population censuses, administrative records, and population projections. The results were contrasted with those calculated by applying the percentage structures of 2010 Population and Housing Census to the figures provided by ENOE at a federal entity level, and with the data in this survey (obtained by direct estimation for those municipalities which had a sufficient sample with acceptable coefficients of variation). It is observed that the results obtained by Small Area Estimation are plausible and register coefficients of variation below 10 percent.