Bayesian Fractional Polynomial Approach to Quantile Regression and Variable Selection with Application in the Analysis of Blood Pressure among US Adults

Hypertension is a highly prevalent chronic medical condition and a strong risk factor for cardiovascular disease (CVD), as it accounts for more than $45\%$ of CVD. The relation between blood pressure (BP) and its risk factors cannot be explored clearly by standard linear models. Although the fractional polynomials (FPs) can act as a concise and accurate formula for examining smooth relationships between response and predictors, modelling conditional mean functions observes the partial view of a distribution of response variable, as the distributions of many response variables such as BP measures are typically skew. Then modelling 'average' BP may link to CVD but extremely high BP could explore CVD insight deeply and precisely. So, existing mean-based FP approaches for modelling the relationship between factors and BP cannot answer key questions in need. Conditional quantile functions with FPs provide a comprehensive relationship between the response variable and its predictors, such as median and extremely high BP measures that may be often required in practical data analysis generally. To the best of our knowledge, this is new in the literature. Therefore, in this paper, we employ Bayesian variable selection with quantile-dependent prior for the FP model to propose a Bayesian variable selection with parametric nonlinear quantile regression model. The objective is to examine a nonlinear relationship between BP measures and their risk factors across median and upper quantile levels using data extracted from the 2007-2008 National Health and Nutrition Examination Survey (NHANES). The variable selection in the model analysis identified that the nonlinear terms of continuous variables (body mass index, age), and categorical variables (ethnicity, gender and marital status) were selected as important predictors in the model across all quantile levels.


Introduction
Over the past three decades, the number of adults aged 30-79 with hypertension has increased from 648 million to 1.278 billion globally (Zhou et al. (2021)).Hypertension is a highly prevalent chronic medical condition and a strong modifiable risk factor for cardiovascular disease (CVD), as it attributes to more than 45% of cardiovascular disease and 51% of stroke deaths (World Health Organization (2013)).The risk of CVD in individuals rises sharply with increasing BP (Ettehad et al. (2016); Bundy et al. (2017); Prospective Studies Collaboration (2002); Navar et al. (2016); Clark et al. (2019)).
Continuous BP measurement has proven to be one of effective incident prevention.This implies that BP is the essential physiological indicator of human body.When the heart beats, it pumps blood to the arteries resulting in changes of BP during the process.When the heart contracts, BP in the vessels reaches its maximum, which is known as systolic BP (SBP).When the heart rests, BP reduces to its minimum, which is known as diastolic BP (DBP).
Linear regression and polynomial regression analyses have been used in assessing the association between BP and risk factors contributing to various diseases (Koh et al. (2022); Liu et al. (2022); Yeo et al. (2022)).It is evident that the polynomial regression models fit the data accurately in some research studies due to its adaptability of nonlinearity property but face high order polynomial approximation.
The fractional polynomials (FPs) proposed by Royston and Altman (1994) act as a concise and accurate formulae for examining smooth relationships between response and predictors, and a compromise between precision and generalisability.FPs are parametric in nature and then intuitive for the interpretation of the analysis results.FP approach has clearly established a role in the nonlinear parametric methodology especially with application by clinicians from various research fields, such as obstetrics and gynecology (Tilling et al. (2014)), gene expression studies in clinical genetics (Tan et al. (2011)) and cognitive function of children (Ryoo et al. (2017)), and other medical applications (Wong et al. (2011); Ravaghi et al. (2020); Frangou et al. (2021) and among others).
However, modelling conditional mean functions observes the partial view of a distribution of response variable, as the distributions of many response variables such as the BP measures are typically skew.
Then 'average' BP may link to CVD but extremely high BP could explore CVD insight deeply and precisely.So, existing mean-based FP approaches for modelling the relationship between factors and BP cannot answer key questions in need.It is attractive to model conditional quantile functions with FPs that accommodates skewness very easily.Quantile regression, introduced by Koenker and Bassett (1978), provides comprehensive relationship between the response variable and its predictors, such as median and extremely high BP measures may be often required in practical data analysis generally.Zhan et al. (2021) suggested quantile regression with FP as a suitable approach for an application, such as age-specific reference values of discrete scales, in terms of model consistency, computational cost and robustness.This approach is also used to derive reference curves and reference intervals in several applications (Chitty and Altman (2003); Bell et al. (2010); Bedogni et al. (2012); Kroon et al. (2017); Casati et al. (2019); Cai et al. (2020); Loef et al. (2020)), which allow quantiles to be estimated as a function of covariates without requiring parametric distributional assumptions.This is essential for data that do not assume normality, linearity and constant variance.Recently, reasonable amount of nonlinear quantile regression analyses have been conducted in medical data analysis (Maidman and Wang (2018); Huang et al. (2023); Wu et al. (2023) and among others).
However, Bayesian approach to quantile regression has advantages over the frequentist approach, as it can lead to exact inference in estimating the influence of risk factors on the upper quantiles of the conditional distribution of BP compared to the asymptotic inference of the frequentist approach (Yu et al. (2005)).It also provides estimation that incorporates parameter uncertainty fully (Yu and frequentist approaches, such as the analysis of risk factors for female CVD patients in Malaysia (Juhan et al. (2020)) and the analysis of risk factors of hypertension in South Africa (Kuhudzai et al. (2022)).
The former revealed that the Bayesian approach has smaller standard errors than that of the frequentist approach.The latter also revealed that credible intervals of the Bayesian approach are narrower than confidence intervals of the frequentist approach.These findings suggest that the Bayesian approach provides more precise estimates than the frequentist approach.
Variable selection in Bayesian quantile regression has been widely studied in the literature (Li et al. (2010); Alhamzawi et al. (2012); Alhamzawi and Yu (2013a); Chen et al. (2013); Adlouni et al. (2018); Alhamzawi et al. (2019); Dao et al. (2022) and among others).It plays an important role in building a multiple regression model, provides regularisation for good estimation of effects, and identifies important variables.Sabanés Bové and Held (2011) combine variable selection and 'parsimonious parametric modelling' of Royston and Altman (1994) to formulate a Bayesian multivariate FP model with variable selection that efficiently selects best fitted FP model via stochastic search algorithm.However, In present, no research studies have been conducted for variable selection in Bayesian parametric nonlinear quantile regression for medical application even though there is a limited amount of studies in case of non-regularised models, such as mixed effect models (Wang (2012); Yu and Yu (2023)).
Therefore, in this paper, we explore a new quantile regression model using FPs and employ Bayesian variable selection with quantile-dependent prior for a more accurate representation of the risk factors on BP measures.The three-stage computational scheme of Dao et al. (2022)  The remainder of this paper is as follows.Section 2 presents the concept of FPs (Royston and Altman (1994)), quantile regression (Koenker and Bassett (1978)) and Bayesian variable selection with quantile-dependent prior (Dao et al. (2022)).The details of the NHANES 2007-2008 dataset used for the analysis are provided in Section 3. Section 4 applies the proposed method to the analysis of the NHANES 2007-2008 dataset, performs comparative analysis with two quantile regression methods and provides all the findings.Section 5 concludes this paper.

