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In spite of its high practical relevance, cluster specific multiple inference for linear mixed model predictors has hardly been addressed so far. While marginal inference for population parameters is well understood, conditional inference for the cluster specific predictors is more intricate. This work introduces a general framework for multiple inference in linear mixed models for cluster specific predictors. Consistent confidence sets for multiple inference are constructed under both, the marginal and the conditional law. Furthermore, it is shown that, remarkably, corresponding multiple marginal confidence sets are also asymptotically valid for conditional inference. Those lend themselves for testing linear hypotheses using standard quantiles without the need of re-sampling techniques. All findings are validated in simulations and illustrated along a study on Covid-19 mortality in US state prisons., 31 pages, 4 figures

Small area estimation is becoming increasingly popular for survey statisticians. One very important program is Small Area Income and Poverty Estimation undertaken by the United States Bureau of the...

Compositional data are vectors of proportions defined on the unit simplex and this type of constrained data occur frequently in Government surveys. It is also possible for the compositional data to be correlated due to the clustering or grouping of the observations within small domains or areas. We propose a new class of the mixed model for compositional data based on the Kent distribution for directional data, where the random effects also have Kent distributions. One useful property of the new directional mixed model is that the marginal mean direction has a closed form and is interpretable. The random effects enter the model in a multiplicative way via the product of a set of rotation matrices and the conditional mean direction is a random rotation of the marginal mean direction. In small area estimation settings, the mean proportions are usually of primary interest and these are shown to be simple functions of the marginal mean direction. For estimation, we apply a quasi-likelihood method whic...

Section 203 of the Voting Rights Act includes provisions requiring the use of election materials in languages other than English for states or political subdivisions, specifically, when a minimum number of voting age U.S. citizens of specified language minority groups who are unable to speak English very well and have obtained less than a fifth-grade education is met. Data on these characteristics are provided by the 2010 Census and the American Community Survey (ACS), a general purpose sample survey designed to produce a large volume of estimates across the spectrum of the nation’s geographic areas and subgroups of the population. This article describes the small-area model and the estimation methods that were developed and applied to create the list of 2011 political subdivisions that were subject to the provisions.

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...

We use data collected in the National Comorbidity Survey - Adolescent (NCS-A) to develop a methodology to estimate the small-area prevalence of serious emotional distress (SED) in schools in the United States, exploiting the clustering of the main NCS-A sample by school. The NCS-A instrument includes both a short screening scale, the K6, and extensive diagnostic assessments of the individual disorders and associated impairment that determine the diagnosis of SED. We fitted a Bayesian bivariate multilevel regression model with correlated effects for the probability of SED and a modified K6 score at the individual and school levels. Our results provide evidence for the existence of variation in the prevalence of SED across schools and geographical regions. Although the concordance between the modified K6 scale and SED is only modest for individuals, the school-level random effects for the two measures are strongly correlated. Under this model we obtain a prediction equation for the rate of SED based on the mean K6 score and covariates. This finding supports the feasibility of using short screening scales like the K6 as an alternative to more comprehensive lay assessments in estimating school-level rates of SED. These methods may be applicable to other studies aiming at small-area estimation for geographical units.

In this article we show how to predict small area means and obtain valid MSE estimators and confidence intervals when the areas represented in the sample are sampled with unequal probabilities that are possibly related to the true (unknown) area means, and the sampling of units within the selected areas is with probabilities that are possibly related to the outcome values. Ignoring the effects of the sampling process on the distribution of the observed outcomes in such cases may bias the inference very severely. Classical design based inference that uses the randomization distribution of probability weighted estimators cannot be applied for predicting the means of nonsampled areas. We propose simple test statistics for testing the informativeness of the selection of the areas and the sampling of units within the selected areas. The proposed procedures are illustrated by a simulation study and a real application of estimating mean body mass index in counties of the U.S.A, using data from the NHANES III survey.

