Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 4. Application to body mass index data

4.1 Body mass index

The performance of our method is studied using the Third National Health and Nutrition Examination Survey, NHANES III. NHANES III is a stratified multistage probability design targeted to obtain a representative sample of the total civilian noninstitutionalized U.S. population age 2 months and older. The sample was selected from households across the United States during the period October 1988 through September 1994. Some individuals area selected with different probabilities. For confidentiality reasons, the final data set for this study uses only the 35 largest counties (from 14 states) 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).

The original sensitive attributes BMI data are transformed to categorical data based on the criteria defined by the Centers for Disease Control (CDC), which are underweight, normal, overweight, obese I, and obese II. If BMI is less than 18.5, it falls within the underweight range. If BMI is 18.5 to $<$ 25, it falls within the normal. If BMI is 25.0 to $<$ 30, it falls within the overweight range. If BMI is 30.0 to $<$ 35, it falls within the obese I range. If BMI is 35.0 or higher, it falls within the obese II range. Our goal is to estimate the proportions of the BMI levels. Table 4.1 gives an illustration of the female BMI data of a few counties, where it can be seen that the cell probability is largest for the normal range and other probabilities roughly tail off on both sides to form the unimodal order restriction. Indeed, there are violations in some counties in the earliest and latest cells.

Thus, for each county, the BMI counts can be assumed to follow a multinomial distribution because each individual person can be assumed to exist independently. Figure 4.1 shows a histogram of all BMI values for females aggregated into a single large sample. It can be clearly seen that the unimodal order restriction holds. Because the data in the individual counties are generally sparse, it is difficult to tell whether the unimodal order restriction holds. However, it is sensible to assume that the same unimodal restriction holds within all the counties. Therefore, we can use multinomial distributions to model the female BMI counts.

Table 4.1
The female BMI in five levels
Table summary
This table displays the results of The female BMI in five levels. The information is grouped by County ID (appearing as row headers), BMI_lvl1, BMI_lvl2, BMI_lvl3, BMI_lvl4 and BMI_lvl5 (appearing as column headers).
County ID	BMI_lvl1	BMI_lvl2	BMI_lvl3	BMI_lvl4	BMI_lvl5
1	3	40	37	13	4
2	1	36	38	15	1
3	3	20	49	13	5
$⋮$	$⋮$	$⋮$	$⋮$	$⋮$	$⋮$
35	1	41	41	9	0
Total	45	1,201	1,318	496	89

Figure 4.1

Description of Figure 4.1

Figure presenting a histogram of the five BMI groups for females aggregated into a single large sample. There are 45 females BMI group 1, 1,201 females BMI group 2, 1,318 females BMI group 3, 496 females BMI group 4 and 89 females BMI group 5.

4.2 Fitting $M_{1},$ $M_{2},$ $M_{3}$ and $M_{4}$

4.2.1 MCMC convergence

For each model, we run 20,000 MCMC iterations, take 10,000 as a “burn in” and use every $10^{th}$ to obtain 1,000 converged posterior samples to maintain consistency. Table 4.2 gives the effective sample sizes of the parameters $μ,$ $τ$ for the model with the order restriction and the general model. The effective sample sizes are almost 1,000. Table 4.3 provides p-values of the Geweke test to check the convergence of the parameters (Geweke, 1991). All p-values are large enough to not reject the null hypothesis that the MCMC is stationary. Then posterior samples can be used for the further inference.

Table 4.2
Effective sample sizes of $μ$ and $τ$
Table summary
This table displays the results of Effective sample sizes of (équation) and (équation) (équation) (appearing as column headers).
	$μ_{1}$	$μ_{2}$	$μ_{3}$	$μ_{4}$	$μ_{5}$	$τ$
$M_{1}$	1,000	1,123.7	1,000	1,000	895.4	1,000
$M_{2}$ (Mode at 2^nd)	1,000	1,000	1,000	1,000	1,150.2	1,000
$M_{3}$ (Mode at 3^rd)	1,000	887	889	1,000	1,173.9	1,000

