Bayesian predictive inference of small area proportions under selection bias
Section 1. Introduction

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In many complex sample surveys, individuals are sampled with differential selection probabilities. For binary responses, if the proportion of positive responses among the sampled individuals differs substantially from those among the nonsampled individuals, there is a selection bias. In some cases a selection bias is obtained by design (e.g., probability proportional to size (PPS) sampling) and we can take care of the selection bias, but in other problems, this is not the case. For example, in non-probability samples the selection probabilities are unknown. In our problem, we assume that the survey weights or selection probabilities can help us to understand the selection bias. We want to make inference about the finite population proportions of the small areas when a possibly biased sample is available from each area. Choi, Nandram and Kim (2017), henceforth CNK, extended the model of Nandram, Bhatta, Bhadra and Shen (2013), henceforth NBBS, who studied a single area, to accommodate inference about small areas. While the CNK model assumes that distribution of the selection probabilities is the same across areas, our new contribution is to assume that the distributions of the selection probabilities over areas are different, but they share an effect.

There are two types of models that can be considered when making inference about small areas. First, we can use an ignorable selection model in which the response variable is not related to the selection probabilities. Such a model will not adjust for the selection bias, and will produce biased estimates if there are no other covariates to act like the selection probabilities. We assume that the only available information is the sampled responses, sampled selection probabilities and area identifiers. Second, in a nonignorable selection model, the response variables are related to the selection probabilities. In our study on binary responses, the distribution of the selection probabilities for the “yes” responses is different from that for the “no” responses, thereby making the response variables and selection probabilities correlated. Official statistics are obtained from many complex surveys, and these surveys are conveniently designed with selection bias (PPS sampling is usually a part of a complex survey design). This study is important in the construction of official statistics from complex surveys.

It is pertinent to give a brief discussion on recent developments. For continuous responses, Pfeffermann (1988), Pfeffermann (1993), Sverchkov and Pfeffermann (2004) and Pfeffermann, Krieger and Rinott (1998) specified a relation between the survey weights and the response variable. Chambers, Dorfman and Wang (1998) assumed that the selection probabilities are related to the continuous responses. A related study within the Bayesian paradigm is given by Ma, Sedransk, Nandram and Chen (2018). There is also an application of this method to calculate the total gas consumption in the US using a PPS sample; see Nandram, Choi, Shen and Burgos (2006). Chen, Elliott and Little (2010) used penalized spline to make a Bayesian predictive inference for PPS sampling. However, because these approaches require some information about the nonsampled selection probabilities, they are difficult to use. The nonsampled selection probabilities are not available to secondary data analysts and we continue to assume that these nonsampled selection probabilities are not available in our work. Pfeffermann and Sverchkov (2007) extended the work of Sverchkov and Pfeffermann (2004) to accommodate small areas with informative sampling of areas and within selected areas. Like Chen et al. (2010), Opsomer, Glaeskens, Ranalli, Kauermann and Breidt (2008) used a penalized spline regression to construct a small-area model, which includes the selection probabilities, in a non-parametric manner. While all these works are for continuous response in small areas, we analyze binary data in this paper.

Our approach to selection bias problems has been to adjust the sample part of a population model. That is, a model is constructed for the entire population, the model is then adjusted to accommodate the sample with the selection probabilities, and the superpopulation parameters are then estimated. Once this is done for the sample, prediction is done for the entire finite population. This approach to selection bias was described nicely by Malec, Davis and Cao (1999). NBBS reviewed and provided a full Bayesian analyis to the method of Malec et al. (1999); Nandram (2007) provided a surrogate sampling procedure within the spirit of Malec et al. (1999). There are many extensions to this approach to accommodate small areas. For example, to analyze continuous responses, Nandram and Choi (2010) included the selection probabilities into a full Bayesian nonignorable model in order to analyze continuous body mass index data, which were discretized in an elegant manner. Malec et al. (1999) used a hierarchical Bayesian model to accommodate a selection mechanism for binary data. As stated by NBBS, there are two possible problems with this model, “First, the ${\theta }_{uy}$ are only weakly identified. Second, the parameters ${\theta }_{uy}$ are never known, and in a Bayesian framework these must also be stochastic. In this paper, in a single attempt, we show how to solve these two problems (weak identifiability and stochastic parameters) for a biased sample drawn from a binary population using information from the survey weights (or selection probabilities)”. Using a nonignorable selection model, NBBS showed how to correct for these two problems for an individual-area model.

Recently, there has been some interesting Bayesian activities for PPS sampling with continuous responses. Zangeneh and Little (2015) used a Bayesian bootstrap to model the measures of size (related to the selection probabilities) and a spline regression of the responses conditional on the measures of size. For the same problem, Si, Pillai and Gelman (2015) poststratified the selection probabilities and used a Gaussian process to model the responses. These are good approaches to Bayesian analyzes of PPS sampling and can be applied to our problem. However, our approach is different because we do not model the nonsampled selection probabilities, rather we adjust a population model.

CNK studied a special case for an extension of NBBS model within the small area context. They assumed that the sample selection probabilities have the same support over the set ${\pi }_u^{*}$ over areas, and the distribution of the selection probabilities given the binary response $y_{ij}$ is

$$P\left({\pi }_{ij}\mathrm{<mo>=</mo>}{\pi }_u^{*}|\theta ,\hspace{0.17em}{y}_{ij}\mathrm{<mo>=</mo>}y\right)\mathrm{<mo>=</mo>}{\theta }_{uy}\mathrm{,}u\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,U\mathrm{,}y\mathrm{<mo>=</mo>}\mathrm{0,}\mathrm{1,}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,n_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots ,l\mathrm{.}$$

This model assumes that the sample selection probabilities have the same support and each area has the same distribution for the selection probabilities; homogeneous nonignorable selection model. However, the homogeneity assumption is strong and might not be true for most of real settings. In practice, the distribution of the sample selection probabilities can be very different across domains due to sampling designs. The distribution of the sample selection probabilities given $y$ can also vary a lot across domains. In this paper, we consider a model under a heterogeneity assumption, which is that the sample selection probabilities have different supports and different distributions over areas; this is the heterogeneous nonignorable selection model.

CNK used a simulation study to show how different a baseline ignorable selection model and the homogeneous nonignorable selection model can be when there is selection bias. But it is well known that design information needs to be included when the sample is not randomly selected. In this paper, we use a simulation study to assess the performance of our heterogeneous nonignorable selection model. We draw data from the homogeneous nonignorable selection model and heterogeneous nonignorable selection model respectively, and fit the three models. Then we compare the performance of the models using several measures.

In this paper, we consider the problem of making inference about the finite population proportions of the small areas when there is likely to be a different selection bias by areas. Specifically we extend the homogeneous nonignorable selection model of CNK to accommodate selection probabilities that have different supports in different areas. In Section 2, we give a review of the ignorable selection model and the homogeneous nonignorable selection model, which were studied previously. In Section 3, we show how to adjust a Bayesian ignorable selection model to incorporate the selection bias in a small area framework when the sample selection probabilities have different distributions by areas. We also describe how to perform the computation. In Section 4, we provide results of an illustrative example on severe activity limitation in the 1995 National Health Interview Survey. We also provide a simulation study to assess the performance of the heterogeneous nonignorable selection model. Section 5 has summary and concluding remarks.

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