Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 6. Concluding remarks

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The Dirichlet multinomial model with mixed order restrictions is an extension of $M_2.$ It increases the robustness and flexibility due to its uncertainty. We have also shown how to acquire samples of the model with mixed order restriction. In our application and simulation, we find that, with the uncertainty, the Dirichlet multinomial model with mixed order restrictions may be the best model for all cases with varied unknown unimodality. For most cases, we could not know the unimodal order restriction, even if we believe it exists. Bringing uncertainty to the model is necessary. We also notice that due to its complexity, it is hard to compute its marginal likelihood. We show a method to estimate the posterior probabilities of the mode location, which is $P\left(L_{\text{pos}}\mathrm{<mo>=</mo>}\text{ℓ}|n\right).$ But there is a precision-efficiency tradeoff.

However, as shown in Figure 4.2 and Figure 4.3, the same unimodal order restriction for all counties may be still strong even with uncertainty. Some counties have more people in the normal BMI level, and some counties have more people in the overweight BMI level. Nandram and Sedransk (1995) and Nandram, Sedransk and Smith (1997) presented a good discussion about unimodal order restriction in a stratified population. With the help of uncertainty, they made inference about the proportion of firms and fish belonging to each of several classes when there are unimodal order relations among the proportions. In that paper, the hyperparameters are specified and they did not have a small area estimation problem; our problem is much more difficult even we consider a similar uncertainty model structure.

In Section 4.2.2, the model with fixed order restrictions is a better model for BMI data because of its largest LPML. But without any background, assuming the modal position is risky and may cause the wrong inference. The multinomial Dirichlet model with order restrictions, incorporating uncertainty, can reduce the risk and is more robust. In the simulation, Model $M_2$ is the best model for the simulated BMI data. Model $M_4$ shows a better consistency for the simulated BMI data and the real BMI data.

The final BMI data set for this study uses only the 35 largest counties with a population of at least 500,000 for selected age categories by sex (male, female) and race (white non-Hispanic, black non-Hispanic, Hispanic, other). We can easily apply our method to the small domains formed by on race, age and sex, such as the male-Hispanic BMI data. But the cells of the multinomial tables will become sparse. We can eliminate some counties that become small or we can combine some counties. However, due to the structures of multinomial-Dirichlet models with order restrictions, we cannot add race, age and sex as covariates into the model.

Since the BMI data are from the survey sampling and individuals are selected with different probabilities, we should not ignore the survey weights. It is possible to incorporate the survey weights into our model as well. Let $W_{ig}$ denote the survey weights, adding up to the population size within each county, $i=\mathrm{1,}\dots \mathrm{,}\text{ℓ},$ sample index $g\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{n}_i$ and cell index $j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}K.$ Yang (2021) provided adjusted weights are

$${\omega }_{ig}\mathrm{<mo>=</mo>}{n}_i\frac{W_{ig}}{{\displaystyle {\sum }_{g=1}^{n_i}{W}_{ig}}},$$

and ${\displaystyle {\sum }_{g\mathrm{<mo>=</mo>}1}^{n_i}{\omega }_{ig}}\mathrm{<mo>=</mo>}{n}_{i\mathrm{.}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^K{n}_{ij}}.$ Yang (2021) used weighted likelihood distributions for a single multinomial model, see also Nandram, Choi and Liu (2021). Yang (2021) found out there is a very small difference between normalized and unnormalized weighed likelihood.

We can transform BMI data using the adjusted weights into adjusted counts. Let $I_{igj}$ be the BMI category indicator for individual $g$ in county $i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}\text{ℓ}$ at cell $j\mathrm{,}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}K.$ We define $I_{igj}\mathrm{<mo>=</mo>}0$ or 1 with ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>}1}^K{I}_{igj}}=1,$ for example, if a person responds in cell $j,$ a one is scored and all other cells have zeros. For simplification, we can have the unnormalized weighted joint posterior distribution as

Our approaches can be applied to the adjusted counts directly.

It is possible to relax the unimodal order restriction somewhat. One can restrict the position of the mode without any ordering on its left or right, we can still have the mode at 2 or 3 for the BMI data to provide a model with uncertainty about the modal position. This can be done in the same spirit as in our current work.

We notice the same unimodal structure across all counties is not satisfied. Borrowing information across those areas may have a negative effect to model inference. Neuenschwander, Wandel, Roychoudhury and Bailey (2016) presented a different approach to increase the model robustness in drug development. They proposed the exchangeability nonexchangeability (EXNEX) approach to reduce the risk of too much shrinkage and excessive borrowing for extreme strata. We can borrow their approach to increase our model robustness. But we believe it is very difficult to make inference using the Dirichlet multinomial model with EXNEX prior because the model complexity increases significantly.

