Small Area Models Research Articles

Regionalisierung von Gesundheitsindikatoren

The representative health surveys conducted by the Department of Epidemiology and Health Reporting weren’t used before to provide estimates for the spatial distribution of health outcomes. We are discussing the possibilities of providing these outcomes using methods for ‘Small-Area-Estimation’. In the study we are using data of the “German Health Update 2009” (GEDA) to analyze regional inequalities for self-assessed health status, smoking and obesity on the district level in Germany. The small area estimates are provided by multilevel logistic regression models using additional regional statistical data from the official INKAR 2009 database of regional indicators for Germany. We are mapping the results of our analysis for the district level (NUTS-3) using simple thematic maps. Afterwards we compared the results of our small area models with conventional estimates that were based on the official German small scale census. The results showed that our estimates are in line with the prevalences of the census. Overall the results suggest that Small-Area-Estimation methods have a big potential to provide regionalized health indicators for the health reporting in Germany.

Best Predictive Small Area Estimation

We derive the best predictive estimator (BPE) of the fixed parameters under two well-known small area models, the Fay–Herriot model and the nested-error regression model. This leads to a new prediction procedure, called observed best prediction (OBP), which is different from the empirical best linear unbiased prediction (EBLUP). We show that BPE is more reasonable than the traditional estimators derived from estimation considerations, such as maximum likelihood (ML) and restricted maximum likelihood (REML), if the main interest is estimation of small area means, which is a mixed-model prediction problem. We use both theoretical derivations and empirical studies to demonstrate that the OBP can significantly outperform EBLUP in terms of the mean squared prediction error (MSPE), if the underlying model is misspecified. On the other hand, when the underlying model is correctly specified, the overall predictive performance of the OBP is very similar to that of the EBLUP if the number of small areas is large. A general theory about OBP, including its exact MSPE comparison with the BLUP in the context of mixed-model prediction, and asymptotic behavior of the BPE, is developed. A real data example is considered. A supplementary appendix is available online.

Small area estimation via M-quantile geographically weighted regression

The effective use of spatial information, that is, the geographic locations of population units, in a regression model-based approach to small area estimation is an important practical issue. One approach for incorporating such spatial information in a small area regression model is via Geographically Weighted Regression (GWR). In GWR, the relationship between the outcome variable and the covariates is characterised by local rather than global parameters, where local is defined spatially. In this paper, we investigate GWR-based small area estimation under the M-quantile modelling approach. In particular, we specify an M-quantile GWR model that is a local model for the M-quantiles of the conditional distribution of the outcome variable given the covariates. This model is then used to define a bias-robust predictor of the small area characteristic of interest that also accounts for spatial association in the data. An important spin-off from applying the M-quantile GWR small area model is that it can potentially offer more efficient synthetic estimation for out of sample areas. We demonstrate the usefulness of this framework through both model-based as well as design-based simulations, with the latter based on a realistic survey data set. The paper concludes with an illustrative application that focuses on estimation of average levels of Acid Neutralising Capacity for lakes in the Northeast of the USA.