Table 4.3
P values of Geweke test for $μ$ and $τ$
Table summary
This table displays the results of P values of Geweke test for (équation) and (équation) (équation) (appearing as column headers).
	$μ_{1}$	$μ_{2}$	$μ_{3}$	$μ_{4}$	$μ_{5}$	$τ$
$M_{1}$	0.623	0.558	0.899	0.767	0.959	0.514
$M_{2}$ (Mode at 2^nd)	0.964	0.705	0.507	0.511	0.837	0.999
$M_{3}$ (Mode at 3^rd)	0.817	0.559	0.580	0.557	0.812	0.516

4.2.2 Model comparison

With the approximate mixture probabilities, we mix posterior samples of $M_{2}$ and $M_{3}$ together to construct samples of $M_{4} .$

We provide posterior mean (PM), posterior standard deviation (PSD) and coefficient of variation (CV) of $θ ’ s$ for all counties, which can be found in Appendix A.4.

To compare model difference visually, we present the posterior densities plots about different counties in those models as Figure 4.2, Figure 4.3, Figure 4.4 and Figure 4.5. We use different colors to indicate five BMI levels and dashed lines for the posterior means. Due to different capability of borrowing information among areas, we can see different flatness of posterior density curves in the models. With different order restriction assumptions, those posterior density curves center at different places and may overlap differently. We mainly focus on density curves of normal BMI and overweight BMI, since the modal position might be second or third.

In Figure 4.2 has posterior density plots for County 2 applying different models. The number of observations with normal BMI level, which is 36, is close to the number of observations with overweight BMI level, which is 38. The unimodal order restriction may not hold in County 2. Maybe for this reason, there is a significant overlap between normal level and overweight level in the first plot after applying $M_{1}$ to our BMI data. The second plot and the third plot show much less overlap in density curves, due to the strong order restriction assumption. The last plot, which is the density curve from $M_{4}$ , is similar to the density curves in $M_{3} .$ Based on the observations in County 2, the order restriction that the modal position is at the third may be reasonable. The density curves in $M_{3}$ and $M_{4}$ may be appropriate for County 2.

In Figure 4.3 has posterior density plots for County 3 applying different models. Unlike in County 2, the density curves of $θ$ from model $M_{1}$ in County 3 shows a very strong unimodality because we have 49 people in overweight BMI level which dominates this county. The second plot from $M_{2},$ which assumes that the mode is at normal BMI level, has a significant overlap. Its order restriction assumption that the modal position is at the second position may not hold in this county. The third plot from $M_{3},$ which assumes that the mode is at overweight BMI level, is similar as the density curve in $M_{1} .$ The posterior mean of normal BMI level probability is higher than in $M_{1} .$ This phenomenon can be considered as an evidence that $M_{3}$ has a stronger borrowing ability than $M_{1} .$ Overall, the modal position among 35 counties may be at the third. $M_{3}$ can borrow more information among those counties than other models. Then the last plot, which is the density curve from $M_{4},$ has a little overlap. But the unimodal pattern is still in $M_{4} .$

In Figure 4.4, they are posterior density plots for County 13 applying different models. Only $M_{2}$ with an assumption that the mode is at normal BMI level does not show a significant overlap. Since more people are at overweight BMI level, that assumption may be validate in County 13.

Figure 4.5 provides posterior density plots for County 35, which has almost same amount of people in normal and overweight BMI level. $M_{2}$ and $M_{3}$ with different unimodal assumptions have opposite conclusion about normal and overweight probabilities. In this county, $M_{1}$ and $M_{4}$ may be better models.

Figure 4.2

Description of Figure 4.2

Figure presenting posterior density plots of θ for County 2 showing different order restrictions under different models. The first graph represents the model without Order Restriction, noted M1. The second graph represents the model with Order Restrictions with normal weight, noted M2. The third graph represents the model with Order Restrictions with overweight, noted M3. The fourth graph represents the model with uncertain order restrictions, noted M4. In each of these graphs, we find the five BMI levels: Underweight, Normal, Overweight, Obese I and Obese II. The vertical dashed lines represent the posterior means.

