Bayesian Analysis of Tuberculosis Cases in Bolgatanga Municipality, Ghana–West Africa

Ernest Zamanah; Suleman Nasiru

doi:10.1515/spp-2025-0023

Article Open Access

Bayesian Analysis of Tuberculosis Cases in Bolgatanga Municipality, Ghana–West Africa

Ernest Zamanah and Suleman Nasiru

Published/Copyright: September 19, 2025

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From the journal Statistics, Politics and Policy Volume 16 Issue 3

Abstract

Tuberculosis (TB) represents a major threat to global health security and is currently the world most deadly infectious disease. TB is noted to affect people of all age groups and exposure to the disease has been shown to exhibit variation in the risk of infection across different factors. This study was therefore carried out to identify possible disease associations of some risk factors in relation to time until onset of TB among TB infected individuals in the Bolgatanga Municipality of the Upper East Region of Ghana. To this end, secondary data on time until onset of TB was obtained from the Bolgatanga Regional Hospital of the Upper East Region of Ghana and used for statistical analysis. In order to obtain appropriate underlying disease associations, the Bayesian approach to the modelling of survival time was employed in the statistical analyses. Thus, different parametric models including; skew normal, lognormal, shifted lognormal, exponential, Weibull and Fréchet models were fitted to the TB data within the Bayesian framework. The Weibull was the overall best fitted model. Based on the estimation results of the Bayesian Weibull regression model; individuals with no education, individuals with diabetes and males were identified as high-risk groups with shorter time frame for the onset of TB in the Bolgatanga Municipality. The study was concluded by indicating the need for Ghana Health Service to institute targeted interventions aimed at preventing and controlling the TB with special emphasis to the high-risk groups in the Bolgatanga Municipality.

Keywords: Bayesian framework; posterior distribution; widely applicable information criterion; Weibull model

1 Introduction

Tuberculosis (TB) is a communicable disease that mostly affects the lungs with mycobacterium tuberculosis, a type of bacteria being the causative agent. As an airborne disease, the bacteria are transmitted when infected people with the disease release bacteria into the air through coughing, sneezing or spitting (Kiani et al. 2021).

TB represents a major public health problem worldwide and poses a significant threat to health security. The incidence of tuberculosis has been shown to exhibit an increasing trend since 2021 with recorded cases exceeding 10 million every year. In 2023, the total number of TB recorded cases was about 10.8 million around the globe with 6.0 million cases attributed to men, 3.6 million cases being women and about 1.3 million cases attributed to children (Deku et al. 2024; World Health Organization [WHO] 2024). Globally, 1.25 million mortalities resulted from TB infections with 1.09 million deaths attributed to HIV-negative people and 161,000 deaths associated with HIV-positive people (WHO 2024). Although some level of progress has been made in the fight against TB, TB has replaced COVID-19 as the current global leading cause of death from a single infectious agent after COVID-19 held the position from 2019 to 2021 (Berida and Lindsley 2024; WHO 2024).

Currently, TB is found everywhere in the world. In 2023, the South-East Asia Region accounted for 45 % of all TB cases in the world. The second largest incidence of TB cases occurred in the African Region accounting for 24 % of all cases whilst Western Pacific Region accounted for 17 % (WHO 2024).

TB has been an old age major health related problem in Africa and Ghana in particular. In Africa, TB is noted as one of the top ten causes of death, outranking HIV/AIDS (WHO 2020). In 2021, the incidence of TB in Ghana was shown to be 136 per 100,0000 with a mortality rate of 36 per 100,000 people (WHO 2023). On annual bases, about 44,000 cases of TB are recorded in Ghana with an estimated mortality of 10,000 cases. Noted as the seventh leading cause of mortality in Ghana, the distribution of TB cases has been shown to exhibit regional variations in Ghana with Upper East Region established as one of the regions with an increased likelihood of TB infection (Iddrisu et al. 2024).

Globally, TB affects people of all age groups and exposure to the disease has been shown to exhibit variation in the likelihood of infection across different risk factors. People with disease conditions such as HIV/AIDS or diabetes that compromise the immune system are noted to be at risk of getting infected with TB (Umeta et al. 2022). Some other covariates include alcohol use, cigarette smoking and educational level among others (Umeta et al. 2022).