Methodology
Regression analysis is a technique that quantifies the relationship between a response variable and predictors.Quantile regression, introduced by Koenker and Bassett (1978), is a method to estimate the quantiles of a conditional distribution of a response variable and such it permits a more accurate portrayal of the relationship between the response variable and predictors.Unlike linear regression analysis, quantile regression analysis gives a better idea about distribution of the data because the latter is robust to outliers.

Quantile Regression
Let τ be the proportion of a sample having data points below the quantile in τ .Given a dataset, {x i , y i } n i=1 and fixed τ , the τ th quantile regression model is represented as where τ is in the range between 0 and 1, and β(τ ) is the vector of unknown parameters of interest and (τ ) is the model error term for the τ th quantile.For the sake of notation simplification, we omit τ from these parameters.
We wish to estimate the unknown parameters, β as β for each τ th quantile, which can be done by minimising the check function over β: with the check function where I ∆≥0 represents the value 1 if ∆ belongs to the set [0, ∞), and the value 0 otherwise.
Minimising (2) is same as maximising a likelihood function.An asymmetric Laplace distribution (ALD) is employed, which is the common choice for the quantile regression analysis (Yu and Moyeed (2001); Yu et al. (2003)).We assume that i ∼ AL(0, σ, τ ), i = 1, . . ., n, where the AL is the ALD with its density Here, ρ τ ( i ) denotes the usual check loss function of Koenker and Bassett (1978).
We are interested in selecting a subset of important predictors which has adequate explanatory and predictive capability.One of the common procedures for simultaneously facilitating the parameter estimation and variable selection is to impose penalty function on the likelihood to arrive at the penalised loss function, which is minimised to obtain the τ th quantile regression estimator.Here, P (β, δ) is a regularisation penalty function and δ is a penalty parameter that controls the level of sparsity.Typically, Bayesian regularised qauantile regression is formulated through the relationship between the check function and the ALD.
Bayesian inference is one of the most popular approaches for the regression analysis since it provides with an entire posterior distribution of a parameter of interest as well as incorporation of parameter uncertainty and prior information about data.So, Bayesian analysis is preferable over frequentist analysis.
Rather than the standard linear model, we will be using the FP model to develop the nonlinear model under Bayesian quantile regression and variable selection.

