Bayesian negative binomial logit hurdle and zero-inflated model for characterizing smoking intensity

Abstract

Smoking invariably has environmental, social, economic and health consequences in Ethiopia. Reducing and quitting cigarette smoking improves individual health and increases available household funds for education, food and better economic productivity. Therefore, this study aimed to apply the Bayesian negative binomial logit hurdle and zero-inflated model to determine associated factors of the number of cigarette smokers per day using the smoking intensity data of 2016 Ethiopia Demographic and Health Survey. The survey was a community-based cross-sectional study conducted from January 18 to June 27, 2016. The survey used two stage stratified sampling design. Bayesian analysis of Negative Binomial Logit Hurdle and Zero-inflated models which incorporate both overdispersion and excess zeros and carry out estimation using Markov Chain Monte Carlo techniques. About 94.2% of them never cigarettes smoked per day and the data were found to have excess zeros and overdispersion. Therefore, after considering both the zero counts and the enduring overdispersion, according to the AIC and Vuong tests, the Zero-inflated Negative Binomial and Negative Binomial Logit Hurdle model best fit to the data. The finding Bayesian estimation technique is more robust and precisely due to that it is more popular data analysis method. Furthermore; using Bayesian Zero-inflation and Zero hurdle model the variable: age, residence, education level, internet use, wealth index, marital status, chewed chat, occupation, the media were the most statistically significant determinate factors on the smoking intensity.

Background

Smoking is currently considered one of the greatest problems in public health worldwide. It is one of the most preventable causes of death and a major known cause of non-communicable diseases [1, 2]. Smoking invariably has social, economic, health, and environmental consequences in Ethiopia. Reducing cigarette smoking improves individual health and increases available household funds for food, education, and better economic productivity [3].

Globally, over 7 million people die each year due to smoking related to this, over 86% of deaths are from direct tobacco use, while around 13% are due to secondhand smoke. The low and middle economical income countries are especially facing an increasing burden of tobacco-related diseases. Africa, particularly sub-Saharan Africa, is experiencing increasing tobacco use [4]. Among Ethiopian men who smoke cigarettes daily, one-quarter (25%) smoke 5–9 cigarettes each day; 6% of daily cigarette smokers smoke 25 or more cigarettes each day. The percentage of men age 15–49 who don't smoke cigarettes has increased slightly since 2011, from 93 to 95% [5]. Despite the presence of studies on tobacco use and associated factors, no study assesses the smoking intensity among smokers. Evidence of smoking intensity is very important for planning and implementing behavior change interventions within the country.

Several studies have been conducted to determine related risk factors of smoking [6, 7]. However, few studies can be found on investigating the intensity of smoking in Ethiopia, which highlights the importance of considering the number of cigarettes smoked per day as a count response variable and investigating. There were no studies which conducted by considering the excess zeros and over dispersion on this outcome variable in Ethiopia. The analysis of count data with many zeros can be done in various fields, including medical and public health. Several models have been developed in recent decades to analyze this count data, especially smoking intensity data [8]. The count data are different from the categorization of count data to be used in crude rate and logistic regression will lead to loss of information.

Furthermore, treating count data is different from ordinal data, continuous variable in linear regression or dichotomous variable in logistic regression models is likely to bias the results [9]. The a Poisson regression or negative binomial regressions are commonly used to model count outcomes assuming Poisson distribution or negative binomial distribution. But the probability of zeroes based on Poisson regression or negative binomial regression cannot account for excess zero counts because excess zeroes will bias the estimation of parameters [10, 11]. Therefore, Hurdle and zero-inflated count models have been the best model so far to solve this issue concerning excess zeroes [12]. The choice of the model either of two is often depend on the nature of the problem and the data collection procedures [13]. The difference is hurdle model includes a mass at zero and a truncated distribution whereas the zero-inflated model is based on a mass at zero and a regular distribution, the inferential results is often very similar. Zero-inflated models and hurdle models provide a way of modeling the excessive proportion of zero values and allow for overdispersion [14, 15].

Consequently; the Negative Binomial–Logit Hurdle (NBLH) and Zero-inflated Negative Binomial (ZINB) Regression model are flexible models for dealing with zero-inflated and over dispersed count data [16]. Bayesian parameter estimation methods can be applied by the Markov chain Monte Carlo (MCMC) simulation that can generate random values with the Gibbs-sampling algorithm. Hence the Bayesian method is more flexible for parameters estimation [17]. Therefore, this study aimed to apply the Bayesian NBLH and Bayesian ZINB regression model to determine the associated factors of the number of cigarettes smoking per day and to identify the optimum model using the smoking intensity data of the 2016 Ethiopia Demographic and Health Survey (EDHS). The finding of this study intended to fill the gap of the model for excess zeros and over dispersed data. This study may use an alarm for policymakers by identifying the associated factors of smoking intensity per day and also it intended to bring to the improvement of the study design and participants of future studies.

Method

Source of data and study design

The dataset used for this study was obtained from 2016 EDHS which conducted from January 18 to June 27, 2016. The survey was a population-based cross-sectional study. The survey used two stage stratified sampling methods. In the first stage, a total of 645 clusters was randomly selected proportional to the household size from the sampling strata, and in the second stage, 28 households per cluster were selected using systematic random sampling. In this survey, a total of 12,645 smoking intensity people selected from 645 clusters.

Variables of the study

The dependent variable of this study was the number of cigarette smokers per day (the Smoking intensity is the number of cigarette sticks smoked in the last 24 h). Since a significant number of the daily smokers used manufactured cigarettes, this study adopted the number of manufactured cigarettes the participants used per day as an outcome variable to assess the smoking intensity in Ethiopia.