Appendix

A.1  Gibbs sampler for $\mu$ and $\tau$ in $M_2$ and $M_3$

We present griddy Gibbs sampler, a Markov chain Monte Carlo (MCMC) algorithm, for $\mu$ with the order restriction and $\tau .$

Liu and Sabatti (2000) presented a comprehensive discussion of the general Gibbs sampler which is more efficient Markov chain Monte Carlo method for Bayesian inference. They explored its connection with the multigrid Monte Carlo method and its use in designing more efficient samplers. Gibbs sampler may be more efficient in our hierarchical model. Therefore we use Gibbs sampler to generate the posterior samples for the Bayesian inference.

We present the modified Gibbs sampler for $\mu \in C_{\mu }$ and $\tau .$ The joint posterior density is

$$\pi \left(\theta \mathrm{,}\mu \mathrm{,}\tau |n\right)\propto {\displaystyle \prod _{i\mathrm{<mo>=</mo>\; 1}}^I\left\{\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_j\tau -1}{I}_C{I}_{C_{\mu }}}}{D\left(\mu \tau \right)C\left(\mu \tau \right)}\right\}\frac{1_{}}{{\left(1+\tau \right)}^2},}$$

where

$$C\left(\mu \tau \right)\mathrm{<mo>=</mo>}{\displaystyle {\int }_{{\theta }_i\in C}\frac{\Gamma \left({\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^K{\mu }_j\tau }\right)}{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K\Gamma \left({\mu }_j\tau \right)}}{\displaystyle \prod _{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{{\mu }_j\tau -1}d{\theta }_i}}\mathrm{.}$$

There is no recognizable conditional distribution of $\mu$ and $\tau$ to generate samples. So we use grid method to draw $\mu$ and $\tau$ from $\pi \left(\mu \mathrm{,}\tau |n\right)$ after integrating with respect to $\theta ,$ we get

Chen and Shao (1997) mentioned that importance sampling could be used to estimate the ratio,

$$\frac{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{{\mu }_j\tau +n_{ij}-1}d{\theta }_i}}}{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{{\mu }_j\tau -1}d{\theta }_i}}}.$$

We consider Dirichlet $\left(r{\overline{n}}_j\right)$ as our importance of all counties function, where $r$ is an adjustable ratio and

$${\overline{n}}_j\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^I{n}_{ij}}}{I}.$$

It combines information together. Since our importance function does not depend on the unknown $\mu$ and $\tau ,$ we can generate one set of numbers for all iterations. In our numerical example, it has been proved as an efficient way to generate posterior samples.

Gibbs sampler steps:

1.    Draw $\tau$ from $\pi \left(\tau |\mu ,n\right);$
2.    For $j$ from $m-1$ to 1, draw ${\mu }_j$ from $\pi \left({\mu }_j|{\mu }^{\left(-j\right)}\mathrm{,}\tau \mathrm{,}n\right),$ where

$$0<{\mu }_j<\min \left\{{\mu }_{j+1}\mathrm{,}\frac{1-{\displaystyle {\sum }_{t\mathrm{<mo>=</mo>\; 1,}t\ne m\mathrm{,}t\ne j}^K{\mu }_t}}{2}\right\};$$

3.    For $j$ from $m+1$ to $K,$ draw ${\mu }_j$ from $\pi \left({\mu }_j|{\mu }^{\left(-j\right)}\mathrm{,}\tau \mathrm{,}n\right),$ where

$$0<{\mu }_j<\min \left\{{\mu }_{j-1}\mathrm{,}\frac{1-{\displaystyle {\sum }_{t\mathrm{<mo>=</mo>\; 1,}t\ne m\mathrm{,}t\ne j}^K{\mu }_t}}{2}\right\};$$

4.    Get ${\mu }_m\mathrm{<mo>=</mo>}1-{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1,}j\ne m}^K{\mu }_j},$ repeat Step 1 to Step 4 until convergence,

$${\mu }^{\left(-j\right)}=\left({\mu }_1\mathrm{,}\dots \mathrm{,}{\mu }_{j-1}\mathrm{,}{\mu }_{j+1}\mathrm{,}\dots \mathrm{,}{\mu }_K\right).$$

A.2   Sampling $\theta$ in $M_2$ and $M_3$

The posterior of $\theta$ has a recognizable distribution, which is the Dirichlet distribution with the order restriction. Instead of drawing samples directly from the Dirichlet distribution with the order restriction, Chen and Nandram (2019) present a direct sampling from truncated Gamma distributions, where Nadarajah and Kotz (2006) offered a method for truncated Gamma.