Fractional Polynomials
Box and Tidwell (1962) introduced the transformation now known as the Box-Tidwell transformation, where a is a real number.Royston and Altman (1997) extend the classical polynomials to a class which they called FPs.
An FP of degree m with powers p 1 ≤ . . .≤ p m and respective coefficients α 1 , . . ., α m is where h 0 (x) = 1 and where j = 1 . . ., m.Note that the definition h j (x) allows the repeated powers.The bracket around the exponent denote the Box-Tidwell transformation (4).For m ≤ 3, Royston and Altman (1994) constrained the set of possible powers p j to the set which encompasses the classical polynomial powers 1, 2, 3 but also offers square roots and reciprocals.Royston and Sauerbrei (2008) argue that this set is sufficient to approximate all powers in internals [−2, 3].The simple example of the FP model is as follows.An FP with m = 3 powers and its power where the last term reflects the repeated power 2.
Generalisation to the case of multiple predictors: This is called the multiple FP model.Suppose we continue examining k continuous predictors x 1 , . . ., x k and content themselves with a maximum degrees of m max ≤ 3 for each f m l l , for instance, 0 ≤ m l ≤ m max for l = 1, . . ., k, where m l = 0 denotes the omission of x l from the model.From the powers set S, m l powers are chosen, which need not be different due to the inclusion of logarithmic terms for repeated powers (4), we now employ the τ th nonlinear quantile regression with the SMN representation of the ALD errors, where the (n a vector of standard Normal random variables and A special way of defining the matrix B is through the use of FPs.In this case, the basis function B(x l ) is chosen as the transformation h lj in (6) (j = 1, . . ., m l ) and the unknown parameter β = (α 1 , . . ., α k ) T , where α l = (α l1 , . . ., α lm l ) for l = 1, . . ., k.The transformation h j are determined by the power vector p 1 , . . ., p k through their definition (4).Note that the p l is empty if the predictor x l is not included in the model (m l = 0).

Bayesian Approach and Variable Selection
Given the model in ( 7), the likelihood function conditional on β, σ, v = (v 1 , . . ., v n ) T can be written as We employ the three-stage algorithm of Dao et al. (2022) for Bayesian nonlinear quantile regression with variable selection.It can be summarised, as follows.
The first stage is the expectation-maximisation algorithm consisting of two main steps: the E-step and the M-step.Dempster et al. (1977) proposed the EM algorithm, which is a statistical simulation method and it aims to solve the complex data analysis problem with missing data.
Repeat E-step and M-step until it meets the required condition, then the final iteration values of the EM algorithm are set as the posterior modes of β and σ, denoted by β and σ, respectively.
The second stage is the Gibbs sampling algorithm.The quantile-specific Zellner's g-prior (Alhamzawi and Yu (2013b)) is used for the prior specification and it is given by where N (•) is the multivariable Normal distribution, g is a scaling factor, V = diag(1/v 1 , . . ., 1/v n ) and This prior specification has an advantage, as it contains information that is dependent upon the quantile levels, which increases posterior inference accuracy.
Given the posterior modes, β and σ as the starting value, we denote β (r−1) and σ (r−1) as the (r−1)th iteration value of parameters β and σ in the Gibbs sampling algorithm.
The algorithm iterates until it reaches the final MCMC iteration indexed at R and discard the burn-in period.
Finally, the third stage is the important re-weighting step.The S samples are drawn from the importance weights without replacement where S < R is the number of importance weighting steps.A random indicator vector γ = (γ 1 , . . ., γ D ) T is introduced to the nonlinear model where B γ is the (n × D γ ) matrix consisting of important predictors and β γ of length D γ is the non-zero parameter vector.The same prior specification in ( 8) is employed along with a prior on γ d , d = 1, . . ., D, and a beta prior on π: where π ∈ [0, 1] is the prior probability of randomly including predictor in the model.Then π is marginalised out from p(γ|π) resulting as The marginal likelihood of y under the model M γ is then obtained by integrating out β and σ resulting as , where t 2n (•) is the multivariate Student t-distribution with 2n degrees of freedom.The posterior probability of M γ is therefore given by p(γ|y, v) ∝ p(y|γ, v)p(γ).Lastly, the independent samples of v from the second stage algorithm are drawn based on the S samples and the important re-weighting step is iterated until the S samples of γ are obtained.Then the posterior inclusion probability is estimated, as follows where S is the number of iterations after discarding the burn-in period.

