Balgobin Nandram

This paper compares two Bayesian small area estimation models, a weighted unit-level model and the Fay–Herriot model, under probability sampling. Posterior inference uses the multiplication rule, with noninformative priors for the unit-level model and a proper shrinkage prior for the Fay–Herriot model. The methodology is illustrated with body mass index data for eight California counties from the National Health and Nutrition Examination Survey and evaluated in two simulation studies without clustering or stratification. The two models produce similar posterior means, but the Fay–Herriot model exhibits larger posterior standard deviations, and the unit-level results are sensitive to the normalization of the survey weights. In the first simulation study, the unit-level credible intervals fall below nominal coverage, which a diagnostic attributes to using the weights as variance scalings when the errors have constant variance, whereas the Fay–Herriot intervals attain or exceed it. The second study, a calibrated Bayes check, shows approximately nominal coverage for both models under their prior predictive distributions, which does not imply nominal coverage at fixed parameter values. These results indicate that smaller posterior uncertainty is not by itself an advantage: while the Fay–Herriot model remains attractive when only area-level summaries are available, posterior uncertainty and interval coverage should be considered together, and the conclusions are conditional on the weighting and variance specifications used.

Non-probability samples (NPS) are increasingly used in practice because they are relatively inexpensive and often contain the study outcome variable. However, NPS may suffer from selection bias and usually do not provide survey weights, making finite-population inference difficult. Probability samples (PS), on the other hand, provide survey weights under known sampling designs but may not contain the study outcome variable. This paper develops a Bayesian approach for integrating probability and non-probability samples under a superpopulation model. The proposed method jointly models latent survey weights and missing probability-sample outcomes within a hierarchical Bayesian framework. The method is compared with a naive Bayesian bootstrap estimator and a weighted regression estimator. A BMI data application and simulation studies show that the proposed method reduces bias relative to the naive estimator while accounting for uncertainty in latent weights and missing outcomes.

1. There are numerous application problems which require fitting models for continuous and positively skewed (CPS) size data from small areas such as consumption, income, insurance claims, stock re...

We consider the problem of integrating a small probability sample (ps) and a non-probability sample (nps). By definition, for the nps, there are no survey weights, but for the ps, there are survey weights. The key issue is that the nps, although much larger than the ps, can lead to a biased estimator of a finite population quantity but with much smaller variance. We begin with a relatively simple problem in which the population is assumed to be homogeneous and there are no common units in the ps and the nps. We assume that there are covariates and responses for everyone in the two samples, and there are no covariates available for the nonsampled units. We use the nps (ps) to construct a prior for the ps (nps). We also introduce partial discounting to avoid a dominance of the prior. We use Bayesian predictive inference for the finite population mean. In our illustrative example on body mass index and our simulation study, we compare the relative performance of alternative procedures and demonstrate that our procedure leads to improved estimates over the ps only estimate.

Many population-based surveys have binary responses from a large number of individuals in each household within small areas. One example is the Nepal Living Standards Survey (NLSS II), in which health status binary data (good versus poor) for each individual from sampled households (sub-areas) are available in the sampled wards (small areas). To make an inference for the finite population proportion of individuals in each household, we use the sub-area logistic regression model with reliable auxiliary information. The contribution of this model is twofold. First, we extend an area-level model to a sub-area level model. Second, because there are numerous sub-areas, standard Markov chain Monte Carlo (MCMC) methods to find the joint posterior density are very time-consuming. Therefore, we provide a sampling-based method, the integrated nested normal approximation (INNA), which permits fast computation. Our main goal is to describe this hierarchical Bayesian logistic regression model and to show that the computation is much faster than the exact MCMC method and also reasonably accurate. The performance of our method is studied by using NLSS II data. Our model can borrow strength from both areas and sub-areas to obtain more efficient and precise estimates. The hierarchical structure of our model captures the variation in the binary data reasonably well.

