An objective stepwise Bayes approach to small area estimation

The term ‘small area’ or ‘small domain’ is commonly used to denote a small geographical area that has a small subpopulation of people within a large area. Small area estimation is an important area in survey sampling because of the growing demand for better statistical inference for small areas in public or private surveys. In small area estimation problems the focus is on how to borrow strength across areas in order to develop a reliable estimator and which makes use of available auxiliary information. Some traditional methods for small area problems such as empirical best linear unbiased prediction borrow strength through linear models that provide links to related areas, which may not be appropriate for some survey data. In this article, we propose a stepwise Bayes approach which borrows strength through an objective posterior distribution. This approach results in a generalized constrained Dirichlet posterior estimator when auxiliary information is available for small areas. The objective posterior distribution is based only on the assumption of exchangeability across related areas and does not make any explicit model assumptions. The form of our posterior distribution allows us to assign a weight to each member of the sample. These weights can then be used in a straight forward fashion to make inferences about the small area means. Theoretically, the stepwise Bayes character of the posterior allows one to prove the admissibility of the point estimators suggesting that inferential procedures based on this approach will tend to have good frequentist properties. Numerically, we demonstrate in simulations that the proposed stepwise Bayes approach can have substantial strengths compared to traditional methods.     

Combining information from multiple surveys by using regression for efficient small domain estimation

Summary.  In sample surveys of finite populations, subpopulations for which the sample size is too small for estimation of adequate precision are referred to as small domains. Demand for small domain estimates has been growing in recent years among users of survey data. We explore the possibility of enhancing the precision of domain estimators by combining comparable information collected in multiple surveys of the same population. For this, we propose a regression method of estimation that is essentially an extended calibration procedure whereby comparable domain estimates from the various surveys are calibrated to each other. We show through analytic results and an empirical study that this method may greatly improve the precision of domain estimators for the variables that are common to these surveys, as these estimators make effective use of increased sample size for the common survey items. The design-based direct estimators proposed involve only domain-specific data on the variables of interest. This is in contrast with small domain (mostly small area) indirect estimators, based on a single survey, which incorporate through modelling data that are external to the targeted small domains. The approach proposed is also highly effective in handling the closely related problem of estimation for rare population characteristics.     

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.     

A nonparametric Bayesian model for inference in related longitudinal studies

Summary.  We discuss a method for combining different but related longitudinal studies to improve predictive precision. The motivation is to borrow strength across clinical studies in which the same measurements are collected at different frequencies. Key features of the data are heterogeneous populations and an unbalanced design across three studies of interest. The first two studies are phase I studies with very detailed observations on a relatively small number of patients. The third study is a large phase III study with over 1500 enrolled patients, but with relatively few measurements on each patient. Patients receive different doses of several drugs in the studies, with the phase III study containing significantly less toxic treatments. Thus, the main challenges for the analysis are to accommodate heterogeneous population distributions and to formalize borrowing strength across the studies and across the various treatment levels. We describe a hierarchical extension over suitable semiparametric longitudinal data models to achieve the inferential goal. A nonparametric random-effects model accommodates the heterogeneity of the population of patients. A hierarchical extension allows borrowing strength across different studies and different levels of treatment by introducing dependence across these nonparametric random-effects distributions. Dependence is introduced by building an analysis of variance (ANOVA) like structure over the random-effects distributions for different studies and treatment combinations. Model structure and parameter interpretation are similar to standard ANOVA models. Instead of the unknown normal means as in standard ANOVA models, however, the basic objects of inference are random distributions, namely the unknown population distributions under each study. The analysis is based on a mixture of Dirichlet processes model as the underlying semiparametric model.     

Penalized Independence Rule for Testing High-Dimensional Hypotheses

We consider a challenging problem of testing any possible association between a response variable and a set of predictors, when the dimensionality of predictors is much greater than the number of observations. In the context of generalized linear models, a new approach is proposed for testing against high-dimensional alternatives. Our method uses soft-thresholding to suppress stochastic noise and applies the independence rule to borrow strength across the predictors. Moreover, the method can provide a ranked predictor list and automatically select “important” features to retain in the test statistic. We compare the performance of this method with some competing approaches via real data and simulation studies, demonstrating that our method maintains relatively higher power against a wide family of alternatives.     

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.     

Inferences from logistic regression models in the presence of small samples,rare events,nonlinearity, and multicollinearity with observational data

The logistic regression model has been widely used in the social and natural sciences and results from studies using this model can have significant policy impacts. Thus, confidence in the reliability of inferences drawn from these models is essential. The robustness of such inferences is dependent on sample size. The purpose of this article is to examine the impact of alternative data sets on the mean estimated bias and efficiency of parameter estimation and inference for the logistic regression model with observational data. A number of simulations are conducted examining the impact of sample size, nonlinear predictors, and multicollinearity on substantive inferences (e.g. odds ratios, marginal effects) when using logistic regression models. Findings suggest that small sample size can negatively affect the quality of parameter estimates and inferences in the presence of rare events, multicollinearity, and nonlinear predictor functions, but marginal effects estimates are relatively more robust to sample size.     