Data Preparation and Data Analysis
This The study variables included SBP and DBP as the response variables.The BP measurements were taken as follows.After a resting period of 5 minutes in a sitting position and determination of maximal inflation level, three consecutive BP readings were recorded.A fourth reading was recorded if a BP measurement is interrupted or incomplete.All the results were taken in Mobile Examination Center.The BP measurements are essential for hypertension screening and disease management, since hypertension is an important risk factor for cardiovascular and renal disease.Then in this study, SBP and DBP were selected as response variables where each was averaged over the second and third readings.Predictor variables were BMI, age, ethnicity, gender and marital status.
We initially included 9,762 participants who have completed both BP and body measure examinations in the study.From 9,762 participants, we exclude those who had not underwent examinations.Then among the remaining 4,612 participants, we further excluded those who refused to reveal their marital status.Finally, 4,609 participants were included for analysis in this study.
The NHANES protocols were approved by the National Center for Health Statistics research ethics review boards, and informed consent was obtained from all participants.The research adhered to the tenets of the Declaration of Helsinki.
The R version 4.2.2 was used to conduct both statistical and Bayesian analyses.Both 'quantreg' and 'Brq' R packages was employed to fit the frequentist and Bayesian approaches of the quantile regression model with FPs, respectively.The source R code was provided from the main author to fit the Bayesian quantile regression with variable selection and FPs via the three-stage algorithm.
This study considers two quantile models at the 50 th , 75 th and 95 th percentiles.When modelling hypertension, it is preferable to model both median and extremely high values of SBP and DBP, which corresponds to the median and upper distributions of SBP and DBP, respectively (Kuhudzai et al. (2022)).The following two quantile models will be used for the analysis for the fixed τ value:

Results
In this section, both descriptive and model analyses are provided for the NHANES 2007-2008 dataset using the proposed model.To evaluate the performance of the proposed model, we included two existing methods, including quantile regression and Bayesian quantile regression, with FP model for a fair comparative analysis.The model comparison is discussed outlining the advantages of the proposed model over these two methods.All the results are provided in this section through tables and figures for each regression analysis.

Descriptive Analysis
For this analysis, continuous variables were collapsed into categorical variables, including SBP, DBP, BMI and age.According to the guidelines of Whelton et al. (2018), the BP variables are divided into three groups: normal (< 120 mmHg for SBP, < 80 mmHg for DBP), pre-hypertension (120 − 139 mmHg for SBP, 80 − 89 mmHg for DBP) and hypertension (≥ 140 mmHg for SBP, ≥ 90 mmHg for DBP).The BMI variable is also divided into six groups: underweight (< 18.5), healthy (18.5 − 24.9), overweight (25 − 29.9), obese (30 − 34.9), very obese (35 − 39.9) and morbidly obese (≥ 40) (Centers for Disease Control and Prevention (2022)).It is evident from Table 1-2 that hypertension was more prevalent in underweight, very obese and morbidly obese participants for both BP measures where the very obese and morbidly obese had the highest prevalence for DBP and SBP measures, respectively.The same trend was observed on the proportions of elevated BP for DBP measure.It was clear that healthy participants had the highest Table 1-2 also showed that men had the highest prevalence of both elevated BP and hypertension for both BP measures.Participants who were separated or divorced and those who became widowed had the highest prevalence of hypertension for DBP and SBP measures, respectively.
Lastly, at the 1% significance level, Table 1-2 exhibited very weak to weak associations between BP measures, BMI and sociodemographic characteristics among the US adults.However, there is a moderate association between SBP measure and age.There is no statistically significant association between DBP measure and marital status at the 5% level.