Independent variable

Age, marital status, residence, educational level, religion, occupational status, wealth index, frequency of media use, current chat chewing behavior and Internet use were the independent variables.

Statistical method

In this study, the variable of interest was count data. When the dependent variable is a count, it is appropriate to use non-linear models to describe the relationship between the dependent variable and a set of predictor variables. For count data, the standard framework for explaining the relationship between the dependent variable and a set of explanatory variables includes the Poisson, negative binomial regression, zero inflated and hurdle models. The advanced models for this study count data are the Bayesian zero-inflated regression model and Bayesian hurdle model [18]. In this study, excess zeros and overdispersion in the number of cigarettes smoked per day data exist. Therefore, the Negative binomial model cannot be used to handle the high number of zeros. To do this, four models are discussed that can deal with the excessive number of zeros, namely, the zero-inflated Poisson, Zero-inflated negative binomial, Poisson logit hurdle and negative binomial logit hurdle models can be alternatively used [19].

Zero inflated regression model

In Poisson data with too many zeros, the variance often exceeds the mean, causing overdispersion. Although the ZIP regression model can handle excess zeros for Poisson data, overdispersion may remain even after modeling excess zeros, and consequently ZIP parameter estimates can be severely biased. In such a case, the use of a zero-inflated negative binomial (ZINB) distribution can be a good alternative regression model [13, 20]. Therefore, the ZIP regression model is more effective for excess zero outcomes than Poisson regression. While the ZINB regression model is more effective for excess zero outcomes in additional overdispersion than negative binomial regression model.

Zero-inflated Poisson regression model

The most commonly employed approach to dealing with heterogeneity associated with excess zeros, is to use a Poisson distribution that has been mixed with a point mass at zero to allow for the inclusion of additional structural zeros. In ZIP regression, the excess zero counts are assumed to occur with probability ${\omega }_{i}$ and follow a Poisson distribution with mean ${\mu }_{i}$, with probability 1 – ${\omega }_{i}$ where i = 0, 1, 2,..., n. ZIP model can thus be seen as a mixture of two-component distributions, a zero part, and no-zero components, given by [12]:

$$ {\text{P}}\left( {{\text{Y}} = {\text{y}}} \right) \, = \left\{ {\begin{array}{*{20}l} {\omega_{i} + \left( {1 - \omega_{i} } \right)e^{{ - \mu_{i} }} , \quad y_{i = 0} } \\ {\left( {1 - \omega_{i} } \right)\frac{{e^{{ - \mu_{i} }} \mu_{i}^{{y_{i} }} }}{{y_{i !} }} , \qquad y_{i} = 1,2,3, \ldots ..} \\ \end{array} } \right. $$

(1)

The first part of the Eq. (1) above is the zero part of the model and the second part is the non-zero count’s part of the model. The mean and variance of the zero-inflated Poisson model are: E(Yi) = ${\mu }_{i}$(1-${\omega }_{i}$) and Var (Yi) = ${\mu }_{i}$ (1 −${\hspace{0.17em}\omega }_{i}$)(1 + ${\mu }_{i}{\omega }_{i}$);

The most natural choice to model the probability of excess zeros is to use a logistic regression model with a logit link and count data excluding excess zeros can be modeled through Poisson regression:

$$ {\text{Logit }}(\omega_{i} ) \, = x_{i}^{T} \beta {\text{ and Log }}(\mu_{i} ) = Z_{i}^{T} \gamma $$

(2)

where ${{x}_{i}}^{T}$ represents a vector of covariates and β a vector of parameters and $\gamma$ are the vector coefficients ${Z}_{i}^{T}$.

Zero-inflated negative binomial regression model

The ZINB model is especially suited to dealing with both over dispersed and zero-inflated data. The ZINB distribution is a mixture distribution assigning a mass of ${\omega }_{i}$ to ‘extra’ zeros and a mass of (1 – ${\omega }_{i}$) to a negative binomial distribution, where 0 ≤ ${\omega }_{i}$  ≤ 1. Based on the probability function of the zero-modified distribution, then the probability mass function for the ZINB regression model is:

$$ {\text{P}}({\text{Y}}_{{\text{i}}} = y_{i} ) = \left\{ \begin{aligned} & \omega_{i} + \left( {1 - \omega_{i} } \right) \left( {\frac{\phi }{{\mu_{i} + \phi }}} \right)^{\phi } { },{ }\quad \quad \qquad \qquad \qquad \quad y_{i} = 0 \hfill \\& \left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y + \phi } \right)}}{{{\Gamma }\left( {{\text{y}} + 1} \right){\Gamma }\left( \phi \right)}}\left( {\frac{\phi }{{ \mu_{i} + \phi }}} \right)^{\phi } \left( {\frac{{\mu_{i} }}{{\mu_{i} + \phi }}} \right)^{{y_{i} }} ,\quad y_{i} = 1,2,3 \ldots \hfill \\ \end{aligned} \right. $$

(3)

where ${\phi }^{-1}$, µ_i and $\Gamma \left(.\right)$ representing dispersion, mean, and gamma function respectively. The mean and variance of the zero-inflated negative binomial regression model are:

$${\text{E }}(y_{i} ) = \left( {1 - \omega _{i} } \right)\mu _{i} {\text{ and Var}}(y_{i} ) = \left( {1 - \omega _{i} } \right)\mu _{i} \left( {1 + ~\emptyset \mu _{i} + \omega _{i} \mu _{i} } \right)$$

Assume that there are predictors for logistic regression function and negative binomial regression function. Hence, the ZINB regression model can be written as follow:

$$ {\text{Logit }}(\omega_{i} ) = x_{i}^{T} \beta {\text{ and Log }}(\mu_{i} ) = Z_{i}^{T} \gamma $$

(4)

where β are the vector coefficients ${X}_{i}^{T}$ and $\gamma$ are the vector coefficients ${Z}_{i}^{T}.$ The standard estimation technique for the ZIP and the ZINB is based on the maximum likelihood estimation. The Newton–Raphson algorithm can be used to maximize the log-likelihood functions [12].