Spatial generalized linear mixed models in small area estimation

In survey sampling, policy decisions regarding the allocation of resources to sub‐groups of a population depend on reliable predictors of their underlying parameters. However, in some sub‐groups, called small areas due to small sample sizes relative to the population, the information needed for reliable estimation is typically not available. Consequently, data on a coarser scale are used to predict the characteristics of small areas. Mixed models are the primary tools in small area estimation (SAE) and also borrow information from alternative sources (e.g., previous surveys and administrative and census data sets). In many circumstances, small area predictors are associated with location. For instance, in the case of chronic disease or cancer, it is important for policy makers to understand spatial patterns of disease in order to determine small areas with high risk of disease and establish prevention strategies. The literature considering SAE with spatial random effects is sparse and mostly in the context of spatial linear mixed models. In this article, small area models are proposed for the class of spatial generalized linear mixed models to obtain small area predictors and corresponding second‐order unbiased mean squared prediction errors via Taylor expansion and a parametric bootstrap approach. The performance of the proposed approach is evaluated through simulation studies and application of the models to a real esophageal cancer data set from Minnesota, U.S.A. The Canadian Journal of Statistics 47: 426–437; 2019 © 2019 Statistical Society of Canada     

Small area estimation of mean price of habitation transaction using time-series and cross-sectional area-level models

In this paper, a new small domain estimator for area-level data is proposed. The proposed estimator is driven by a real problem of estimating the mean price of habitation transaction at a regional level in a European country, using data collected from a longitudinal survey conducted by a national statistical office. At the desired level of inference, it is not possible to provide accurate direct estimates because the sample sizes in these domains are very small. An area-level model with a heterogeneous covariance structure of random effects assists the proposed combined estimator. This model is an extension of a model due to Fay and Herriot [5], but it integrates information across domains and over several periods of time. In addition, a modified method of estimation of variance components for time-series and cross-sectional area-level models is proposed by including the design weights. A Monte Carlo simulation, based on real data, is conducted to investigate the performance of the proposed estimators in comparison with other estimators frequently used in small area estimation problems. In particular, we compare the performance of these estimators with the estimator based on the Rao–Yu model [23]. The simulation study also accesses the performance of the modified variance component estimators in comparison with the traditional ANOVA method. Simulation results show that the estimators proposed perform better than the other estimators in terms of both precision and bias.     

On prediction of population and subpopulation characteristics for future periods

The estimation or prediction of population characteristics based on the sample information is the key issue in survey sampling. If the sample sizes in subpopulations (domains) are large enough, similar methods as used for the whole population can be used to estimate or to predict subpopulations characteristics as well. To estimate or to predict characteristics of domains with small or even zero sample sizes, small area estimation methods “borrowing strength” from other subpopulations or time periods are widely used. We extend this problem and study methods of prediction of future population and subpopulations’ characteristics based on the longitudinal data.     

A BLUP Synthetic Versus an EBLUP Estimator: An Empirical Study of a Small Area Estimation Problem

Model-based estimators are becoming very popular in statistical offices because Governments require accurate estimates for small domains that were not planned when the study was designed, as their inclusion would have produced an increase in the cost of the study. The sample sizes in these domains are very small or even zero; consequently, traditional direct design-based estimators lead to unacceptably large standard errors. In this regard, model-based estimators that 'borrow information' from related areas by using auxiliary information are appropriate. This paper reviews, under the model-based approach, a BLUP synthetic and an EBLUP estimator. The goal is to obtain estimators of domain totals when there are several domains with very small sample sizes or without sampled units. We also provide detailed expressions of the mean squared error at different levels of aggregation. The results are illustrated with real data from the Basque Country Business Survey.     

Prediction of response values in linear regression models from replicated experiments

This paper considers the problem of prediction in a linear regression model when data sets are available from replicated experiments. Pooling the data sets for the estimation of regression parameters, we present three predictors — one arising from the least squares method and two stemming from Stein-rule method. Efficiency properties of these predictors are discussed when they are used to predict actual and average values of response variable within/outside the sample. Received: November 17, 1999; revised version: August 10, 2000     

Prediction under uncertainty of degree of polynomial in growth curve models

Random coefficient polynomial regression model has been considered for prediction purpose when there is uncertainty about the degree of the polynomialo Expressions for mean square errors of two predictors based on simple estimators have been derived and their perfomaiices have been compared when parameters are estimated from the sample. A modified predictor has also been suggested when parameters in the predicting equations are to be estimated from the sample. Perform-ance ofseveral predictors haife been compared by cross validation technique from a real set of data.     