This article shows how to benchmark small area estimators, produced by fitting separate state-space models within the areas, to aggregates of the survey direct estimators within a group of areas. State-space models are used by the U.S. Bureau of Labor Statistics (BLS) for the production of the monthly Employment and Unemployment State estimates. The computation of the benchmarked estimators and their variances is accomplished by incorporating the benchmark constraints within a joint model of the direct estimators in the different areas, which requires the development of a new filtering algorithm for state-space models with correlated measurement errors. No such algorithm has been developed before. The properties and implications of the use of the benchmarked estimators are discussed and illustrated using BLS unemployment series. The problem of Small Area Estimation is how to produce reliable estimates of area (domain) characteristics, when the sample sizes within the areas are too small to warrant the use of traditional direct survey estimates. This problem is commonly handled by borrowing strength from either neighboring areas and/or from previous surveys, using appropriate cross-sectional/time series models. In order to protect against possible model breakdowns and for consistency in publication, it is often required to benchmark the area model dependent estimates to the direct survey estimate in a group of areas for which the survey estimate is sufficiently accurate. The latter estimate is a weighted sum of the direct estimates in the areas included in the group, so that the benchmarking process defines another way of borrowing strength across the areas.

This article describes a Bayesian hierarchical model for factor analysis of spatially correlated multivariate data. The first level specifies, for each area on a map, the distribution of a vector of manifest variables conditional on an underlying latent factor; at the second level, the area-specific latent factors have a joint distribution that incorporates spatial correlation. The framework allows for both marginal and conditional (e.g., conditional autoregressive) specifications of spatial correlation. The model is used to quantify material deprivation at the census tract level using data from the 1990 U.S. Census in Rhode Island. An existing and widely used measure of material deprivation is the Townsend index, an unweighted sum of four standardized census variables (i.e., Z scores) corresponding to area-level proportions of unemployment, car ownership, crowding, and home ownership. The Townsend and many related indices are computed as linear combinations of measured census variables, which motivates t...

In the small area estimation literature, the sampling error variances are customarily assumed to be known or to depend on a finite number of parameters. We consider the empirical best linear unbiased predictor (EBLUP) obtained by using the individual directly estimated variance for each small area. An approximation for the mean squared error (MSE) of the EBLUP that recognizes the impact on the predictors of estimation of the variance components is derived. Simulation studies show that the theoretical expressions are good approximations for the MSE of the predictors unless the between-area variance component is very small (relative to the within-area variance). An improved estimator of the MSE is developed that has smaller overestimation than the original estimator when the between-area variance component is small. The robustness of the MSE estimator is studied and predictors for nonnormal sampling errors are proposed. An example from the National Resources Inventory that motivated the development of the t...

In the National Crime Survey (NCS), data on victimization can be poststratified into domains determined by urbanization, type of place, and poverty level. There is much difficulty in the analysis of binary data with substantial nonresponse. We consider three Bayesian hierarchical models for binary nonresponse data, like those from the NCS, which are clustered within a number of domains or areas. As in small area estimation, one key feature is that each model “borrows strength” across the areas through the selection approach to nonresponse. This is necessary to estimate the parameters with the least association to the observed data (i.e., weakly identified parameters). The first model assumes that the nonresponse mechanism is ignorable, and the second model assumes that it is nonignorable. We argue that a discrete model expansion (a probabilistic mixture) may be inappropriate for modeling uncertainty about ignorability. Therefore, we propose a third model through a continuous model expansion on an odds rat...

Bayesian methods have been used quite extensively in recent years for solving small-area estimation problems. Particularly effective in this regard has been the hierarchical or empirical Bayes approach, which is especially suitable for a systematic connection of local areas through models. However, the development to date has mainly concentrated on continuous-valued variates. Often the survey data are discrete or categorical, so that hierarchical or empirical Bayes techniques designed for continuous variates are inappropriate. This article considers hierarchical Bayes generalized linear models for a unified analysis of both discrete and continuous data. A general theorem is provided that ensures the propriety of posteriors under diffuse priors. This result is then extended to the case of spatial generalized linear models. The hierarchical Bayes procedure is implemented via Markov chain Monte Carlo integration techniques. Two examples (one featuring spatial correlation structure) are given to illu...

