Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 1. Introduction

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The term “small area” generally refers to a small geographical area such as a county. It can be described as the sub-population of interest in a large sample survey. Sample survey data certainly can be used to derive reliable estimators of totals and means for large areas or domains. However, using the same survey, sample data for small areas are typically small and likely to yield unacceptably large standard errors (Ghosh and Rao, 1994). Considering the cost and feasibility of conducting new sample survey for small areas, there is a growing demand for reliable small area statistics using the current large sample survey. Pooling information from related areas to find more accurate estimates is key in small area estimation (Rao and Molina, 2015).

With the pooling information feature, Bayesian hierarchical models for small area estimation have lots of potential in small area estimation. It automatically incorporates all sources of uncertainty associated with an inference problem; see, for example, Nandram, Erciulescu and Cruze (2019), Trevisani and Torelli (2007), and You and Rao (2002). In the small area context, multinomial Dirichlet models as one of Bayesian hierarchical models have been widely used for modeling categorical data. Maples (2019) propose a pair of Dirichlet-Multinomial small area models to jointly estimate relevant school-aged child population and poverty. Wang, Berg, Zhu, Sun and Demuth (2018) develop a spatial hierarchical model based on the generalized Dirichlet distribution to construct small area estimators of compositional proportions in several mutually exclusive and exhaustive landcover categories. We focus on extensions of the multinomial Dirichlet model. Recently, there are extensive researches considering constrained inference for small area estimation, for example, Wu, Meyer and Opsomer (2016) and Heck and Davis-Stober (2019). Nandram (1997) provided a clear discussion about a hierarchical Bayesian approach for taste-testing experiment and appropriate methods for the model. To select the best population, he studied three criteria based on the distribution of random variables representing values on a hedonic scale using the simple tree order. Nandram (1998) pooled data from several multinomial populations using a three-stage multinomial Dirichlet model.

In many statistical problems, it is necessary to take into account the order restrictions of the unknown parameters of interest. Based on the characteristic of data, incorporating order restrictions on cell probabilities of count data can improve the accuracy of estimation. Our major task is to assume the same unimodal order restrictions across areas in the multinomial Dirichlet model. A lot of discussion have been done about the multinomial Dirichlet model with order restrictions.

Sedransk, Monahan and Chiu (1985) described a Bayesian method for estimation of finite population parameters in general population surveys. They added order restrictions to the model to capture the unimodal smoothness relationships among cell probabilities $\left(p_1\mathrm{,}\dots \mathrm{,}{p}_J\right),$ such as

$$p_1\le \dots \le p_k\ge p_{k+1}\ge \dots \ge p_J.$$

But their model cannot pooling information among areas and is not intended for small area estimation.

Gelfand, Smith and Lee (1992) provided very-detailed Gibbs sampler structures for Bayesian analysis of constrained parameters. They suggested that a Dirichlet prior should be used for ordered multinomial parameters, such as $p_1\le p_2\le \dots \le p_k\ge \dots \ge p_J.$ They noted that the Gibbs sampler cannot be employed directly when $k$ is unknown and prior $\text{Pr}\left(k\mathrm{<mo>=</mo>}j\right)\mathrm{<mo>=</mo>}{w}_j\mathrm{,}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}K.$ But the marginal posterior for $k$ can be calculated directly, taking the from

$$\text{Pr}\left(k\mathrm{<mo>=</mo>}j|Y\right)\mathrm{<mo>=</mo>}\frac{C\left({\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_J\mathrm{,}j\right)w_j/C\left({\beta }_1+Y_1\mathrm{,}\dots \mathrm{,}{\beta }_J+Y_J\mathrm{,}j\right)}{{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{J}C\left({\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_J\mathrm{,}j\right)w_j}/C\left({\beta }_1+Y_1\mathrm{,}\dots \mathrm{,}{\beta }_J+Y_J\mathrm{,}j\right)}\mathrm{,}$$

where $C(\dots )$ are normalization constant of the Dirichlet distribution with order restrictions. They showed Bayesian inference on order parameters can have higher precision. However, their Dirichlet multinomial model with the ordered parameters does not consider stratification and cannot borrow information among areas either.

Nandram and Sedransk (1995) showed the precision of inference about ${\pi }_{ij},$ the proportion of firms in stratum $i$ belonging to SR class $j,$ can be dramatically increased by using Dirichlet multinomial model with appropriate order restrictions on ${\pi }_{ij},$ within stratum $i,$ $R^{\left({\text{ℓ}}_s\right)}\mathrm{<mo>=</mo>}\left\{{\pi }_{ij}\mathrm{<mo>:</mo>}{\pi }_{i1}\le \dots \le {\pi }_{i{\text{ℓ}}_s}\ge \dots \ge {\pi }_{iJ}\right\}.$ Their order restriction is complicated due to the stratification. They also consider the case where there is uncertainty about the vector of modal positions $L,$ which can take $g$ possible values, ${\text{ℓ}}_1\mathrm{,}{\text{ℓ}}_2\mathrm{,}\dots \mathrm{,}{\text{ℓ}}_g\mathrm{,}g\le J.$ The position probabilities are given below,

$$\text{Pr}\left(L\mathrm{<mo>=</mo>}{\text{ℓ}}_s\right)\mathrm{<mo>=</mo>}{w}_s\mathrm{,}s\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}g\text{,}\text{ where}{w}_s\text{ are specified and}{\displaystyle \sum _{s\mathrm{<mo>=</mo>\; 1}}^g{w}_s\mathrm{<mo>=</mo>}1}\mathrm{.}$$

They directly applied Monte Carlo integration to estimate the posterior $w_s\mathrm{<mo>=</mo>}\text{Pr}\left(L\mathrm{<mo>=</mo>}{\text{ℓ}}_s|n\right).$ Adopting a Bayesian view, they showed that the posterior variances can be dramatically reduced by including order restrictions among ${\pi }_{ij},$ both within and between the strata. However, their model cannot borrow information among strata and their order restriction assumption is totally different from ours.