Denote $\beta \mathrm{<mo>=</mo>}\left({\beta }_1\mathrm{,}\dots \mathrm{,}{\beta }_K\right),$ if $0\le {\theta }_1\le {\theta }_2\le \dots \le {\theta }_m\ge \dots \ge {\theta }_K$ and the mode is ${\theta }_m,$ then we assume $0\le {\beta }_1\le {\beta }_2\le \dots \le {\beta }_m\ge \dots \ge {\beta }_K,$ the mode is ${\beta }_m.$

Steps of sampling $\theta$ from Dirichlet $\mathrm{<mo\; stretchy="false">(</mo>}{\alpha }_1\mathrm{,}\dots \mathrm{,}{\alpha }_K\mathrm{<mo\; stretchy="false">)</mo>}:$

1.    Draw ${\beta }_m~\text{Gamma}\left({\alpha }_m\mathrm{,}1\right),$ where $0\le {\beta }_m<\infty ;$
2.    Draw from ${\beta }_{m-1}$ to ${\beta }_1,$

${\beta }_{m-1}~\text{Truncated Gamma}\left({\alpha }_{m-1}\mathrm{,}1\right),\text{where}0\le {\beta }_{m-1}\le {\beta }_m,$

$\dots$

${\beta }_1~\text{Truncated Gamma}\left({\alpha }_1\mathrm{,}1\right),\text{where}0\le {\beta }_1\le {\beta }_2;$

3.    Draw from ${\beta }_{m+1}$ to ${\beta }_K,$

${\beta }_{m+1}~\text{Truncated Gamma}\left({\alpha }_{m+1}\mathrm{,}1\right),\text{where}0\le {\beta }_{m+1}\le {\beta }_m,$

$\dots$

${\beta }_K~\text{Truncated Gamma}\left({\alpha }_K\mathrm{,}1\right),\text{where}0\le {\beta }_K\le {\beta }_{K-1}.$

Then,

$${\theta }_1\mathrm{<mo>=</mo>}\frac{{\beta }_1}{{\beta }_1+{\beta }_2+\dots +{\beta }_K}\mathrm{,}\dots \mathrm{,}{\theta }_{K-1}\mathrm{<mo>=</mo>}\frac{{\beta }_{K-1}}{{\beta }_1+{\beta }_2+\dots +{\beta }_K}\mathrm{,}{\theta }_K\mathrm{<mo>=</mo>}1-{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^{K-1}{\theta }_i}\mathrm{.}$$

A.3   Bayesian diagnostics of $M_2,$ $M_3,$ and $M_4$

Since the only difference between $M_2$ and $M_3$ is the order restriction assumption and the CPOs of $M_2$ and $M_3$ are similar, we only present the CPO of $M_2$ here,

       $$\begin{array}{ll}{\widehat{\text{CPO}}}_{i \left(M_2\right)}\hfill & \mathrm{<mo>=</mo>}{\left[{\displaystyle \frac{1}{M}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>1}}^M\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{n}_{ij}!}}{n_{i\mathrm{.}}\mathrm{<mo>!</mo>}}}{\displaystyle \frac{D\left({\mu }^{(h)}{\tau }^{(h)}\right)C\left({\mu }^{(h)}{\tau }^{(h)}\right)}{D\left(n_i+{\mu }^{(h)}{\tau }^{(h)}\right)C\left(n_i+{\mu }^{(h)}{\tau }^{(h)}\right)}}\right]}^{-1}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left[{\displaystyle \frac{1}{M}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>1}}^M\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{n}_{ij}!}}{n_{i\mathrm{.}}\mathrm{<mo>!</mo>}}}{\displaystyle \frac{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{{\mu }_{(h)}{\tau }_{(h)}-1}d{\theta }_i}}}{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}d{\theta }_i}}}}\right]}^{-1}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left[{\displaystyle \frac{1}{M}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>1}}^M\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{n}_{ij}!}}{n_{i\mathrm{.}}\mathrm{<mo>!</mo>}}}{\displaystyle {\int }_{{\theta }_i\in C}\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{{\mu }_{(h)}{\tau }_{(h)}-1}}}{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}}}\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}}}{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}}}}d{\theta }_i}\right]}^{-1}\mathrm{,}\hfill \end{array}$$         

where

$$\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}}}{{\displaystyle {\int }_{{\theta }_i\in C}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{\theta }_{ij}^{n_{ij}+{\mu }_{(h)}{\tau }_{(h)}-1}d{\theta }_i}}}$$

is the density function of ${\theta }_i,$ and ${\theta }_i\in C.$

We notice ${\mu }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}$ and ${\tau }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}$ are the posterior samples from Section 7.2. For each pair of ${\mu }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}$ and ${\tau }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}},$ we can draw ${\theta }_i$ from Dirichlet $\left(n_i+{\mu }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}{\tau }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}\right),$