Hurdle model

A hurdle model is a class of statistical models where a random variable is modelled using two parts, the first which is the probability of attaining value zero (a point mass at zero), and the second part models the probability of the non-zero values (a distribution that generates non-zero counts) from a zero-truncated distribution. The use of hurdle models is often motivated by an excess of zeros in the data, that is not sufficiently accounted for in more standard statistical models [4]. All zeros in the hurdle model are assumed to be “structural” zeros, i.e., they are generated from a single process, and are observed since the condition is absent. We explore two zero-truncated count distributions for the hurdle model specification [18]. The Hurdle Model of count data can be expressed as follows for the Poisson and Negative Binomial distribution. We consider a Poisson hurdle regression model in which the response variable y has the distribution:

$$ {\text{P}}\left( {{\text{Y}}_{{\text{i}}} = {\text{y}}_{{\text{i}}} } \right) = \left\{ {\begin{array}{*{20}l} {\omega_{i} , \quad \qquad \qquad \qquad y_{i = 0} } \\ {\left( {1 - \omega_{i} } \right)\frac{{e^{{ - \mu_{i} }} \mu_{i}^{{y_{i} }} }}{{y_{i !} \left( {1 - e^{{ - \mu_{i} }} } \right)}} , \quad y_{i} = 1,2,3, \ldots } \\ \end{array} } \right. $$

(5)

A negative binomial hurdle distribution is given by:

$$ {\text{P}}\left( {{\text{Y}}_{{\text{i}}} = {\text{y}}_{{\text{i}}} } \right) = \left\{ {\begin{array}{*{20}l} {\omega_{i} ,\qquad \qquad \qquad \qquad \qquad \qquad \quad y_{i} = 0 } \\ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y_{i} + \phi } \right)}}{{{\Gamma }\left( {y_{i} + 1} \right){\Gamma }\left( \phi \right)}} \frac{{\mu^{{y_{i} }} \phi^{{y_{i} }} \left( {1 + \mu \phi } \right)^{{ - \left( {y_{i} + \phi } \right)}} }}{{1 - \left( {1 + \emptyset \mu } \right)^{\phi } }} ,\quad y_{i} = 1,2,3 \ldots } \\ \end{array} } \right. $$

(6)

where $\phi$ ≥ 0 is a dispersion parameter that is assumed not to depend on covariates. Zero and truncated hurdle model:

$$ {\text{Logit }}(\omega_{i} ) = x_{i}^{T} \beta {\text{ and Log }}(\mu_{i} ) = Z_{i}^{T} \gamma $$

(7)

where β are the vector coefficients ${X}_{i}^{T}$ and $\gamma$ are the vector coefficients ${Z}_{i}^{T}.$ The parameter $\varnothing$ is a measure of dispersion.

Tests for ZIP and ZINB regression models

The test of overdispersion in ZIP regression model against ZINB alternatives, H₀: a = 0 vs. H₁: a > 0, can be performed using likelihood ratio test (LRT), T = 2(ln L1−ln L0), where lnL₁ and lnL₀ are the models ‘log-likelihood under their respective hypotheses. Since the null hypothesis is on the boundary of parameter space, the LRT statistic is asymptotically distributed as a half of probability mass at zero and a half of chi-square with one degree of freedom. The Maximum Likelihood Estimation (MLE) method is used to estimate parameters in the count models. This study includes ZIP, ZINB, Hurdle Poisson, and NBLH to accommodate the excess zeros for the number of cigarettes smoked per day count data. In this paper, Akaike’s information criteria (AIC) and log‐likelihood values are used for model selection measures. It is also used dispersion parameters to test for overdispersion. AIC and log‐likelihood are basic methods for assessing the performance of the models and model selection [12]. Feature Selection plays a very important role in machine learning algorithms. The feature selection of techniques tunes certain parameters to select only few features which are most essential and relevant and far away from the redundant information [21]. But in this article, one of important count regression variable selection by using forward variable selection algorithm.

Bayesian ZINB and negative binomial–logit hurdle model

The number of cigarette smokers daily is a count variable. For the modeling of count data, two-part models are applied in the presence of excessive zeros. Therefore, for a better fit an over-dispersed model that incorporates excessive zeros, i.e., the ZINB regression model is used. If the data is under dispersed, the ZINB model should not be used. The hurdle model is also flexible and can handle both under-dispersion and over-dispersion problem. The NBLH model is used on data with either excessive zero counts in the response or at times too few zero counts. In the case where there are too few zero counts, a zero-inflated model cannot be used. The hurdle model is a good way to deal with such data [18]. It uses two-part. The first part estimates zero elements the dependent variable are zero hurdle model and the second part estimates not zero elements (non-negative integer) from the dependent variable is called truncated negative binomial models [22]. Suppose that Y₁,..., Y_n is a random sample from either ZINB or NBLH distribution. Then the probability mass functions of ZINB or NBLH regression models are given respectively by:

$$ {\text{P}}({\text{Y}}_{{\text{i}}} = ) = y_{i} \left\{ {\begin{array}{*{20}l} {\omega_{i} + \left( {1 - \omega_{i} } \right) \left( {\frac{\phi }{{\mu_{i} + \phi }}} \right)^{\phi } , \qquad \quad \qquad \quad y_{i} = 0} \\ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y + \phi } \right)}}{{{\Gamma }\left( {{\text{y}} + 1} \right){\Gamma }\left( \phi \right)}}\left( {\frac{\phi }{{ \mu_{i} + \phi }}} \right)^{\phi } \left( {\frac{{\mu_{i} }}{{\mu_{i} + \phi }}} \right)^{{y_{i} }} , \quad y_{i} = 1,2,3 \ldots } \\ \end{array} } \right. $$