Figure 4.3

Description of Figure 4.3

Figure presenting posterior density plots of θ for County 3 showing different order restrictions under different models. The first graph represents the model without Order Restriction, noted M1. The second graph represents the model with Order Restrictions with normal weight, noted M2. The third graph represents the model with Order Restrictions with overweight, noted M3. The fourth graph represents the model with uncertain order restrictions, noted M4. In each of these graphs, we find the five BMI levels: Underweight, Normal, Overweight, Obese I and Obese II. The vertical dashed lines represent the posterior means.

Figure 4.4

Description of Figure 4.4

Figure presenting posterior density plots of θ for County 13 showing different order restrictions under different models. The first graph represents the model without Order Restriction, noted M1. The second graph represents the model with Order Restrictions with normal weight, noted M2. The third graph represents the model with Order Restrictions with overweight, noted M3. The fourth graph represents the model with uncertain order restrictions, noted M4. In each of these graphs, we find the five BMI levels: Underweight, Normal, Overweight, Obese I and Obese II. The vertical dashed lines represent the posterior means.

Figure 4.5

Description of Figure 4.5

Figure presenting posterior density plots of θ for County 35 showing different order restrictions under different models. The first graph represents the model without Order Restriction, noted M1. The second graph represents the model with Order Restrictions with normal weight, noted M2. The third graph represents the model with Order Restrictions with overweight, noted M3. The fourth graph represents the model with uncertain order restrictions, noted M4. In each of these graphs, we find the five BMI levels: Underweight, Normal, Overweight, Obese I and Obese II. The vertical dashed lines represent the posterior means.

Overall, the model with order restrictions, $M_{2}$ and $M_{3},$ can borrow more information among areas than the model without order restriction, $M_{1} .$ The model with uncertain order restriction, $M_{4},$ borrow less information among areas than $M_{2}$ or $M_{3} .$ For this reason, $M_{2}$ and $M_{3}$ have sharper posterior density curves than $M_{1},$ $M_{4}$ has slightly flatter posterior density curves than $M_{2}$ and $M_{3} .$ For the same reason, as shown in Table 4.4, $M_{1}$ has the largest total variance, which is the sum of posterior variance of all counties’ cell probabilities. $M_{2}$ and $M_{3}$ have the smallest variance due to its strong unimodal order restriction assumption. $M_{4} ’ s$ variance is between $M_{1}$ and $M_{3}$ (or $M_{2})$ since $M_{4}$ is a mixture of $M_{2}$ and $M_{3} .$

Table 4.4
Total variance of $θ$
Table summary
This table displays the results of Total variance of (équation). The information is grouped by (équation) (appearing as row headers), (équation) (mode at normal), (équation) (mode at overweight) and (équation) (appearing as column headers).
$M_{1}$	$M_{2}$ (mode at normal)	$M_{3}$ (mode at overweight)	$M_{4}$
0.172	0.063	0.069	0.107

Figure 4.6 and Figure 4.7 are boxplots of $θ ’ s$ posterior samples. The first (Underweight) and last (Obese II) blocks show that different models do not have much difference in estimating the cell probabilities of underweight, normal, and obese I. In the box plots, short line segments from $M_{2},$ $M_{3},$ and $M_{4}$ and long line segments from $M_{1}$ show that the models with order restrictions $(M_{2}, M_{3}, M_{4})$ have smaller variances than the model without order restriction $(M_{1}) .$ The models with order restrictions can borrow more information than the model without order restriction. The differences between each box of $M_{1}$ are larger than the differences in $M_{2},$ $M_{3},$ and $M_{4} .$ In other word, the differences between posterior mean of each county in $M_{1}$ are larger than other models’. It proves that the models with order restrictions borrow more information among areas than the model without order restriction.