Sample surveys play a significant role in obtaining reliable estimators of finite population quantities, and survey weights are used to deal with selection bias and non-response bias. The main idea of this research is to compare the performance of nine methods with differently constructed survey weights, and we can use these methods for non-probability sampling after weights are estimated (e.g. quasi-randomization). The original survey weights are calibrated to the population size. In particular, the base model does not include survey weights or design weights. We use original survey weights, adjusted survey weights, trimmed survey weights, and adjusted trimmed survey weights into pseudo-likelihood function to build unnormalized or normalized posterior distributions. In this research, we focus on binary data, which occur in many different situations. A simulation study is performed and we analyze the simulated data using average posterior mean, average posterior standard deviation, average relative bias, average posterior root mean squared error, and the coverage rate of 95% credible intervals. We also performed an application on body mass index to further understand these nine methods. The results show that methods with trimmed weights are preferred than methods with untrimmed weights, and methods with adjusted weights have higher variability than methods with unadjusted weights.

There are many models in the current statistical literature for making inferences based on samples selected from a finite population. Parametric models may be problematic because statistical inference is sensitive to parametric assumptions. The Dirichlet process (DP) prior is very flexible and determines the complexity of the model. It is indexed by two hyper-parameters: the baseline distribution and concentration parameter. We address two distinct problems in the article. Firstly, we review the current sampling methods for the concentration parameter, which use the continuous baseline distribution. We compare three different methods: the adaptive rejection method, the mixture of Gammas method and the grid method. We also propose a new method based on the ratio of uniforms. Secondly, in practice, some survey responses are known to be discrete. If a continuous distribution is adopted as the baseline distribution, the model is misspecified and standard inference may be invalid. We propose a discrete baseline approach to the DP prior and sample the unobserved responses from the finite population both using a Polya urn scheme and a Multinomial distribution. We applied our discrete baseline approach to a Phytophthora data set.

The use of hierarchical Bayesian small area models, which take survey estimates along with auxiliary data as input to produce official statistics, has increased in recent years. Survey estimates for small domains are usually unreliable due to small sample sizes, and the corresponding sampling variances can also be imprecise and unreliable. This affects the performance of the model (i.e., the model will not produce an estimate or will produce a low-quality modeled estimate), which results in a reduced number of official statistics published by a government agency. To mitigate the unreliable sampling variances, these survey-estimated variances are typically modeled against the direct estimates wherever a relationship between the two is present. However, this is not always the case. This paper explores different alternatives to mitigate the unreliable (beyond some threshold) sampling variances. A Bayesian approach under the area-level model set-up and a distribution-free technique based on bootstrap sampling are proposed to update the survey data. An application to the county-level corn yield data from the County Agricultural Production Survey of the United States Department of Agriculture’s (USDA’s) National Agricultural Statistics Service (NASS) is used to illustrate the proposed approaches. The final county-level model-based estimates for small area domains, produced based on updated survey data from each method, are compared with county-level model-based estimates produced based on the original survey data and the official statistics published in 2016.

In the production of US agricultural official statistics, certain inequality and benchmarking constraints must be satisfied. For example, available administrative data provide an accurate lower bound for the county-level estimates of planted acres, produced by the U.S. Department of Agriculture’s (USDA) National Agricultural statistics Services (NASS). In addition, the county-level estimates within a state need to add to the state-level estimates. A sub-area hierarchical Bayesian model with inequality constraints to produce county-level estimates that satisfy these important relationships is discussed, along with associated measures of uncertainty. This model combines the County Agricultural Production Survey (CAPS) data with administrative data. Inequality constraints add complexity to fitting the model and present a computational challenge to a full Bayesian approach. To evaluate the inclusion of these constraints, the models with and without inequality constraints were compared using 2014 corn planted acres estimates for three states. The performance of the model with inequality constraints illustrates the improvement of county-level estimates in accuracy and precision while preserving required relationships.