(8)

$$ {\text{P}}\left( {{\text{Yi}} = {\text{yi}}} \right) = \left\{ {\begin{array}{*{20}l} {\omega_{i} ,\qquad \qquad \qquad \qquad \qquad \qquad \quad y_{i} = 0 } \\ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y_{i} + \phi } \right)}}{{{\Gamma }\left( {y_{i} + 1} \right){\Gamma }\left( \phi \right)}} \frac{{\mu^{{y_{i} }} \phi^{{y_{i} }} \left( {1 + \mu \phi } \right)^{{ - \left( {y_{i} + \phi } \right)}} }}{{1 - \left( {1 + \emptyset \mu } \right)^{\phi } }} , \quad y_{i} = 1,2,3 \ldots } \\ \end{array} } \right. $$

(9)

where ϕ μ, and Γ(.) representing the dispersion parameter, mean, and gamma function, respectively. The most natural choice to model the probability of excess zeros is to use the logit link function and a negative binomial and truncated negative binomial model with log link function respectively.

$$ {\text{Logit }}(\omega_{i} ) \, = X_{i}^{T} \beta {\text{ and Log }}(\mu_{i} ) = Z_{i}^{T} \gamma $$

(10)

where β are the vector coefficients ${X}_{i}^{T}$ and $\gamma$ are the vector coefficients ${Z}_{i}^{T}.$ The parameter $\varnothing$ is a measure of dispersion. When $\varnothing$ = 0, the NBLH and ZINB model reduces to the Poisson regression model. For $\varnothing$ > 0, the NBLH and ZINB models can be used to fit over dispersed count data. When $\varnothing$ < 0, the NBLH model can be used to fit under dispersed count data. Suppose that Y₁,..., Y_n is a random sample from either a ZINB or NBLH distribution. Then the likelihood functions of the two models are given respectively by

$$ L_{1} \left( {\beta ,\gamma ,\emptyset } \right) = \mathop \prod \limits_{i = 1}^{n} \left[ {I\left( {y_{i} = 0} \right)\left\{ {\omega_{i} + \left( {1 - \omega_{i} } \right) \left( {\frac{\phi }{{\mu_{i} + \phi }}} \right)^{\phi } } \right\} + I\left( {y_{i} > 0} \right){ }\left\{ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y + \phi } \right)}}{{{\Gamma }\left( {{\text{y}} + 1} \right){\Gamma }\left( \phi \right)}}\left( {\frac{\phi }{{ \mu_{i} + \phi }}} \right)^{\phi } \left( {\frac{{\mu_{i} }}{{\mu_{i} + \phi }}} \right)^{{y_{i} }} } \right\}} \right] $$

(11)

$$ L_{2} \left( {\beta ,\gamma ,\emptyset } \right) = \mathop \prod \limits_{i = 1}^{n} \left[ {I\left( {y_{i} = 0} \right)\{ \omega_{i} { }\} + I\left( {y_{i} > 0} \right){ }\left\{ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y_{i} + \phi } \right)}}{{{\Gamma }\left( {y_{i} + 1} \right){\Gamma }\left( \phi \right)}} \frac{{\mu^{{y_{i} }} \phi^{{y_{i} }} \left( {1 + \mu \phi } \right)^{{ - \left( {y_{i} + \phi } \right)}} }}{{1 - \left( {1 + \emptyset \mu } \right)^{\phi } }}} \right\}} \right] $$

(12)

Let β and γ are the set of parameters for the above-mentioned model. We assume independent priors for these parameters. Since there is no prior information from historical data or previous experiments, then all parameters will use conjugate non-informative priors. The prior distribution for β and γ is assumed to be normal, while ϕ is assumed to be gamma-distributed. So, the joint prior distribution for either ZINB or NBLH regression parameters is:

$$ f\left( {\beta ,\gamma ,\emptyset } \right) = f\left( \beta \right) \times f\left( \gamma \right) \times f\left( \emptyset \right) $$

$$ f\left( {\beta ,\gamma ,\emptyset } \right) = \mathop \prod \limits_{j = 1}^{p} \left[ {\frac{1}{{\sigma_{{\beta_{j } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\beta_{j} - \mu_{{\beta_{j} }} } \right]^{2} }}{{2\sigma_{{\beta_{j} }}^{2} }}}} } \right] \times \mathop \prod \limits_{k = 1}^{p} \left[ {\frac{1}{{\sigma_{{\gamma_{k } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\gamma_{k} - \mu_{{\gamma_{k} }} } \right]^{2} }}{{2\sigma_{{\gamma_{k} }}^{2} }}}} } \right] \times \frac{1}{{b^{a} {\Gamma }\left( {\upalpha } \right)}}\emptyset^{a - 1} e^{{ - \frac{\emptyset }{b}}} $$