                                       ${\widehat{\text{CPO}}}_{i\left(M_2\right)}$ $\mathrm{<mo>=</mo>}{\left[{\displaystyle \frac{1}{M}}{\displaystyle \sum _{h\mathrm{<mo>=</mo>}1}^M\frac{{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^K{n}_{ij}!\mathrm{}}}{n_{i\mathrm{.}}!\mathrm{}}}\left({\displaystyle \frac{1}{M^{\prime }}}{\displaystyle \sum _{h^{\prime }\mathrm{<mo>=</mo>}1}^{M^{\prime }}{\displaystyle \prod _{j\mathrm{<mo>=</mo>}1}^K{\theta }_{ij}^{{\left(h^{\prime }\right)}^{-n_{ij}}}}}\right)\right]}^{-1}\mathrm{,}$

where ${\theta }_i^{\left(h^{\prime }\right)}~\text{Dirichlet}\left(n_i+{\mu }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}{\tau }^{\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">)</mo>}}\right)$ with order restriction. Then we get the LPML as $\widehat{\text{LPML}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^I\text{log}}\left({\widehat{\text{CPO}}}_i\right)$ .

However, it is not easy to compute ${\text{CPO}}_i$ or ${\widehat{\text{CPO}}}_i$ of $M_4$ directly. We present how to use the known CPOs, such as ${\text{CPO}}_{i\left(M_2\right)}$ and ${\text{CPO}}_{i\left(M_3\right)},$ to compute ${\text{CPO}}_{i\left(M_4\right)},$

$$ CPO i   ( M 4 )     =     f   ( n i   |   n ( i ) )     =     ( f   ( n ( i ) ) f   ( n ) ) − 1                                                     = [ ∑ ℓ = 1 K   P   ( L = ℓ )   ∫ ∫   f   ( n ( i )   |   μ ,   τ ,   L   =   ℓ )   f   ( μ ,   τ   |   L = ℓ )     d   μ   d τ                                             f   ( n )                       ] − 1                                                     = [ ∑ ℓ = 1 K   P   ( L = ℓ )   ∫ ∫ f   ( n ( i )   |   μ ,   τ ,   L   =   ℓ )   f   ( μ ,   τ   |   L = ℓ )   f   ( n )         d   μ   d τ ]                       − 1                                                   = [ ∑ ℓ = 1 K   P   ( L = ℓ )   ∫ ∫ f   ( n i   |   μ ,   τ ,   L   =   ℓ )   f   ( n ( i )   |   μ ,   τ ,   L   =   ℓ ) f   ( μ ,   τ   |   L = ℓ )     f   ( n i   |   μ ,   τ ,   L   =   ℓ )   f   ( n )                   d   μ   d τ ]                         − 1                                                   = [ ∑ ℓ = 1 K   P   ( L = ℓ )   ∫ ∫ f   ( n   |   μ ,   τ ,   L   =   ℓ )     f   ( μ ,   τ   |   L = ℓ )   f   ( n i   |   μ ,   τ ,   L   =   ℓ )   f   ( n )           d   μ   d τ ]                         − 1                                                   = [ ∑ ℓ = 1 K   P   ( L = ℓ )   ∫ ∫                                   f   ( n   |   L   =   ℓ )                                               f   ( n i   |   μ ,   τ ,   L   =   ℓ )   f   ( n )         f   ( n   |   μ ,   τ ,   L   =   ℓ )   f   ( μ ,   τ   |   L = ℓ )   f   ( n   |   L   =   ℓ )             d   μ   d τ ]                         − 1                                                   = [ ∑ ℓ = 1 K   P   ( L = ℓ )   f   ( n   |   L   =   ℓ )                                                 f   ( n )                               ∫ ∫   1 f   ( n i   |   μ ,   τ ,   L   =   ℓ )         f   ( n   |   μ ,   τ ,   L   =   ℓ )   f   ( μ ,   τ   |   L = ℓ )   f   ( n   |   L   =   ℓ )             d   μ   d τ ]                         − 1                                     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then ${\widehat{\text{CPO}}}_{i \left(M_4\right)}\approx {\left[{\displaystyle {\sum }_{\text{ℓ}\mathrm{<mo>=</mo>}1}^{K}P}\left(\widehat{L=\text{ℓ}} \text{|} n\right)\frac{1}{{\widehat{\text{CPO}}}_{i \left(L_{\text{pos}}=\text{ℓ}\right)}}\right]}^{-1},$ where ${\widehat{\text{CPO}}}_{i \left(L_{\text{pos}}=\text{ℓ}\right)}$ are known, such as ${\widehat{\text{CPO}}}_{i \left(M_2\right)}$ and ${\widehat{\text{CPO}}}_{i \left(M_3\right)}$ . Without extra computation, taking advantage of known CPOs from $M_2$ and $M_3,$ we can easily acquire the CPO of $M_4.$

A.4   Posterior summary of $\theta$