Using IPTW Marginal Structural Modeling to Balance Confounding by Indication

S (ACE) AEP Vol. 20, No. 9 September 2010: 691–724 718 categories ranging from documented clinical diagnosis of Autism to suspicionofASDwithnodiagnostic or educational documentation. Changes in the distribution of cases across categories both by time and presence of IntellectualDisability were modeled using Bayesian multinomial regression. RESULTS: We observed an increase in the proportion of children with a clinical diagnosis of Autistic Disorder from 2002 onwards and a decrease in the proportion suspected of ASD with no clinical or school documentation. Among children without an ID, there was a similarly timed increase in the proportion receiving school services for ASD without a clinical diagnosis. CONCLUSION: Using a Bayesian approach provided a number of analytic and interpretive advantages including direct estimation of uncertainty concerning the true classification probabilities through the posterior distributions and the ability to evaluate changes in the the classification probabilities without the interpretive difficulty associated with sampling-based inference in the more common frequentist approach. P76 A DEMONSTRATION OF MULTILEVEL, SMALL AREA MODELING: ESTIMATING HPV VACCINE COVERAGE IN TEXAS COUNTIES JM Eberth, MDM Hossain, SW Vernon, JA Tiro, University of Texas School of Public Health, Houston TX PURPOSE: Local health data can be used in policy development, resource allocation, program planning, hypothesis generation, and health disparities research. Methods to produce local estimates using data collected for larger regions are known as small area estimation (SAE). Here, we demonstrate the use of model-based SAE to estimate HPV vaccine coverage. METHODS: Graphically and quantitatively, we show the steps involved in the development and evaluation of amultilevel, small area (SA) model for estimating HPV vaccine coverage among girls aged 11–17. The outcome ( 1 dose Gardasil or HPV4) and level 1 predictors were obtained from the 2008 Texas Behavioral Risk Factor Surveillance System. Level 2 predictors and county population counts were obtained from auxiliary datasets. RESULTS: The predicted probabilities from our SAmodel were linked with demographic-specific population counts from the U.S. Census to derive county level estimates of HPV vaccine coverage. Preliminary analyses show geographic variability in HPV vaccine coverage at the county level, with estimates ranging from w10–40%. Only 1 county (i.e. Loving County) had too few residents to estimate HPV vaccine coverage among girls aged 11–17. The 2008 National Immunization Survey estimates 32% of girls aged 13-17 in TX have obtained at least 1 dose of HPV4. CONCLUSION: Although geographic variability in HPV vaccination has been previously reported, our study is the first to predict coverage rates at the county level using model-based SAE.

A Demonstration of Multilevel, Small Area Modeling: Estimating HPV Vaccine Coverage in Texas Counties

A Demonstration of Multilevel, Small Area Modeling: Estimating HPV Vaccine Coverage in Texas Counties

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.

An adjusted maximum likelihood method for solving small area estimation problems

For the well-known Fay–Herriot small area model, standard variance component estimation methods frequently produce zero estimates of the strictly positive model variance. As a consequence, an empirical best linear unbiased predictor of a small area mean, commonly used in small area estimation, could reduce to a simple regression estimator, which typically has an overshrinking problem. We propose an adjusted maximum likelihood estimator of the model variance that maximizes an adjusted likelihood defined as a product of the model variance and a standard likelihood (e.g., a profile or residual likelihood) function. The adjustment factor was suggested earlier by Carl Morris in the context of approximating a hierarchical Bayes solution where the hyperparameters, including the model variance, are assumed to follow a prior distribution. Interestingly, the proposed adjustment does not affect the mean squared error property of the model variance estimator or the corresponding empirical best linear unbiased predictors of the small area means in a higher order asymptotic sense. However, as demonstrated in our simulation study, the proposed adjustment has a considerable advantage in small sample inference, especially in estimating the shrinkage parameters and in constructing the parametric bootstrap prediction intervals of the small area means, which require the use of a strictly positive consistent model variance estimate.

Bayesian Small Area Models for Assessing Wildlife Conservation Risk in Patchy Populations

Species conservation risk assessments require accurate, probabilistic, and biologically meaningful maps of population distribution. In patchy populations, the reasons for discontinuities are not often well understood. We tested a novel approach to habitat modeling in which methods of small area estimation were used within a hierarchical Bayesian framework. Amphibian occurrence was modeled with logistic regression that included third-order drainages as hierarchical effects to account for patchy populations. Models including the random drainage effects adequately represented species occurrences in patchy populations of 4 amphibian species in the Oregon Coast Range (U.S.A.). Amphibian surveys from other locations within the same drainage were used to calibrate local drainage-scale effects. Cross-validation showed that prediction errors for calibrated models were 77% to 86% lower than comparable regionally constructed models, depending on species. When calibration data were unavailable, small area and regional models performed similarly, although poorly. Small area estimation models complement wildlife ecology and habitat studies, and can help managers develop a regional picture of the conservation status for relatively rare species.