Bayesian Random Segmentation Models to Identify Shared Copy Number Aberrations for Array CGH Data

Array-based comparative genomic hybridization (aCGH) is a high-resolution high-throughput technique for studying the genetic basis of cancer. The resulting data consists of log fluorescence ratios as a function of the genomic DNA location and provides a cytogenetic representation of the relative DNA copy number variation. Analysis of such data typically involves estimation of the underlying copy number state at each location and segmenting regions of DNA with similar copy number states. Most current methods proceed by modeling a single sample/array at a time, and thus fail to borrow strength across multiple samples to infer shared regions of copy number aberrations. We propose a hierarchical Bayesian random segmentation approach for modeling aCGH data that utilizes information across arrays from a common population to yield segments of shared copy number changes. These changes characterize the underlying population and allow us to compare different population aCGH profiles to assess which regions of the genome have differential alterations. Our method, referred to as BDSAcgh (Bayesian Detection of Shared Aberrations in aCGH), is based on a unified Bayesian hierarchical model that allows us to obtain probabilities of alteration states as well as probabilities of differential alteration that correspond to local false discovery rates. We evaluate the operating characteristics of our method via simulations and an application using a lung cancer aCGH data set.     

A simulation study of estimators for generalized linear measurement error models

The purpose of this paper is to examine the properties of several bias-corrected estimators for generalized linear measurement error models, along with the naive estimator, in some special settings. In particular, we consider logistic regression, poisson regression and exponential-gamma models where the covariates are subject to measurement error. Monte Carlo experiments are conducted to compare the relative performance of the estimators in terms of several criteria. The results indicate that the naive estimator of slope is biased towards zero by a factor increasing with the magnitude of slope and measurement error as well as the sample size. It is found that none of the biased-corrected estimators always outperforms the others, and that their small sample properties typically depend on the underlying model assumptions.     

Small domain estimation by empirical bayes and kalman filtering procedures - a case study

An application of empirical Bayes and Kalman filtering tecniques is reported, using live data from Indian Statistical Institute (ISI), Calcutta . to illustrate how initial small domain estimators may be vastly improved upon. A stratified two stage sampling procedure is adopted, allowing selection of first stage units with unequal probabilities but of second stage units with equal probabilities. Standard design-based estimators for domain totals are initialized based on domain specific survey data alone. Strength is then borrowed across domains and from past surveys. The resulting gains in efficacy are numlerically demonstrated, through replicated sampling from official records.     

Cure Rate Model with Measurement Error

This simulation study focuses on the relative small sample properties of some widely applied predictors in regression with AR(1) errors where there errors are allowed to follow normal and non-normal distributions. The conclusions are: all predictors considered are significantly unbiased; the relative performances of predictors, from the efficiency point of view, seemed insensitive to the nature of the error distribution; and the standard errors of predictors computed from the asymptotic formulas are very useful for purposes of inference in small sample and under all assumed distributions.     

多源高维数据的多分类纵向整合分析及应用

多分类数据分析在实证研究中具有重要意义。然而,由于高维数、小样本及低信噪比等原因,现有的多分类方法仍面临信息量不足而导致的效果不佳问题。为此,学者们通过收集更多信息源 数据以更全面地刻画实际问题。不同于收集相同自变量的不同源样本,目前较为流行的多源数据收集了相同样本的不同源自变量,它们的独立性和相关性为统计建模带来了新的挑战。本文提出基于典型变量回归的多分类纵向整合分析方法,其中利用惩罚技术实现变量选择,并独特地考虑不同源数据间的关联结构,提出高效的ADMM算法进行模型优化。数值模拟结果表明,该方法在变量选择和分类预测 上均具有优越性。基于我国上证50的多源股票数据,利用该方法对2019年股票日收益率的影响因素进行了实证探究。研究表明,本文提出的多分类整合分析在筛选出具有解释意义变量的同时具有更好的预测效果。     

A two-stage procedure to pool information across quantile levels in linear quantile regression

In linear quantile regression, the regression coefficients for different quantiles are typically estimated separately. Efforts to improve the efficiency of estimators are often based on assumptions of commonality among the slope coefficients. We propose instead a two-stage procedure whereby the regression coefficients are first estimated separately and then smoothed over quantile level. Due to the strong correlation between coefficient estimates at nearby quantile levels, existing bandwidth selectors will pick bandwidths that are too small. To remedy this, we use 10-fold cross-validation to determine a common bandwidth inflation factor for smoothing the intercept as well as slope estimates. Simulation results suggest that the proposed method is effective in pooling information across quantile levels, resulting in estimates that are typically more efficient than the separately obtained estimates and the interquantile shrinkage estimates derived using a fused penalty function. The usefulness of the proposed method is demonstrated in a real data example.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.