Estimation of finite population means is considered when samples are collected using a stratified sampling design. Finite populations for different strata are assumed to be realizations from different superpopulations. The true means of the observations lie on a regression surface with random intercepts for different strata. The true sampling variances are also different and random for different strata. The strata are connected through two common prior distributions, one for the intercepts and another for the sampling variances for all the strata. The model is appropriate in two important survey situations. First, it can be applied to repeated surveys where the physical characteristics of the sampling units change slowly over time. Second, the model is appropriate in small-area estimation problems where a very few samples are available for any particular area. Empirical Bayes estimators of the finite population means are shown to be asymptotically optimal in the sense of Robbins. The proposed emp...

This article develops a general methodology for small domain estimation based on data from repeated surveys. The results are directly applied to the estimation of median income of four-person families for the 50 states and the District of Columbia. These estimates are needed by the U.S. Department of Health and Human Services (HHS) to formulate its energy assistance program for low income families. The U.S. Bureau of the Census, by an informal agreement, has provided such estimates to HHS through a linear regression methodology since the latter part of the 1970s. The current method is an empirical Bayes method (EB) that uses the Current Population Survey (CPS) estimates as well as the most recent decennial census estimates updated by the per capita income estimates of the Bureau of Economic Analysis. However, with the existing methodology, standard errors associated with these estimates are not easy to obtain. The EB estimates, when used naively, can lead to underestimation of standard errors. Mo...

This article develops an empirical Bayes-type approach for small area estimation based only on the specification of a set of conditionally independent hierarchical mean and variance functions describing the first two moments of the process generating the data. The objective of the analysis is to draw inference about the intermediate or the random means. We combine the quasi-likelihood functions to construct the quasi-posterior density of the random means conditional on the data and the marginal parameters. We describe a method for estimating the marginal parameters that are then substituted in the quasi-posterior density of the random means. The empirical quasi-posterior density so derived is used to obtain the quasi-empirical Bayes estimates of the random parameters. We apply the methodology to estimate the utilization rates of cancer chemotherapy for the selected counties in the state of Washington.

Many finite populations which are sampled repeatedly change slowly over time. Then estimation of finite population characteristics for the current occasion, l, may be improved by the use of data from previous surveys. In this article we investigate the use of empirical Bayes procedures based on two superpopulation models. Each model has the same first stage: The values of the population units on the ith occasion are a random sample from the normal distribution with mean μ i and variance σ 2 i . At the second stage we assume that either (a) μ 1, …, μ l are a random sample from the normal distribution with mean θ and variance δ 2, or (b) given σ 2 i and τ, μ i has the normal distribution with mean θ and variance σ 2 i τ (independently for each i), whereas the σ 2 i are a random sample from the inverse gamma distribution with parameters η/2 and κ/2. In (a) the σ 2 i , θ, and σ 2 are assumed to be unknown, whereas in (b) θ, τ, and κ are unknown. We develop empirical Bayes point estimators and confide...

The estimation of a treatment contrast from experimental data and the estimation of a small-area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two-part mixed linear model. In this article a Bayesian approach to point and interval prediction is presented and its computational requirements are examined. Differences between the Bayesian approach and the traditional (classical) approach are discussed in general terms and, in addition, in terms of two examples taken from the literature: (1) the comparison of drug formulations in a bioavailability trial (Westlake) and (2) the estimation of corn-crop areas using satellite data (Battese, Harter, and Fuller). Some deficiencies in the classical approach are pointed out, and the Bayesian approach is considered from a frequentist perspective. It is shown, via a Monte Carlo study, that, for certain (noninformative) choices of the prior distribution, the frequentist prop...