In Figure 4.8, we have some regression lines to show the overall posterior standard deviation comparison among those models. The black dashed line is a reference line whose slope is one. The first plot shows a comparison between $M_{1}$ and $M_{3}$ (mode at overweight). All of regression lines are above the reference line, which means that $M_{3}$ (mode at overweight) has smaller standard deviation. We gain higher precision on estimation of cell probabilities among 35 counties in $M_{3} .$ The second plot shows a comparison between $M_{2}$ (mode at normal) and $M_{3}$ (mode at overweight). The regression lines about underweight, Obese I and Obese II are around the reference line. Only the regression line about overweight shows significant difference. It means $M_{3}$ (mode at overweight) is slightly better than $M_{2}$ (mode at normal). In other word, the assumption that overweight BMI probability is the highest may be more reasonable. The last two plots in Figure 4.8 is a comparison between $M_{2}$ (mode at normal) and $M_{4},$ $M_{3}$ (mode at overweight) and $M_{4} .$ $M_{4} ’ s$ performance is slightly worse than $M_{3}$ and $M_{2} .$

In Figure 4.9, we use different symbols to represent different models’ CPO for all 35 counties. In BMI data, County 4 has the largest population, which shows lowest CPO value among others. It is known that low CPO values suggest possible outliers, high-leverage and influential observations. Due to the borrowing feature from the models, County 3 has a low CPO which may be affected by County 4. For most counties, the model with order restriction which assumes the mode is at overweight position can have large CPO, compared with other models. As a summary, in Table 4.5, $M_{2}$ (mode at overweight) has the largest LMPL, which should the “best” model for our BMI data.

Figure 4.6

Description of Figure 4.6

Figure presenting the comparison by box plots of $θ ’ s$ posterior samples for counties 1 to 18 for the 5 BMI levels (Underweight, Normal, Overweight, Obese I and Obese II) under different models (M1, M2, M3 and M4). In the box plots, short line segments from M2, M3, and M4 and long line segments from M1 show that the models with order restrictions (M2, M3 and M4) have smaller variances than the model without order restriction (M1).

Figure 4.7

Description of Figure 4.7

Figure presenting the comparison by box plots of $θ ’ s$ posterior samples for counties 19 to 35 for the 5 BMI levels (Underweight, Normal, Overweight, Obese I and Obese II) under different models (M1, M2, M3 and M4). In the box plots, short line segments from M2, M3, and M4 and long line segments from M1 show that the models with order restrictions (M2, M3 and M4) have smaller variances than the model without order restriction (M1).

Figure 4.8

Description of Figure 4.8

Figure presenting the overall posterior standard deviation comparison between those models to show improvement. The black dashed line is a reference line whose slope is one. The first plot shows a comparison between M1 and M3. All of regression lines are above the reference line, which means that M3 has smaller standard deviation. We gain higher precision on estimation of cell probabilities among 35 counties in M3. The second plot shows a comparison between M2 and M3. The regression lines about underweight, Obese I and Obese II are around the reference line. Only the regression line about overweight shows significant difference. It means M3 is slightly better than M2. In other words, the assumption that overweight BMI probability is the highest may be more reasonable. The last two plots in Figure 4.8 is a comparison between M2 and M4, and M3 and M4. M4’s performance is slightly worse than M3 and M2.

Figure 4.9

Description of Figure 4.9

Figure presenting the conditional predictive ordinates (CPOs) for the 35 counties under models M1, M2, M3 and M4. Different symbols are used to represent different models’ CPO for all 35 counties. A lower CPO suggests possible outliers, high-leverage and influential observations.

Table 4.5
LPML, comparison of the four models using LPML
Table summary
This table displays the results of LPML. The information is grouped by (équation) (appearing as row headers), (équation) (mode at normal), (équation) (mode at overweight) and (équation) (appearing as column headers).
$M_{1}$	$M_{2}$ (mode at normal)	$M_{3}$ (mode at overweight)	$M_{4}$
-326.76	-331.88	-318.26	-329.58

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Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 4. Application to body mass index data

4.1 Body mass index

4.2 Fitting $M_{1},$ $M_{2},$ $M_{3}$ and $M_{4}$

4.2.1 MCMC convergence

4.2.2 Model comparison

Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction Section 4. Application to body mass index data

4.1 Body mass index

4.2.1 MCMC convergence

4.2.2 Model comparison

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Bayesian inference for multinomial data from small areas incorporating uncertainty about order restriction
Section 4. Application to body mass index data