Considerable progress has been made in relation to the treatment and elimination of TB in Ghana (Deku et al. 2024). However, there is paucity of data in relation to assessing the age of onset of TB across different risk factors in the Bolgatanga Municipality of the Upper East Region of Ghana. Bayesian estimation procedure is a useful method in modelling the time until onset of infectious diseases (Rastegar et al. 2024). Thus, the primary objective of this study was to evaluate possible risk factors associated with the time frame until onset of TB and determine possible disease associations in the Bolgatanga Municipality of the Upper East Region of Ghana by applying the Bayesian approach.

The structure for the rest of the article is as follows: In Section 2, the materials and methods are presented whilst Section 3 outlines the results and discussions. Section 4 finally presents the conclusions and recommendations of the study.

2 Materials and Methods

2.1 Study Design

Secondary data on tuberculosis was obtained for this study through a retrospective cohort study design.

2.2 Study Area

The Bolgatanga Municipality of the Upper East Region of Ghana represented the study area of this work. As the administrative capital of the Upper East Region of Ghana, the Bolgatanga Municipality is home to the Bolgatanga Regional Hospital. The hospital serves as a referral center for all private health facilities within the Municipality and also provides healthcare services to all patients within the Municipality and beyond.

2.3 Study Population

The participants representing the study population included recorded history of tuberculosis patients from 2015 to 2024 at the Bolgatanga Regional Hospital.

2.4 Inclusion and Exclusion Criteria

The medical records of all tuberculosis cases from 2015 to 2024 were included in the study. Medical records of tuberculosis cases were however excluded from the study where information on patients age were unavailable.

2.5 Data Collection

Secondary data on tuberculosis were obtained from the Bolgatanga Regional Hospital of the Upper East Region of Ghana by reviewing the medical histories of all tuberculosis cases recorded from 2015 to 2024. The Records Department of the Bolgatanga Regional Hospital is staffed with qualified and well trained personnel. Thus, the review of the medical records was done by the staff of the Records Department of the hospital. In all, the data were obtained from the records of 343 patients infected with TB in the Bolgatanga Municipality. The time frame representing the age of the patients from birth until onset of tuberculosis was the variable of interest in the study. Other variables such as diabetes status of patients at the time of TB diagnosis, sex of patient, marital status, employment status, religion and educational level were used as independent variables in the study.

2.6 Ethical Approval

The data were provided and approved for its use by the Records Department of the Bolgatanga Regional Hospital upon written request for the data. Personal information and identification numbers of patients were excluded from the data given out for this study. This, ensured confidentiality in the use of the data that were obtained for the study.

2.7 Study Variables

2.7.1 Dependent Variable

Time until the onset of TB represented the response variable for the study. Thus, the age of patient in years from birth until onset of TB was the variable of interest.

Starting time : Starting time is the age at birth of the TB patient.

End time : is the age at which the event occurred. That is, the age of TB diagnosis.

2.7.2 Independent Variables

Sex : As a prognostic factor, sex refers to male and female categories of the patients.

Marital Status : Three categories (married, single and divorced/widowed) were defined for marital status of the patients as an independent variable.

Employment Status : Two categories (Employed and Unemployed) were defined for the employment status of the patients.

Religion : Three categories (Muslim, Christian and Traditionalist) were defined for the religion of patients.

Educational Level : Four categories (No education, Primary, Secondary and Tertiary) were defined for educational level of the patients.

Diabetes Status : For the diabetes status of the patients, two categories (Yes and No) were defined.

2.8 Statistical Analysis

2.8.1 Bayesian Inference

Consider a random sample of independently and identically distributed survival times t = t 1 , t 2 , . . . , t n ′ , with a probability function denoted by P(t _j|η) and survival function denoted by S(t _j). The likelihood function of the sample according to Umeta et al. (2022) given that there are censored observations in the data is defined by

L = ∏ j = 1 n P t j η δ j S t j 1 − δ j ,

where

δ j = 0 , c e n s o r e d 1 , o b s e r v e d

is an indicator variable.