(13)

where $\varnothing$~Gamma (a, b) with a = 0.001 and b = 0.001. but our a priori judgment was that knowledge of the slope parameter ${\varvec{\upgamma}}$ does not provide any information about $\beta$. Given that the prior distribution for parameters has been assessed, the next procedure is to combined the likelihood function prior to performing Bayesian inference. The posterior distribution can be written as

$$ {\text{P}}(\theta /y) \propto L\left( {\theta ;y} \right)p\left( \theta \right) $$

where $\theta$ = ($\beta ,\gamma , \varnothing,$), $L(\theta ;y,X)$ are the likelihood function and $p\left(\theta \right)$ is the prior distribution. Then the fully conditional posterior distributions of parameters on the ZINB are given by:

$$ \begin{aligned} & P_{1} \left( {{\upbeta },{\upgamma },{ }\emptyset ,{\text{y X}}} \right) \propto \mathop \prod \limits_{i = 1}^{n} \left[ {\begin{array}{*{20}l} {I\left( {y_{i} = 0} \right)\{ \omega_{i} + \left( {1 - \omega_{i} } \right) \left( {\frac{\phi }{{\mu_{i} + \phi }}} \right)^{\phi } \} + } \\ {I\left( {y_{i} > 0} \right) \left\{ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y + \phi } \right)}}{{{\Gamma }\left( {{\text{y}} + 1} \right){\Gamma }\left( \phi \right)}}\left( {\frac{\phi }{{ \mu_{i} + \phi }}} \right)^{\phi } \left( {\frac{{\mu_{i} }}{{\mu_{i} + \phi }}} \right)^{{y_{i} }} } \right\}} \\ \end{array} } \right] \\ & \; \times \mathop \prod \limits_{j = 1}^{p} \left[ {\frac{1}{{\sigma_{{\beta_{j } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\beta_{j} - \mu_{{\beta_{j} }} } \right]^{2} }}{{2\sigma_{{\beta_{j} }}^{2} }}}} } \right] \times \mathop \prod \limits_{k = 1}^{p} \left[ {\frac{1}{{\sigma_{{\gamma_{k } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\gamma_{k} - \mu_{{\gamma_{k} }} } \right]^{2} }}{{2\sigma_{{\gamma_{k} }}^{2} }}}} } \right] \\ & \; \times \frac{1}{{b^{a} {\Gamma }\left( {\upalpha } \right)}}\emptyset^{a - 1} e^{{ - \frac{\emptyset }{b}}} \\ \end{aligned} $$

(14)

The fully conditional posterior distributions of the parameter on the NBLH are given by ${P}_{2}\left(\upbeta ,\upgamma , \varnothing, \text{y X}\right)$

$$ \begin{gathered} \propto \mathop \prod \limits_{i = 1}^{n} \left[ {I\left( {y_{i} = 0} \right)\{ \omega_{i} { }\} + I\left( {y_{i} > 0} \right){ }\left\{ {\left( {1 - \omega_{i} } \right)\frac{{{\Gamma }\left( {y_{i} + \phi } \right)}}{{{\Gamma }\left( {y_{i} + 1} \right){\Gamma }\left( \phi \right)}} \frac{{\mu^{{y_{i} }} \phi^{{y_{i} }} \left( {1 + \mu \phi } \right)^{{ - \left( {y_{i} + \phi } \right)}} }}{{1 - \left( {1 + \emptyset \mu } \right)^{\phi } }}} \right\}} \right] \hfill \\ \times \mathop \prod \limits_{j = 1}^{p} \left[ {\frac{1}{{\sigma_{{\beta_{j } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\beta_{j} - \mu_{{\beta_{j} }} } \right]^{2} }}{{2\sigma_{{\beta_{j} }}^{2} }}}} } \right] \times \mathop \prod \limits_{k = 1}^{p} \left[ {\frac{1}{{\sigma_{{\gamma_{k } }} \sqrt {2\pi } }}e^{{ - \frac{{\left[ {\gamma_{k} - \mu_{{\gamma_{k} }} } \right]^{2} }}{{2\sigma_{{\gamma_{k} }}^{2} }}}} } \right] \hfill \\ \frac{1}{{b^{a} {\Gamma }\left( {\upalpha } \right)}}\emptyset^{a - 1} e^{{ - \frac{\emptyset }{b}}} \hfill \\ \end{gathered} $$

(15)

This procedure is implemented using the MCMC algorithm. The MCMC method is a general simulation method for sampling from posterior distributions and computing posterior quantities of interest. Each sample depends on the previous one, hence the notion of the Markov chain. The Markov chain method has been quite successful in modern Bayesian computing [23]. The two most common Markov Chain Monte Carlo algorithms are the Metropolis–Hastings and the Gibbs samplers. These two algorithms have many variants and extensions that have been developed. These forms and extensions are more advanced and sometimes more peculiar to some problems. In this section, we discuss in detail the Gibbs sampler together with its variants and extensions. The Gibbs sampling algorithm is a special case of the Metropolis–Hastings algorithm in which parameters are drawn from distributions with a 100% acceptance rate. It is an alternating conditional sampling. The joint posterior distribution is decomposed into a sequence of simpler conditional distributions, where the target is to generate a data point from the conditional distribution of each parameter, conditional on the current values of the other parameter. The Gibbs algorithm implements MCMC methods using the R software.