Two-stage informative cluster sampling—estimation and prediction with applications for small-area models

This paper considers the effects of informative two-stage cluster sampling on estimation and prediction. The aims of this article are twofold: first to estimate the parameters of the superpopulation model for two-stage cluster sampling from a finite population, when the sampling design for both stages is informative, using maximum likelihood estimation methods based on the sample-likelihood function; secondly to predict the finite population total and to predict the cluster-specific effects and the cluster totals for clusters in the sample and for clusters not in the sample. To achieve this we derive the sample and sample-complement distributions and the moments of the first and second stage measurements. Also we derive the conditional sample and conditional sample-complement distributions and the moments of the cluster-specific effects given the cluster measurements. It should be noted that classical design-based inference that consists of weighting the sample observations by the inverse of sample selection probabilities cannot be applied for the prediction of the cluster-specific effects for clusters not in the sample. Also we give an alternative justification of the Royall [1976. The linear least squares prediction approach to two-stage sampling. Journal of the American Statistical Association 71, 657–664] predictor of the finite population total under two-stage cluster population. Furthermore, small-area models are studied under informative sampling.

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.

Estimating poverty in the Italian provinces using small area estimation models

Sample survey data are broadly used to provide direct estimates of poverty for the whole population and large areas or domains. That is one of the main deficiencies of poverty analysis at a sub-national level (i.e., related either to regions, or provinces). As they are considered very small geographical areas, since the domain-specific sample is not large enough to support direct estimates of adequate precision, they are likely to produce large standard errors, due to the unduly small size of the sample in that area (Ghosh & Rao, 1994). The aim of our paper is to improve the estimation process quality, in terms of efficiency, of some poverty measures for Italian provinces (NUTS3). The adopted approach deals with Area Level Random Effect Model (Fay & Herriot, 1979) which relates small area direct estimators to domain specific covariates, considering the random area effects as independent. Under that model, the Empirical Best Linear Unbiased Predictor (EBLUP) is obtained. We extend the analysis beyond the conventional measures of income poverty that simply dichotomise the population into the "poor" and the "non poor" by a threshold value and we also consider a fuzzy monetary measure treating poverty as a matter of degree (Cheli & Lemmi, 1995; Cheli, 1995). Through such an analysis, we determine some of the socio-economic factors contributing to poverty levels and living standards, and we investigate in depth the territorial perspective. In order to evaluate the performance of the estimation process through small area models and, consequently, the contribution of auxiliary information to composite poverty estimates, we have defined some outcome measures and some quality indicators (Rao, 2003) have been computed. They allow us to test the extent to which the modelling modifies the input direct estimates and the degree of improvement in the accuracy level of the estimates provided by modelling and, more generally, to evaluate the performance of small area estimators.

Analysis of patient flows for orthopedic procedures using small area analysis in Switzerland

BackgroundIn general cantons regulate and control the Swiss health service system; patient flows within and between cantons are thereby partially disregarded. This paper develops an alternative spatial model, based upon the construction of orthopedic hospital service areas (HSAOs), and introduces indices for the analysis of patient streams in order to identify areas, irrespective of canton, with diverse characteristics, importance, needs, or demands.MethodsHSAOs were constructed using orthopedic discharge data. Patient streams between the HSAOs were analysed by calculating three indices: the localization index (% local residents discharged locally), the netindex (the ratio of discharges of nonlocal incoming residents to outgoing local residents), and the market share index (% of local resident discharges of all discharges in local hospitals).ResultsThe 85 orthopedic HSAOs show a median localization index of 60.8%, a market share index of 75.1%, and 30% of HSAOs have a positive netindex. Insurance class of bed, admission type, and patient age are partially but significantly associated with those indicators. A trend to more centrally provided health services can be observed not only in large urban HSAOs such as Geneva, Bern, Basel, and Zurich, but also in HSAOs in mountain sport areas such as Sion, Davos, or St.Moritz. Furthermore, elderly and emergency patients are more frequently treated locally than younger people or those having elective procedures.ConclusionThe division of Switzerland into HSAOs provides an alternative spatial model for analysing and describing patient streams for health service utilization. Because this small area model allows more in-depth analysis of patient streams both within and between cantons, it may improve support and planning of resource allocation of in-patient care in the Swiss healthcare system.