The Bayesian approach utilizes prior distributions of the unknown parameters η of the likelihood function in the modelling process. Before the sample t _j, is observed, the prior distribution gives a reflection of existing knowledge about the parameter space η. Considered as stochastic, appropriate distributions known as prior distributions are assigned to all the parameters η within the Bayesian framework (Blangiardo and Cameletti 2015).

In a posterior inference, a probability distribution known as the posterior distribution is produced by combining the likelihood function of the sample and prior distributions (Blangiardo and Cameletti 2015). Given the observed data, the posterior distribution, denoted by P(η|t _j) is the conditional distribution of the parameters. Defined as the product of the likelihood function and the prior distributions P(η), the posterior distribution is given by

(1) P η t j = P t j η P η ∫ η L t j η P η d η .

Alternatively, the posterior distribution can be expressed as

(2) P η t j ∝ P t j η P η ,

with ∫ η L t j η P η d η being a normalizing constant. By assigning a distribution for each parameter under the assumption of the parameter estimates being stochastic could produce informative priors resulting in misleading statistical inferences (Hahn 2014). Conjugate or non-informative priors allow for more efficiency in the parameter estimates of the posterior distributions within the Bayesian framework (Robert and Rousseau 2023). Thus, non-informative priors were utilized in this study.

To assess the adequacy of the Bayesian model, trace and density plots of the parameter estimates of posterior distributions were utilized through Markov Chain Monte Carlo (MCMC) procedures to determine model convergence in this study.

2.8.2 Bayesian Analysis Using Weibull Model

Consider that the distribution of the random sample of survival times t = t 1 , t 2 , . . . , t n ′ , is that of the Weibull distribution with parameters γ and η. The PDF and cumulative distribution function (CDF) of the Weibull distribution due to Weibull (1951) are given respectively by

(3) f t j = γ η t η γ − 1 e 1 η γ , t j ≥ 0 ,

and

(4) F t j = 1 − e − 1 η γ , t j ≥ 0 .

The survival function of the Weibull distribution is given by

(5) S t j = e − 1 η γ , t j ≥ 0 .

In building the Weibull regression, the log link function is used to introduce covariates to the regression structure such that

E t j = exp W φ ,

where W′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

η Γ 1 + 1 / γ = exp W φ .

Thus,

η = exp W φ Γ 1 + 1 / γ .

This therefore gives the likelihood function as

(6) L = ∏ j = 1 n γ exp w j φ / Γ 1 + 1 / γ × t j exp w j φ / Γ 1 + 1 / γ γ − 1 × e − t exp w j φ / Γ 1 + 1 / γ γ δ j × e − t j exp w j φ / Γ 1 + 1 / γ γ 1 − δ j .

In building the posterior distribution, the prior distributions for the regression coefficients and shape parameter of the model are specified. In the current study, a Gaussian prior was chosen for the regression coefficients such that φ i ∼ N 0 , σ 0 2 with a half Cauchy prior chosen for the shape parameter such that γ∼HC(0,k ₀).

By applying the Bayes theorem, the joint posterior distribution is given by

(7) P γ , φ W , t j ∝ L t j γ , φ , W P γ P φ .

That is,

(8) P γ , φ W , t j ∝ ∏ i = 1 n γ exp w j φ / Γ 1 + 1 / γ × t j exp w j φ / Γ 1 + 1 / γ γ − 1 × e − t j exp w j φ / Γ 1 + 1 / γ γ δ j × e − t j exp w j φ / Γ 1 + 1 / γ γ 1 − δ j × 2 × k 0 P γ 2 + k 0 2 × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.8.3 Bayesian Analysis Using Exponential Model

Consider that the distribution of the random sample of survival times t = t 1 , t 2 , . . . , t n ′ , is that of the exponential distribution with parameter η. The PDF of the exponential distribution as cited by Rather and Rather (2017) and Umeta et al. (2022) is given by

(9) f t j = η exp − η t j , t j ≥ 0 .