The convergence of the algorithm

Flexible software for Bayesian analysis of complex statistical models by using MCMC methods. We use these tools to estimate the ZINB and NBLH regression models. MCMC is based on a combination of Markov chain and Monte Carlo estimation which eventually converges to the target distribution (the posterior distribution). If a chain becomes convergent means the produced sample from the target distribution has been obtained correctly. The Markov chain Monte Carlo (MCMC) method is a general simulation method for sampling from posterior distributions and computing posterior quantities of interest. MCMC methods sample successively from a target distribution. Several other aspects of the Markov chain method also contributed to its success. Most importantly, if the simulation algorithm is implemented correctly, the Markov chain is guaranteed to converge to the target distribution [24,25,26,27]. MCMC technique depends on the approximate distribution which is improved by a simulation of each step until a convergence of the posterior distribution is achieved. Appropriate diagnostics such as; the Heidelberger Welch (stationarity test) and Heidelberger-Welch (halfwidth test).

Results

Information on the number of cigarettes smoked per day obtained from a total of 12,645 respondents in Ethiopia was studied. Figure 1 showed the percentage distribution of the number of cigarettes smoked in Ethiopia based on information from 12,645 respondents. In this study, 94.2% of them never were cigarettes smoked per day, whereas 0.6% of cigarettes smoking only once per day, 1.0% of cigarette smoking twice per day, 1.2% of cigarette smoking three times per day and 3.3% of cigarettes smoking at least four per day.

Fig. 1

Histogram of the number of cigarettes smoked per day in Ethiopia

This indicates excess zero outcomes were large in numbers. However large numbers of cigarettes smoked per day were observed less frequently. This leads to a positively skewed distribution. It indicates that the data could be fitted better by a zero-inflated and hurdle model which takes into account excess zeroes. We visualized that an over-dispersion of the response variable. Since the histogram was highly peaked at zero, from this we can state that the overdispersion comes due to an excess zero. Due to a large number of excess zero outcomes, the histogram was highly picked at the very beginning (about the zero values).

Test of overdispersion

The Vuong test was used to test between the pairs of non-nested models. In this study, we compared ZIP versus PLH and ZINB versus NBLH to test if the over-dispersion in the smoke occurrence data was attributable to high frequencies of zero counts (excessive zeros). This to investigate if the excessive zeros were due to two sources (structure and sampling) or only one source (structure). Table 1 summary of the model comparisons based on Vuong’s statistics for the four count regression models explored. States that if the corresponding p-value is bigger than a pre-specified critical value such as 0.05, then one can conclude that the four models fit the data equally well with no preference given to either model. But, if |V| yields a p-value smaller than the thresholds 0.05, then one of the models is better. Therefore, the ZINB, ZIP, NBLH, PLH was chosen as the best model. There was no difference between ZINB and NBLH models, indicating all models the excessive zeros equally well. The ZIP and ZINB models are nested so they can be compared by using the likelihood test for overdispersion to test H₀:$\varnothing$ = 0, which provide evidence for preferring the ZINB over the ZIP (p-value < 2.2e−16).

Table 1 The likelihood ratio test (LRT) and Vuong test among the ZIP, ZINB, PLH, and NBLH models

In Fig. 2 zero-inflated and hurdle models captured almost all zero values. Based on predicted probabilities, the differences in model fit between the four models were remarkable. Still, the ZIP model and the PLH model did not fit the data reasonably well. The NBLH and ZINB models gave a good prediction of zeros. Ninety four percent of the observed zeros were predicted by both models.

Fig. 2

Comparison of the densities of each model fits a–c number of cigarette smoking

Figure 3 illustrates the observed frequency of smoking occurrence in the 12,645-validation data points (bar chart) and predicted frequencies of smoking occurrence from each of the four models. It was clear that the ZINB, PLH, and NBLH model yielded better predictions for both zero counts and positive counts than did the other models.

Fig. 3

Comparison of the densities of each model fits a–d observed and predicted frequencies of smoke occurrence for ZIP, ZINB, PLH, NBLH models

In Table 2 to compare the model improved convergence of model parameters, decreases the deviance and increases the predictive power. we can state that the zero-inflated Poisson model is better than the standard Poisson model, and also the zero-inflated negative binomial model is better then the negative binomial model. Since the ZINB and the NBLH model have the closest AIC values, we can conclude that these models perform best for our data set.

Table 2 Model comparison using AIC and log-likelihood

Results of Bayesian ZINB and NBLH Model

The main goal of this paper was to present a diagnosis of MCMC convergence and to investigate the significant predictors as well as characterizing smoking intensity in Ethiopia. Bayesian approach results, it is needed to check the convergence assessment, which involves checking that the sequence or chain has converged to and provides a representative sample from the posterior distribution. The Heidelberg and Welch diagnostic calculates a test statistic (based on the Cramer-von Mises test statistic) to accept or reject the null hypothesis that the Markov chain Monte Carlo is from a stationary distribution. The diagnostic consists of two parts. In part one; if the null hypothesis is rejected, discard the first 10% of the chain. Repeat until the null hypothesis is accepted or 50% of the chain is discarded. If the test still rejects the null hypothesis, then the chain fails the test and needs to be run longer. If the chain passes the first part of the diagnostic, then it takes the part of the chain not discarded from the first part to test the second part. The halfwidth test calculates half the width of the 95% credible interval around the mean. If the ratio of the half-width and the mean is significant, the chain passes the test. Table 3 shows the Heidelberger and Welch stationarity tests for the Bayesian Markov chain Monte Carlo. It shows the stationarity and convergence during the burn-in period.

Table 3 Heidelberger and Welch Stationarity and half-width tests for the Bayesian chains used in the diagnosis of MCMC

Application of Bayesian negative binomial-logit hurdle model

Bayesian count model coefficients

In Table 4 the estimated Bayesian count model coefficient odds ratio comparing the urban, considering other variables are held constant in the model. The expected number of the smoker who takes manufactured cigarettes was 0.851 times less than for rural to urban, remaining all other variables constant in the model. A smoker affiliated with the Muslim religion smoked on average was 1.237 smoke cigarettes per day times higher than other religions, remaining all other variables constant in the model. The result also revealed that the expected number of smokers who were used chewed chat was 1.247 times higher than people who have not used chewed chat. The expected counts of smokers would be expected to increase by 1.161 times for those who have an occupation to the group who have not occupation.