Nandram, Sedransk and Smith (1997) improved estimation of the age composition of the population of Atlantic cod with the help of order-restricted Bayesian estimation. Their work was inspired by Sedransk, Monahan and Chiu (1985) and Gelfand, Smith and Lee (1992). Let ${\pi }_{ij}$ denote as cell probabilities that a fish belong to a length stratum $i$ and an age class $j.$ To simplify the analysis, the likelihood of $\pi$ is

$$\text{ℓ}\left(\pi |n\right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>}1}^I{\displaystyle \prod _{j\mathrm{<mo>=</mo>}1}^J{\pi }_{ij}^{n_{ij}}}}\mathrm{.}$$

They took independent Dirichlet distributions as prior; that is

$$f\left(\pi |\alpha \right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>}1}^I{\displaystyle \prod _{j\mathrm{<mo>=</mo>}1}^J{\pi }_{ij}^{{\alpha }_{ij}-1}}}\mathrm{,}$$

where ${\alpha }_{ij}>0$ is a fixed quantity, within stratum $i,$ ${\pi }_{i1}\le \dots \le {\pi }_{i{k}_i}\ge \dots \ge {\pi }_{i{u}_i}$ for some $k_i\in Z_i.$

In their Atlantic cod study, let $i\mathrm{<mo>=</mo>}1$ correspond to the stratum with the shortest fish and $j\mathrm{<mo>=</mo>}1$ correspond to the youngest fish. It is expected that as $i$ increases, the relative values of the $\left\{{\pi }_{ij}\mathrm{<mo>:</mo>}j\in Z_i\right\}$ will change. The order restrictions are not just within strata, but also among strata, such as

$${\pi }_{i1}\le \dots \le {\pi }_{it}\ge \dots \ge {\pi }_{iK}\mathrm{,}$$

$${\pi }_{j1}\le \dots \le {\pi }_{j{t}^{*}}\ge \dots \ge {\pi }_{jK}\text{ where}i<j\text{ and}t<t^{*}\mathrm{.}$$

They presented uncertainty about both the locations of the modes and the unimodality itself is included as part of the probabilistic specification, as an extension of their work. They considered the case where there is uncertainty about the vector of modal position $L,$

$$\text{Pr}\left(L\mathrm{<mo>=</mo>}{\text{ℓ}}_s\right)\mathrm{<mo>=</mo>}{w}_s\text{,}s\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}g\mathrm{.}$$

They showed the joint posterior distribution of $\pi$ and $L$ is

$$f\left(\pi \mathrm{,}L\mathrm{<mo>=</mo>}{\text{ℓ}}_s|n\right)\mathrm{<mo>=</mo>}\frac{w_s{C}_{{\text{ℓ}}_s}\left(\alpha \right){\displaystyle {\prod }_{i\mathrm{<mo>=</mo>\; 1}}^I{g}_{n_i}\left({\pi }_i\right)}}{{\displaystyle {\sum }_{s^{\prime }\mathrm{<mo>=</mo>}1}^g{w}_s^{\prime }{C}_{{\text{ℓ}}_s^{\prime }}\left(\alpha \right)}/C_{{\text{ℓ}}_s^{\prime }}\left(n\right)}\mathrm{.}$$

Their order restriction assumption is not the same across strata, which makes their model is different from ours.

Chen and Nandram (2019), which appeared in the Proceedings of the American Statistical Association, proposed a multinomial Dirichlet model with order restrictions. They considered similar unimodal structure within each area. They showed how to use the Gibbs sampler to sample the posterior distribution. A huge improvement for estimating the cell probabilities has been shown in their model application. Chen and Nandram (2021) have an overview for this type of order-restricted problem for small area estimation. Their overview cover model selection, sampling from posterior distribution, model diagnostics.

We notice the same unimodal order restrictions may not hold for some cases. Incorporating uncertainty about the order restrictions may solve the issue, see Nandram, Sedransk, and Smith (1997). In our work, to increase the model flexibility, we add uncertainty to the unimodal order restriction. Areas have similar unimodal order restrictions on parameters of interest, but not the same modal position. Our order restrictions occur within areas, not across them and the restriction may not be similar across area. They create a difficult problem that will be discussed in the paper.

The article is organized as follows. In Section 2, we present a brief review of the multinomial Dirichlet model and the multinomial Dirichlet model with order restrictions. In Section 3, we incorporate uncertainty about order restriction into the model. We present the estimation method and show how to use the conditional predictive ordinate as Bayesian diagnostics. In Section 4, for illustrative purpose, we show how to analyze the body mass index (BMI) data using the model incorporating uncertainty about order. We demonstrate how much improvement there is under the order restrictions. In Section 5, we also demonstrate that incorporating uncertainty about order restrictions to the model can improve the robustness of the model. Section 6 has a summary and the future work.

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