The CDF and survival function of the exponential distribution are given respectively by

(10) F t j = 1 − e − η t j , t j ≥ 0 .

and

(11) S t j = e − η t j , t j ≥ 0 .

Using the log link function, the exponential regression is defined such that

E t i = exp W φ ,

where W′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

1 / η = exp W φ .

Thus,

η = 1 exp W φ

= exp − W φ .

This therefore gives the likelihood function as

(12) L = ∏ j = 1 n e − w j φ + t j exp − w j φ δ j × e − t j exp − w j φ 1 − δ j

By choosing a Gaussian prior for the regression coefficients such that φ i ∼ N 0 , σ 0 2 , the joint posterior distribution is obtained by applying the Bayes theorem. This gives

(13) P φ W , t j ∝ L t j φ , W P φ .

That is,

(14) P φ W , t j ∝ ∏ j = 1 n e − w j φ + t j exp − w j φ δ j × e − t j exp − w j φ 1 − δ j × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.8.4 Bayesian Analysis Using Lognormal Model

Consider that the distribution of the random sample of independently and identically distributed survival times t = t 1 , t 2 , . . . , t n ′ , is that of the lognormal with parameters μ and σ. The PDF and CDF of the lognormal distribution as cited in Umeta et al. (2022) are given respectively by

(15) f t j = 2 π − 1 / 2 t j σ − 1 exp − 1 2 σ 2 log t j − μ 2 , μ ∈ R , σ ≥ 0 ,

and

(16) F t j = Φ log t j − μ σ .

The survival function of the lognormal distribution is given by

(17) S t j = 1 − Φ log t j − μ σ .

Using the log link function, the lognormal regression is defined such that

E t i = exp W φ ,

where W′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

e μ + σ 2 / 2 = exp W φ .

Thus,

μ = W φ − σ 2 2

This therefore gives the likelihood function as

(18) L = ∏ j = 1 n 2 π − 1 / 2 t j σ − 1 exp − 1 2 σ 2 log t j + σ 2 / 2 − W φ 2 δ j × 1 − Φ log t j + σ 2 / 2 − W φ σ 1 − δ j

By choosing a Gaussian prior for the regression coefficients such that φ i ∼ N 0 , σ 0 2 and a half Cauchy prior for the scale parameter such that σ∼HC(0,k ₀), the joint posterior distribution is obtained by applying the Bayes theorem. This gives

(19) P σ , φ W , t j ∝ L t j σ , φ , W P σ P φ .

That is,

(20) P σ , φ W , t j ∝ ∏ j = 1 n 2 π − 1 / 2 t j σ − 1 exp − 1 2 σ 2 log t j + σ 2 / 2 − W φ 2 δ j × 1 − Φ log t j + σ 2 / 2 − W φ σ 1 − δ j × 2 × k 0 P σ 2 + k 0 2 × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.8.5 Bayesian Analysis Using Skew Normal Model

Consider that the distribution of the random sample of independently and identically distributed survival times t = t 1 , t 2 , . . . , t n ′ , is that of the skew normal with parameters μ, σ and γ. The PDF of the skew normal distribution due to Azzalini (1985) as cited in Wei et al. (2025) is given by

(21) f t j = 2 σ ϕ t j − μ σ Φ γ t j − μ σ , μ , γ ∈ R , σ > 0 ,

where Φ(⋅) and ϕ(⋅) represent the CDF and PDF of the standard normal distribution respectively. The CDF is

and

(22) F t j = Φ t j − μ σ − 2 T t j − μ σ , γ ,

where T(u,v) is Owen’s T function.

The survival function of the skew normal distribution is given by

(23) S t j = 1 − Φ t j − μ σ − 2 T t j − μ σ , γ .

Using the log link function, the skew normal regression is defined such that

E t i = exp W φ ,

where W′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

μ + γ σ 1 + γ 2 2 π = exp W φ .

Thus,

μ = exp W φ − γ σ 1 + γ 2 2 π .