Table 4 Summary statistic of the posterior distribution of the model parameters

Bayesian-zero hurdle model coefficients

This finding is the estimated Bayesian-zero hurdle regression coefficient’s odds ratio comparing the age group below 24, considering other variables are held constant in the model. Estimated the odds of the non-zero smokers for age 25–34 years and $\ge 35$ years were 2.779 and 4.195 times higher than as compared to age $\le 24$ years respectively. The Zero hurdle model indicated that the estimated odds of the number of non-zero smokers of people who lived in rural was 0.492 times less than those who lived in urban. Additionally, the effects of education level on the smoke of people, we found that education level people were (OR = 1.446; HPD CrI 0.174, 0.573) times more likely to the number of non-zero smokers compare to no educated people. Furthermore, as the level of education level increases, the odds of the number of non-zero smoke also increased by 1.446.

Though Orthodox are found to be highly likely to non-zero smoke than their counterparts. Compared to men who are affiliated with other religious groups, those affiliated with Muslims and other religions are associated with having a lower non-zero smoking probability. The results indicate that people in lower socioeconomic have a higher likelihood of non-zero smoke. Additionally, the people in the higher and middle wealth category are reported to be less likely to non-zero smoke than their counterparts in the lower wealth category.

Regarding the determinants of cigarette smoking intensity, the study finds that single in marital status who non-zero smoke cigarette have a 0.69 times lower likelihood of non-zero smoke quantities of a cigarette than their counterparts. The estimated odds that the number of non-zero smoke with people chewed chat was 8.339 times more as compared people non chewed chat. Media use is found one of the important significant predictors of smoke cigarettes. The estimated number of non-zero smoke cigarettes for people who were used media is about 0.797 times lower than people who were not used. Similarly, the estimated odds of the number of non-zero smoke those have occupation people had decreased by 31.8% as compared to the on the occupation (see Table 4).

Application of Bayesian ZINB model

Bayesian count model coefficients

The finding of this study also revealed that residence had a significant factor in the number of cigarette smokers. The expected number of daily cigarette smokers with rural was 0.844 times less than that of the expected number of cigarette smokers daily for urban smokers while holding all other variables in the model. This means urban smokers and rural smokers are not certain zero. But urban smokers are higher than rural smokers. Besides, the expected number of daily cigarette smokers with have occupation was 1.164 times higher than that of the expected number of cigarette smokers daily for not occupation smokers. Thus, the occupation, the higher the predicted daily. Revealed that religion has a significant factor in the number of cigarette smokers. The expected number of cigarette smokers for Muslims was 1.237 times higher than the expected number of cigarette smokers for other religions controlling other variables in the model. And also, the expected number of cigarette smokers for chewed chat increased by 24.7% as compared to the expected number of cigarette smokers for not chewed chat controlling other variables in the model (see Table 4).

Bayesian zero-inflation model coefficient

The zero-inflated negative binomial part also indicated that the estimated odds that the number of cigarette smokers becomes zero with chewed chat decreased by 87.7% (OR = 0.123; HPD of 95% CrI − 2.315, − 1.875) as compared to not used chewed chat. In other words, the lower the used chewed chat, the less likely the cigarette smokers are a certain zero. As shown place of residence has a significant effect on the probability of being the always not cigarette smokers. The odds of being in the always not cigarette smokers for rural was 2.008 times higher than compared to those in urban centers controlling other variables in the model. The analysis further indicated that the estimated odds the number of cigarette smokers becomes zero with age 25–34 and age ≥ 35 were 0.36 and 0.24 times less than that of age 15–24 respectively. Revealed that education has a significant factor in the probability of being an excess zero cigarette smokers. The odds of being in the always zero cigarette smokers decreased by a factor of 0.685 for educated as compared to not educated holding all other variables constant. According to the findings of this study, the wealth index of the household has a significant influence on the probability of being an excess zero cigarette smokers. The odds of being in the always zero cigarette smokers for medium and rich households were increased by a factor of 1.358 and 2.433 times higher than that of poor households, respectively while holding all other variables in the model constant. As a show that currently working had statistically significant on the number of excess zero cigarette smokers. The estimated odds that the number of cigarette smokers becomes zero with currently working increased by factor 1.501 as compared to not currently working. Similarly, that the estimated odds that the number of cigarette smokers becomes zero in media was 1.26 times that of the not use media (see Table 4).

Discussion

Smoking intensity is defined usually as the number of cigarettes smoked per day. It can be considered as an important factor in establishing many serious smoking-related diseases. Count regression models are the first-line models that may be wanting to determine factors related to smoking intensity as a counter response. As a result, we want to construct a model that may handle the existence of excess zero counts and also the over-dispersed phenomenon. Motivated by these facts, with excess zeros and high variability of non-zero outcomes, the NBLH and ZINB model was found to be better fitted. Therefore, there is no or little difference in AIC between the ZINB model and the NBLH model [28, 29]. However, it should be noted that NBLH which allows for over-dispersion and also accommodates the presence of excess zeros, is more appropriate among all zero-adjusted models [30].