Using small area models to estimate the total area occupied by olive trees

This article aims to estimate the total area occupied by olive trees in a region called Comarca IV, located in a central region of Navarra, Spain. Traditionally, small area linear mixed models have been used for similar purposes using regular quadrats (also called segments) as sampling units, and assuming that the majority of segments are fully included in the study domain. When this does not happen, the sampling units are of different size, and there exists an extra variability that can be very different within areas. In this case it is advisable to include model weights in the model. In this article, we propose a weighted unit level linear mixed model where both the variance components and the coefficients of the model are estimated using these weights. We also discuss, the model performance and compare it with alternatives.

Combining sampling and model weights in agriculture small area estimation

AbstractThis paper is focussed on agriculture small area models for predicting minor crops. In the application considered here, the study domain is often poor in the crop of interest leading to irregular and sparsely distributed plots, usually smaller than the segments used in the sampling design. Hence, the variability among the sampled units becomes large in those areas with a high number of segments. To date, models including weights to account for heteroscedasticity, as well as models considering sampling weights to achieve design‐consistency have been proposed to derive estimators of small area means or totals. In this work, we discuss extensions of these models and the convenience of using both types of weighting. The models performance is illustrated for predicting the total area occupied by olive trees in a region of Navarra, Spain. Copyright © 2006 John Wiley & Sons, Ltd.

On Parametric Bootstrap Methods for Small Area Prediction

SummaryThe particularly wide range of applications of small area prediction, e.g. in policy making decisions, has meant that this topic has received substantial attention in recent years. The problems of estimating mean-squared predictive error, of correcting that estimator for bias and of constructing prediction intervals have been addressed by various workers, although existing methodology is still restricted to a narrow range of models. To overcome this difficulty we develop new, bootstrap-based methods, which are applicable in very general settings, for constructing bias-corrected estimators of mean-squared error and for computing prediction regions. Unlike existing techniques, which are based largely on Taylor expansions, our bias-corrected mean-squared error estimators do not require analytical calculation. They also have the property that they are non-negative. Our prediction intervals have a high degree of coverage accuracy, O(n−3), where n is the number of areas, if double-bootstrap methods are employed. The techniques do not depend on the form of the small area estimator and are applicable to general two-level, small area models, where the variables at either level can be discrete or continuous and, in particular, can be non-normal. Most importantly, the new methods are simple and easy to apply.

Weighted Estimation of Variance Components and Fixed Effects in Small Area Models

Weighted Estimation of Variance Components and Fixed Effects in Small Area Models

Worldwide evaluation of a defibrillation lead with a small geometric electrode surface for high-impedance pacing