This therefore gives the likelihood function as

(24) L = ∏ j = 1 n 2 σ ϕ t j − exp W φ + γ σ 1 + γ 2 2 π σ Φ γ t j − exp W φ + γ σ 1 + γ 2 2 π σ δ j × 1 − Φ t j − exp W φ + γ σ 1 + γ 2 2 π σ − 2 T t j − exp W φ + γ σ 1 + γ 2 2 π σ , γ 1 − δ j

By choosing a Gaussian prior for the regression coefficients such that φ i ∼ N 0 , σ 0 2 , a half Cauchy prior for the parameters (σ,γ) such that σ∼HC(0,k ₀) and γ∼HC(0,k ₁), the joint posterior distribution is obtained by applying the Bayes theorem. This gives

(25) P σ , γ , φ W , t j ∝ L t j σ , γ , φ , W P σ P γ P φ .

That is,

(26) P σ , φ W , t j ∝ ∏ j = 1 n 2 σ ϕ t j − exp W φ + γ σ 1 + γ 2 2 π σ Φ γ t j − exp W φ + γ σ 1 + γ 2 2 π σ δ j × 1 − Φ t j − exp W φ + γ σ 1 + γ 2 2 π σ − 2 T t j − exp W φ + γ σ 1 + γ 2 2 π σ , γ 1 − δ j × 2 × k 0 P σ 2 + k 0 2 × 2 × k 1 P γ 2 + k 1 2 × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.8.6 Bayesian Analysis Using Shifted Lognormal Model

Consider that the distribution of the random sample of independently and identically distributed survival times t = t 1 , t 2 , . . . , t n ′ , is that of the shifted lognormal with parameters μ, σ and λ. The PDF of the shifted lognormal distribution due to Aitchison and Brown (1969) as cited in Dabet et al. (2018) is given by

(27) f t j = 1 σ t j − λ 2 π e ln t j − λ − μ 2 / 2 σ 2 , μ , σ > 0 , t j ≥ λ .

The CDF of the shifted lognormal distribution is given by

(28) F t j = 1 2 1 − sgn − ln t j − λ + μ / σ erf − ln t j − λ + μ / σ / 2 .

The survival function of the shifted lognormal distribution is given by

(29) S t j = 1 − 1 2 1 − sgn − ln t j − λ + μ / σ erf − ln t j − λ + μ / σ / 2 .

Using the log link function, the shifted lognormal regression is defined such that

E t j = exp W φ ,

where W′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

exp μ = exp W φ / 1 + exp σ 2 − 1 exp σ 2 .

Thus,

μ = W φ − σ 2 .

This therefore gives the likelihood function as

(30) L = ∏ j = 1 n 1 σ t j − λ 2 π e ln t j − λ − W φ + σ 2 2 / 2 σ 2 δ j × 1 − 1 2 1 − sgn − ln t j − λ + W φ − σ 2 / σ erf − ln t j − λ + W φ − σ 2 / σ / 2 1 − δ j

By choosing a Gaussian prior for the regression coefficients such that φ i ∼ N 0 , σ 0 2 , a half Cauchy prior for the parameters (σ,λ) such that σ∼HC(0,k ₀) and λ∼HC(0,k ₁), the joint posterior distribution is obtained by applying the Bayes theorem. This gives

(31) P σ , λ , φ W , t j ∝ L t j σ , λ , φ , W P σ P λ P φ .

That is,

(32) P σ , φ W , t j ∝ ∏ j = 1 n 1 σ t j − λ 2 π e ln t j − λ − W φ + σ 2 2 / 2 σ 2 δ j × 1 − 1 2 1 − sgn − ln t j − λ + W φ − σ 2 / σ erf − ln t j − λ + W φ − σ 2 / σ / 2 1 − δ j × 2 × k 0 P σ 2 + k 0 2 × 2 × k 1 P λ 2 + k 1 2 × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.8.7 Bayesian Analysis Using Fréchet Model

Consider that the distribution of the random sample of survival times t = t 1 , t 2 , . . . , t n ′ , is that of the Fréchet distribution with parameters α > 0 and λ > 0. The PDF and CDF of the Fréchet distribution due to Fréchet (1927) as cited in Zamanah et al. (2022) are given respectively by

(33) f t j = α λ t j − λ − 1 e − α t j − λ , t j > 0 .

and

(34) F t j = e − α t j − λ .