Model Analysis
Table 3-4 provides the coefficients for predictors relating to SBP and DBP responses for three quantile regression models with FPs at three quantile levels (τ = 0.50, 0.75, 0.95), including one frequentist and two Bayesian approaches with one using variable selection.Bayesian parameters are obtained via posterior men.The 95% confidence intervals are provided for the frequentist approach, whilst the 95% credible intervals are provided for the Bayesian approaches.Either 95% confidence interval or 95% credible interval indicates that the user is 95% confident that the population mean is within the interval.We denote the frequentist approach as the QR-FP model, and two Bayesian approaches as the BQR-FP and BQRVS-FP models where the latter uses variable selection.
For the BQR-FP model, the algorithm was implemented for 10,000 MCMC iterations and 1,000 MCMC iterations were discarded as a burn-in period.For the BQRVS-FP model, the first stage algorithm ran for 1,000 EM iterations and repeated for 2 replications.Then 5,000 MCMC iterations were drawn for the second stage algorithm while discarding 2,500 MCMC iterations as a burn-in period.Finally, the last algorithm ran for 1,250 important re-weighting steps of which 500 steps were discarded as a burn-in period.The value of g is selected as 1,000 for all implementations of the variable selection model.
It is evident from Table 3 that all the risk factors except both linear and nonlinear terms of age were found to have statistically significant associations with SBP across the two upper quantile levels according to their 95% confidence intervals containing no zero value under the QR-FP model.Looking at the median level, the linear term had association with SBP under the same approach.When looking at the BQR-FP and BQRVS-FP models, only the nonlinear term of age did not have a statistically significant association for all quantile levels.On the other hand, Table 4 observed that all the risk factors including nonlinear terms had statistically significant associations with DBP across all quantile levels for all model approaches.When looking at the median level under the QR-FP model, it revealed that the marital status did not have statistically significant association.It is also shown from Figure 1-2 that each risk factor of SBP and DBP across all the quantile levels has increasingly random stationary posterior distribution although at the 95 th percentile, the trend has a slower decreasing rate.marginal inclusion probability (MIP) of at least 0.9, respectively.The risk factors selected lie above the cutoff of 0.9 of MIP.Across all the quantile levels for both SBP and DBP models, the same important risk factors were consistently selected where the DBP model selected all the risk factors including the nonlinear terms except marital status at the median level.The SBP model selected all except the nonlinear term of age.This mostly agreed with findings on 95% credible intervals from

Model Comparison
Observing at the 95% confidence intervals of frequentist approach and the 95% credible intervals of two Bayesian approaches from Table 3-4, the BQRVS-FP model has tighter intervals compared to the QR-FP model having wider intervals.
Another finding is from the diagnostic plots that the autocorrelation plots of BQRVS-FP model have a faster decreasing trend rate across all the quantile levels, whereas those of the BQR-FP model have a slower rate.This is evident that the BQRVS-FP model has more random stationary posterior distributions of interest.
When looking at Table 3-4 and Figure 5-6, the BQRVS-FP model selected the important predictors coinciding with statistically significant associations between SBP, DBP and their risk factors based on their 95% credible intervals.
These findings suggest that the Bayesian variable selection approach to quantile regression model with FPs obtained more precise estimates than the frequentist and Bayesian approaches.The nonlinear terms were selected as important variables in both SBP and DBP models indicating that FP model was necessary to examine the nonlinear relationship between SBP, DBP and risk factors.
National Health and Nutrition Examination Survey (NHANES).The descriptive analysis showed that the prevalence of hypertension increases by age and the hypertension is highly prevalent among very obese and morbidly obese participants.In particular, it is more prevalent in men than women.Moreover, there is a statistically significant moderate association between SBP and age based on the Cramer's V value, whilst the remaining associations were weaker for both BP measures.However, there is no association between DBP and marital status.
The analysis motivates a new Bayesian nonlinear quantile regression model under fractional polynomials (FPs) and variable selection with quantile-dependent prior where the quantile regression analysis investigates how the relationships differ across the median and upper quantile levels.The use of FPs allows for the relationships to be nonlinear parameterically.The variable selection investigates for important predictors that contribute to the nonlinear relationships via the Bayesian paradigm.The model analysis suggested that the proposed model provides better estimates because in comparison of two methods, the frequentist and Bayesian approaches of quantile regression model, the 95% credible intervals were narrower and the autocorrelation plots have faster decreasing rate of correlated posterior samples.
The analysis of the data showed that nonlinear relations do exist because the proposed model identified the nonlinear terms of continuous variables, including BMI and age as important predictors in the model across all the quantile levels.On the other hand, the nonlinear term of age is not selected under the SBP model.The marital status is not selected as an important risk factor for the DBP model at the median level.This agreed with findings of both descriptive and model analyses.Moreover, the data analysis suggested that the quantile based FP approaches have goodness of fit comparing to mean-based FP approaches.Thus, the importance of the nonlinear quantile model with FPs is significant for modelling of BP measures.
is employed as a variable selection method due to its fast convergence rate, low approximation error and guaranteed posterior consistency under model misspecification.So, we propose a Bayesian variable selection with nonlinear quantile regression model to assess how body mass index (BMI) among the United States (US) influences BP measures, including SBP and DBP.The objective of this paper is to examine a nonlinear relationship between BP measures and their risk factors across median and upper quantile levels.The dataset used in this paper is the 2007-2008 National Health and Nutrition Examination Survey (NHANES), including the information on BP measurements, body measures and sociodemographic questionnaires.
for any a, b > 0, letting a = 1/ √ 2σ & b = / √ 2σ and multiplying a factor of exp(−(2τ − 1) /2σ), to express the probability density function (pdf) of the ALD errors as its scale mixture of Normals (SMN) representation, study is based on the data of the National Health and Nutrition Examination Survey (NHANES) during 2007-2008.The survey conducted by the National Center for Health Statistics of the Centers for Disease Control and Prevention used a complex, stratified, multistage sampling design to select a representative sample of noninstitutionalized population in the United Status civilians to participate in a series of comprehensive health-related interviews and examinations.In total, 12,943 people participated in NHANES 2007-2008.