Small-area estimation has received considerable attention in recent years because of a growing demand for reliable small-area statistics. The direct-survey estimators, based only on the data from a given small area (or small domain), are likely to yield unacceptably large standard errors because of small sample size in the domain. Therefore, alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency. These estimators use models, either implicitly or explicitly, that connect the small areas through supplementary (e.g., census and administrative) data. For example, simple synthetic estimators are based on implicit modeling. In this article, three small-area models, of Battese, Harter, and Fuller (1988), Dempster, Rubin, and Tsutakawa (1981), and Fay and Herriot (1979), are investigated. These models are all special cases of a general mixed linear model involving fixed and random effects, and a small-area mean can be expr...

An adaptation of the James-Stein estimator is applied to sample estimates of income for small places (i.e., population less than 1,000) from the 1970 Census of Population and Housing. The adaptation incorporates linear regression in the context of unequal variances. Evidence is presented that the resulting estimates have smaller average error than either the sample estimates or an alternate procedure of using county averages. The new estimates for these small places now form the basis for the Census Bureau's updated estimates of per capita income for the General Revenue Sharing Program.

The purpose of this study is to investigate methodologies for constructing intercensal estimates of various characteristics of the population for small areas. The proposed methodology is illustrated mainly in the context of unemployment estimates, with one section utilizing dilapidated housing estimates. Alternative synthetic estimates of unemployment based on the 1970 Census 20-percent sample are investigated and their relative error is analyzed. The reliability of the synthetic estimates is discussed in the context of dilapidated housing estimates. Two types of regression models are studied, and the improvements obtained by excluding outliers from the regression are discussed.

Knowledge of the area under different crops is important to the U.S. Department of Agriculture. Sample surveys have been designed to estimate crop areas for large regions, such as crop-reporting districts, individual states, and the United States as a whole. Predicting crop areas for small areas such as counties has generally not been attempted, due to a lack of available data from farm surveys for these areas. The use of satellite data in association with farm-level survey observations has been the subject of considerable research in recent years. This article considers (a) data for 12 Iowa counties, obtained from the 1978 June Enumerative Survey of the U.S. Department of Agriculture and (b) data obtained from land observatory satellites (LANDSAT) during the 1978 growing season. Emphasis is given to predicting the area under corn and soybeans in these counties. A linear regression model is specified for the relationship between the reported hectares of corn and soybeans within sample segments in...

This article considers simultaneous estimation of means from several strata, where each stratum contains a finite number of elements. For example, quite often estimates of annual incomes, unemployment rates, and so forth must be made simultaneously for many areas. Our results find application in small area estimation where very few samples are available from an individual area, and an estimate of a certain area mean or simultaneous estimates of several area means can be improved by incorporating information from similar neighboring areas. Ghosh and Meeden (1986) considered empirical Bayes estimation of the finite population mean assuming a normal superpopulation model. Their estimators generalize automatically when one estimates a vector of stratum means. This article relaxes the normality assumption and assumes instead that the posterior expectation of any stratum mean is a linear function of the sample observations. This is the so-called “posterior linearity” property as described in Ericson (1...

Two approaches to estimating a finite population mean from unequal probability samples are contrasted. In the prediction approach a model is specified for the population values and is used to predict the nonsampled values. In the generalized regression approach the Horvitz-Thompson estimator of the population mean is modified to allow the introduction of covariate information (Sarndal 1980). Generalized regression estimates have the desirable property of asymptotic design consistency, which is not always enjoyed by estimates in the prediction class. However, it is suggested that the prediction approach is the more principled method of estimation. If asymptotic design consistency is a desirable property in a given application, then only models that yield asymptotically design-consistent prediction estimates should be used. Two classes of models with this property are suggested, namely fixed and random-effects models that allow a separate intercept for subclasses of the population indexed by the pr...