The survival function of the Fréchet distribution is given by

(35) S t j = 1 − e − α t j − λ .

Using the log link function, the Fréchet regression is defined such that

E t i = exp W φ ,

where W ^′ is a k × 1 vector of explanatory variables and φ is a k × 1 vector of regression coefficients. That is,

E t j = exp W φ .

Further,

α Γ 1 − 1 / λ = exp W φ .

Thus,

α = exp W φ Γ 1 − 1 / λ .

This therefore gives the likelihood function as

(36) L = ∏ j = 1 n λ t j − λ − 1 e w j φ − t j − λ exp w j φ / Γ 1 − 1 / λ Γ 1 − 1 / λ δ j × e − t j − λ exp w j φ / Γ 1 − 1 / λ 1 − δ j .

By choosing a Gaussian prior for the regression coefficients such that φ i ∼ N 0 , σ 0 2 and a half Cauchy prior for the shape parameter such that λ∼HC(0, k ₀), the joint posterior distribution is obtained by applying the Bayes theorem. This gives

(37) P λ , φ W , t j ∝ L t j λ , φ , W P λ P φ .

That is,

(38) P γ , φ W , t j ∝ ∏ j = 1 n λ t j − λ − 1 e w j φ − t j − λ exp w j φ / Γ 1 − 1 / λ Γ 1 − 1 / λ δ j × e − t j − λ exp w j φ / Γ 1 − 1 / λ 1 − δ j × 2 × k 0 P λ 2 + k 0 2 × ∏ i = 1 I 1 σ o 2 π exp − 1 2 × σ 0 2 φ i 2 .

2.9 Model Selection Criteria

To determine the model of best fit to the TB data, leave one out cross-validation (LOO-CV) criterion and the widely applicable information criterion (WAIC) were used in the study. The LOO-CV due to Vehtari et al. (2017) is a method that utilizes the log-likelihood evaluated at the posterior simulations of the parameter values to estimate pointwise out-of-sample prediction accuracy from a fitted Bayesian model. The LOO-CV is given as

(39) elpd loo = ∑ j = 1 n log p w j w − j ,

where

p w j w − j = ∫ p w j φ p φ w − d φ

is the LOO predictive density for data point w _j given the remaining data points w _-j and φ represents the parameter space of the model.

As an information-based model selection criterion, the WAIC is given by

(40) WAIC = − 2 ∑ j = 1 n log p p o s t w j + 2 p D ,

where ∑ j = 1 n log p p o s t w j is the sum of predictive density for each data point and pD is the effective number of parameters (Umeta et al. 2022). The model with least values of LOO-CV and WAIC represents the best model (Farhin and Khan 2022).

3 Results and Discussions

3.1 Summary Statistics

Figure 1 presents the histogram and kernel density plots of the time until onset of TB. The results show that the ages of the patients until onset of TB are skewed to the right. That is, majority of the TB patients experienced the onset of TB at a younger age on average.

Figure 1:

Histogram and kernel density plots of the tuberculosis data.

Table 1 shows the descriptive statistics illustrating the distribution of the patients across various categories of the predictor variables. Table 1 shows that among the different sexes, TB infections were high among males accounting for 72.3 % of all the infections. Also, majority of the patients representing 64.1 % were married followed by 30.3 % of the patients being single. Further, the section of the patients who were unemployed represented a majority of 70.0 % compared to the employed group. Across the various religions, majority of the TB infections representing 56.6 % were shown to be among Christians. Across various educational levels also, the level of TB infection was higher among those with no form of formal education accounting for 43.1 % of the infections. Finally, the descriptive statistics showed diabetes as a comorbidity among 27.4 % of the patients infected with TB.

Table 1:

Descriptive statistics of patients based on some predictor variables.