In this article considered several count data models to examine the factors associated with the number of cigarette smokers using a dataset with over-dispersion and inflated with zeros from 2016 EDHS. The NB model has more flexibility to capture additional variability and it fits the highly variable cigarette data better than the Poisson. However, the model underestimated zeros in the data, suggesting a model appropriate for zero-inflated data and hurdle model may be indicated. The dispersion parameters from the ZINB and NBLH were significant, suggesting the equidispersion assumption in the count portions of ZIP and PLH was violated. Due to this, after considering both the zero counts and the enduring overdispersion, the ZINB and NBLH fit the data best according to the AIC and Vuong tests [31].

We used two models to fit the data and computed Bayesian estimates for existing parameters with the non-informative prior. The competing models were the NBLH and ZINB models. We used the Gibbs sampler to obtain samples from the full conditional distributions. The algorithm was run for 10,000 iterations after the initial 5000 iterates were discarded as a burn-in. We computed the posterior means and 95% highest posterior density (HPD) intervals of the regression coefficients and other relevant parameters [8].

The ZINB and NBLH models deal with zero-inflation and over-dispersion at the same time. These models have become a bit popular lately and to analyze the number of cigarettes smoked per day. The best thing about using these models more specific when handling zero-inflation is that they are doing decrease biases that result from the acute non-normality [32]. Interpretation between the Bayesian ZINB and NBLH are similar but with an important distinction, particularly concerning the logit component. Estimates from the NBLH logit suggest increased age was significantly associated with non-zero cigarette smoking, whereas the ZINB logit model suggests age was positively associated with being a structural zero that is, belonging to the non-smokers latent class. Therefore, in this study older age daily smokers had a higher likelihood to smoke cigarettes intensively, with the number of cigarette smokers becomes zero with older age were less than that of age 15–24 respectively. The age increases with an increased number of cigarette smokers in Ethiopia [33]. So, policymakers and programmers need to select and implement interventions reaching elderly smokers.

According to the findings of this study, the wealth index of the household had a significant influence on the probability of being an excess zero cigarette smokers. Zero cigarette smokers for medium and rich households were higher than that of poor households. This finding is consistent with those reported in Ethiopia, Ghana, and Kenya [34,35,36]. Generally, improving the economic status of the community is one of a positive policy direction.

As shown place of residence has a significant effect on the probability of being the always not cigarette smokers. The odds of being always not cigarette smokers for rural was higher than compared to those in urban. This finding is in line with a study done in Ethiopia [34]. This could be explained by the fact the differences in the availability and accessibility of manufactured cigarettes between the urban and rural areas.

The zero-inflated negative binomial part also indicated that the number of cigarette smokers becomes zero with chewed chat was less likely than not used chewed chat. In other words, the lower the used chewed chat, the less likely the cigarette smokers are a certain zero. This finding is similar with the study done in Ethiopia [37]. This study result revealed that tobacco control interventions need to be tailored to address other substance use behaviors.

The association of religion with smoking. Patterns of smoking are known to vary significantly by religion but less is known about how this association is affected by other factors. The link between religion and smoking can vary significantly across different religious communities and must be deployed with careful attention to community norms if it is to be effective. Revealed that religion has a significant factor in the number of cigarette smokers. The number of cigarette smokers for Muslims was higher than the number of cigarette smokers for other [38].

On the other hand, Mass media exposure is found one of the important significant predictors of smoke cigarettes. The estimated number of non-zero smoke cigarettes for people who were used media is lower than people who were not used. The estimated number of non-zero smoke cigarettes for people who were used media is lower than people who were not used. This finding is similar with the study done Nigeran [40]. Entertainment media outlets, for example television, movies, and radio are influential in initiating or sustaining the smoking habits of consumers.

Besides, smokers who had education smoked a higher number of cigarettes than those who were no education level. This is similar with a study done in Ghana and different from studies done in Nepal [36, 41]. In the same way number of cigarette smokers becomes zero with currently working was higher than not currently working.

Strengths and limitations

The main strength of the Bayesian negative binomial logit Hurdle and zero-inflated model used in this case accounts for survey design features such as weighting, clustering and stratification since a failure to account for design features leads to invalid statistical inference such as standard errors and over or under estimation. In general, hurdle and zero-inflated models have a wide range of application, and a Bayesian approach should appeal to investigators seeking more flexible alternatives to classical model-fitting procedures. The data on the number of cigarettes smoked per day was self-reported which may have recall bias and social desirability bias, leading to an underreporting of the rates. Finally, the study used the number of manufactured cigarettes used per day to measure the smoking intensity.

Conclusion

In this article the count regression models are the first-line models that can be used to determine factors associated with the number of cigarette smoking using a set with overdispersion and inflated with zeros from 2016 EDHS. Four count regression models were compared in terms of AIC and voung test. The model comparison identified that ZINB and NBLH models were better fitted for modeling the observed data with excess zeros and overdispersion. In this study, we proved that that the Bayesian negative binomial logit hurdle (NBLH) and zero inflated negative binomial (ZINB) models are more appropriate methods for the analysis of data with an excess of zeros and overdispersion. In estimating the parameters, Bayesian methods were used. Complex calculations using the Bayesian method in the estimation parameters can be solved by the Markov Chain Monte Carlo simulation that can generate random values with the Gibbs-sampling algorithm.

Furthermore, using Bayesian ZINB and NBLH helps to select the most significant factor. The variable: age, residence, education level, internet use, wealth index, marital status, chewed chat, occupation, media were the most determinate factors on the smoking intensity in Ethiopia. It is suggested that future research considers other proportions of zeros and event stage distributions, under dispersion adjustments, different optimization procedures. This research should also be extended to an approach for direct Bayesian marginal inference. The aim is to develop a Bayesian marginalized model for zero-inflated univariate count outcome in the presence of overdispersion.