Figure 1 :
Figure 1: Trace, density and autocorrelation plots for the risk factors of SBP at three quantile levels (τ = 0.5, 0.75, 0.95) under the Bayesian quantile regression model with FPs.
with guidelines given byRea and Parker (2014): 0.00 to under 0.10 = very weak association, 0.10 to under 0.20 = weak association, 0.20 to under 0.40 = moderate association and 0.40 and above = strong association.

Figure 2 :
Figure 2: Trace, density and autocorrelation plots for the risk factors of DBP at three quantile levels (τ = 0.5, 0.75, 0.95) under the Bayesian quantile regression model with FPs.

Figure 3 :
Figure 3: Trace, density and autocorrelation plots for the risk factors of SBP at three quantile levels (τ = 0.5, 0.75, 0.95) under the Bayesian quantile regression model with FPs and variable selection.

Figure 4 :
Figure 4: Trace, density and autocorrelation plots for the risk factors of DBP at three quantile levels (τ = 0.5, 0.75, 0.95) under the Bayesian quantile regression model with FPs and variable selection.

Figure 1 -
Figure 1-2 present the trace, density and autocorrelation plots for each risk factor of SBP and DBP,

Figure 5 :
Figure 5: The selected predictors and cutoff thresholds (dashed lines) of the NHANES dataset for the SBP model via the BQRVS-FP approach at τ = 0.50, τ = 0.75 and τ = 0.95.

Figure 3 -
Figure 3-4 also present the trace, density and autocorrelation plots for each risk factor of SBP and DBP, respectively under the BQRVS-FP model.All the plots show stationarity, good Markov chain mixing rate and good convergence.Each autocorrelation plot indicated that their stationary distribution became random and less correlated with the initial values at a faster rate.

Figure 5 -
Figure 5-6 provide the BQRVS-FP model determined by risk factors of SBP and DBP having the

Figure 6 :
Figure 6: The selected predictors and cutoff thresholds (dashed lines) of the NHANES dataset for the DBP model via the BQRVS-FP approach at τ = 0.50, τ = 0.75 and τ = 0.95.
n, are treated as a function of fixed predictors.Maximum likelihood estimates (MLE) can be obtained by maximising log-likelihood function log f (β, σ|y, v) of the

Table 1 :
SBP among United Status Adults by BMI and Sociodemographic Characteristics.

Table 2 :
DBP among United Status Adults by BMI and Sociodemographic Characteristics.

Table 3 :
One Frequentist and Two Bayesian Quantile Regression Analyses for Relationship between SBP and Risk Factors.

Table 4 :
One Frequentist and Two Bayesian Quantile Regression Analyses for Relationship between DBP and Risk Factors.

Table 1 -
2 present SBP and DBP proportions among US adults by demographic and lifestyle characteristics, including BMI, age, ethnicity, gender and marital status.The Cramer's V value is used to measure the magnitude of the association between SBP, DBP, sociodemographic characteristics and BMI of the participants.Their values with p-values are also presented in Table1-2 and compared with

Table 3
also observed that the BMI, nonlinear term of age and gender have negative associations with SBP, whilst the nonlinear term of BMI, age and gender have negative associations with DBP from