Variable	Category	Frequency (%)
Sex	Female	95 (27.7 %)
Sex	Male	248 (72.3 %)
Marital status	Divorced/Widowed	19 (5.5 %)
	Married	220 (64.1 %)
	Single	104 (30.3 %)
Employment status	Unemployed	103 (30.0 %)
Employment status	Employed	240 (70.0 %)
Religion	Christian	194 (56.6 %)
	Muslim	30 (8.7 %)
	Traditional	119 (34.7 %)
Educational level	No education	148 (43.1 %)
	Primary	58 (16.9 %)
	Secondary	108 (31.5 %)
	Tertiary	29 (8.5 %)
Diabetes status	Yes	94 (27.4 %)
Diabetes status	No	249 (72.6 %)

3.2 Model Selection Results

To evaluate the effects of the predictor variables on the time until onset of TB in the Bolgatanga Municipality, Bayesian parametric models were fitted to the TB data and their performances compared using the LOO-CV criterion and WAIC. Table 2 shows the accuracy measures of the fitted Bayesian regression models. The results showed the Weibull model as the best fitted model with least values of LOO-CV and WAIC.

Table 2:

Model accuracy results on the tuberculosis data.

Model	LOO estimates		WAIC
Model	Elpd-diff	se-diff	WAIC
Weibull	0.0000	0.0000	2,643.6000
Lognormal	−10.4000	7.5000	2,664.4000
Shifted lognormal	−11.7000	8.0000	2,666.9000
Skew normal	−31.0000	9.0000	2,705.5000
Fréchet	−56.3000	14.4000	2,754.5000
Exponential	−317.1000	12.9000	3,277.8000

Bolded means best based on selection criteria.

Table 3 shows the parameter estimates of the posterior distribution for the Weibull regression model. Among the predictors, the results showed sex, religion, educational level and diabetes status to be statistically significant at 95 % confidence level.

Table 3:

Parameter estimates of Weibull Bayesian regression on the tuberculosis data.

Parameter		Estimates	Standard error	95 % Credible interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		3.86	0.07	(3.74, 4.00)	1.00	5,513	4,212
Sex	Male	−0.26	0.04	(−0.33, −0.19)	1.00	6,392	4,596
Sex	Female (Ref)
Marital status	Married	0.01	0.06	(−0.11, 0.12)	1.00	5,582	4,343
	Single	−0.09	0.06	(−0.22, 0.02)	1.00	5,361	3,992
	Divorced (Ref)
Employment status	Unemployed	−0.04	0.03	(−0.10, 0.02)	1.00	6,760	4,359
Employment status	Employed (Ref)
Religion	Muslim	0.02	0.05	(−0.01, 0.18)	1.00	7,320	4,139
	Traditional	0.17	0.03	(0.10, 0.21)	1.00	7,416	4,609
	Christian (Ref)
Educational level	Primary	0.22	0.04	(0.14, 0.29)	1.00	6,925	4,878
	Secondary	0.22	0.03	(0.15, 0.28)	1.00	6,880	5,101
	Tertiary	0.28	0.05	(0.18, 0.38)	1.00	6,976	4,966
	None (Ref)
Diabetes status	Yes	−0.15	0.03	(−0.21, −0.09)	1.00	7,099	4,315
Diabetes status	No (Ref)

3.3 Model Diagnostic Checks

This section investigates the adequacy of the fitted Weibull Bayesian regression model to determine whether there was satisfactory model convergence. Figures 2–5 show the trace and density plots of the parameters of the fitted Weibull Bayesian regression model. The results showed an overlay of the chains illustrating a good mixing of the three chains for all the parameters. The results confirm that the Monte Carlo Markov Chain simulation converged satisfactorily to the target posterior distribution in the estimation of the parameters. The results in Table 3 also support this finding with the Rhat values being 1 for all the parameters of the model.

Figure 2:

Density and Trace plot of Weibull Bayesian regression parameters.

Figure 3:

Density and Trace plot of Weibull Bayesian regression parameters.

Figure 4:

Density and Trace plots of some parameters for Weibull Bayesian regression parameters.

Figure 5:

Density and Trace plot of Weibull Bayesian regression parameters.

Further, the posterior predictive plot in Figure 6 shows an overlap of the posterior and replicated distributions having the same trend. The results confirm that the posterior predictive capability of the Weibull regression model was adequate in modelling the time until onset of TB in the Bolgatanga Municipality.