Background Pacing leads with a small electrode surface for high-impedance stimulation have been shown to prolong pacemaker longevity, but no sufficient data is available on the safety and feasibility of a defibrillation lead with this novel design. Methods We evaluated the clinical performance of a tined, steroid-eluting defibrillation lead with a small electrode surface area (model 6944) in a prospective multicenter study. A total of 542 patients with conventional indications for an implantable cardioverter defibrillator were randomized 1:1 to receive either the model 6944 or a tined, steroid-eluting defibrillation lead with a conventional sized electrode surface area (model 6942). Device performance and electrical parameters were evaluated at implant and 1, 3, 6, and 12 months thereafter (mean follow-up 11.3 ± 5.6 months). Results Baseline characteristics, lead implant success rates, and defibrillation thresholds did not differ significantly between the 2 groups. While pacing thresholds did not differ significantly during follow-up, pacing impedance was approximately twice as high in the model 6944 as in the model 6942 lead ( P < .0001). Mean R-wave amplitudes were smaller in patients with a 6944 (9.1 ± 3.1 mV vs 9.8 ± 3.6 mV for model 6942, P < .05), but remained stable within both groups throughout the observation period. The total number of ventricular lead-related adverse events and patient survival did not differ significantly between the 2 groups. Conclusions The use of a defibrillation lead with a small electrode surface for high-efficiency pacing is safe and feasible and increases pacing impedance without significantly compromising clinical performance.

반복조사에서 소지역자료 베이지안 분석

Rao와 Yu(1994)는 소지역 추정(small area estimation) 문제를 해결하기 위한 방법으로 추정 시점과 인접지역 정보 등 보조정보와 과걱의 표본조사 결과를 모두 이용하는 모형과 그 모형으로 부터 경험적최량선형비편향추정량(Empirical Best Unbiased Predictor)을 제안하였다. 본 논문에서는 Rao와 Yu의 모형에서 미지의 모수에 대한 사전확률분포를 가정한 계층적 베이즈 추정량을 제안하고, 이를 미국의 주별 4인가족 소득추정문제에 적용하여 그 효율을 미국의 Census Bureau에서 사용하고 있는 경험적 베이즈추정량 및 이전에 제안된 다른 추정량들과 비교하였다. In this paper, we consider the HB estimators of small area means with repeated survey. mao and Yu(1994) considered small area model with repeated survey data and proposed empirical best linear unbiased estimators. We propose a hierachical Bayes version of Rao and Yu by assigning prior distributions for unknown hyperparameters. We illustrate our HB estimator using very popular data in small area problem and then compare the results with the estimator of Census Bureau and other estimators previously proposed.

Hydrodechlorination of dichlorodifluoromethane over palladium model catalysts and a comparison with the hydrodechlorination of 1,1-dichlorotetrafluoroethane

The hydrodechlorination of dichlorodifluoromethane (CFC-12) was performed on small surface area model catalysts, including Pd(111) and Pd (110)single crystals and polycrystalline palladium foil. Reactions were performed in a batch reactor, at atmospheric pressure and in the temperature rangeof 423–523 K. The main products were CH2F2 and CH 4. Ethane could be detected at higher temperatures. Lower deactivation ratesand a higher activation energy for CH2F2 formation were foundfor Pd(111). In a comparison to our previous results with dichlorotetrafluoroethane (CFC-114a), the hydrodechlorination rate of CCl 2F2 was 2 orders of magnitude lower, the mono-dechlorinated product (CHClF2) was not produced, and the bulk palladium hydrogen activity was not detected. An overview of the hydrodechlorination reaction is presented.

Liquid-Phase Hydrogenation of Cyclohexene over Pt Foil Catalysts

A liquid-phase reaction cell, coupled to an ultrahigh-vacuum surface analysis chamber, was built to study liquid-phase hydrogenation reactions on small area model catalyst surfaces. The hydrogenation of cyclohexene to cyclohexane was studied at about 1.5 atm total pressure of H2and as a function of temperature in the 313–333 K range in liquid cyclohexene. The hydrogenation rate of cyclohexene on a clean Pt surface increased with the increasing circulation velocity of the liquid, indicating that the reaction rate was controlled by hydrogen diffusion to the surface. When the surface reaction rate was reduced by the deposition of hydrocarbon fragments on the platinum surface, the rate of cyclohexene hydrogenation became independent of hydrogen diffusion and became controlled by the kinetics of the surface reaction. The estimated activation energy of the reaction is 6 kcal/mol for cyclohexene hydrogenation on a platinum foil that was partly covered with carbonaceous deposits.