Figure 6:

Posterior predictive plot of Weibull Bayesian regression.

3.4 Discussions

The time frame for onset of TB in the Bolgatanga Municipality was shown to be skewed to the right from the summary results of the study thereby making classical regression analysis inappropriate for application. Also, the summary results showed that 27.4 % of the patients infected with TB had diabetes as a comorbidity.

As a robust modelling approach, the Bayesian framework was utilized in modelling time until onset of TB as a response variable in this study. Varied Bayesian parametric models that were empirically evaluated included: skew normal, lognormal, shifted-lognormal, exponential, Weibull and Fréchet Bayesian regression models. The Weibull Bayesian regression model was the best performing model with least values of LOO-CV and WAIC.

The posterior estimation results of the Weibull Bayesian regression identified sex, religion, educational level and diabetes status as statistically significant prognostic factors at 95 % confidence level. However, marital status and employment status showed no statistical significance.

Thus, at the 95 % confidence level, the results illustrated a statistically significant difference in the time frame for the onset of TB between males and females. Specifically, it was empirically established that the time frame for onset of TB is shorter in males than females. This shows that men get TB at a younger age than women. Thus, males have a higher risk of getting infected with TB at a younger age than that of females in the Bolgatanga Municipality. This observation is consistent with Humayun et al. (2022) that males have a higher risk of getting infected with TB at a younger age than that of females.

Again, the findings showed a statistically significant difference in the time frame for the onset of TB between Christians and Traditionalists. The findings specifically confirmed that Traditionalists have a longer time frame for the onset of TB compared to Christians. Between Christians and Muslims however, the results showed no significant difference in the time frame for the onset of TB. The difference could be as a result of the mass gatherings associated with both Christian and Muslim worship compared to Traditional worship. This finding is consistent with the observations of other studies such as Hoang and Gautret (2018) and Yezli et al. (2023) that mass gathering, as practiced by Muslims and Christians, is associated with an increased risk of TB infection.

Across different educational levels, the findings indicated a statistically significant difference in the time frame for the onset of TB between individuals with no form of formal education and those with higher educational levels. Specifically, the results confirmed that individuals with higher educational levels (primary, secondary and tertiary) have a longer time frame for the onset of TB compared to those with no formal education. That is, individuals with lower levels of education get infected with TB at a younger age compared to individuals with higher levels of education. This finding is consistent with Medhi and Mahanta (2008) that individuals with lower levels of education get infected with TB at a younger age compared to individuals with higher levels of education.

The findings further established a statistically significant difference in the time frame for the onset of TB between individuals infected with diabetes and individuals without diabetes. Specifically, the findings confirmed that individuals infected with diabetes have a shorter time frame for the onset of TB compared to individuals without diabetes. The finding is consistent with Boadu et al. (2024) and Buasroung et al. (2022) that individuals with diabetes have a higher risk of TB infection.

4 Conclusions and Recommendations

4.1 Conclusions

The study was designed to model the time until onset of TB in the Bolgatanga Municipality of the Upper East Region of Ghana and identify possible disease associations of some risk factors. To this end, skew normal, lognormal, shifted lognormal, exponential, Weibull and Fréchet models were fitted to the TB data using the Bayesian procedure. The Weibull model was established as the best performing model in comparison with the other models. Based on the posterior estimation results of the Weibull model, sex, religion, educational level and diabetes status were established as observed characteristics that significantly influence the time frame for the onset of TB in the Bolgatanga Municipality. Specifically, it was established that males have a shorter time frame for the onset of TB compared to females. It was also established that Traditionalists have a longer time frame for the onset of TB compared to Christians. Between Christians and Muslims, no significant difference was however detected in the time frame for the onset of TB in the Bolgatanga Municipality. Across different educational levels, it was established that individuals with higher educational levels have a longer time frame for the onset of TB compared to individuals with no education. Again, it was established that individuals with diabetes have a shorter time frame for the onset of TB compared to individuals without diabetes.

4.2 Recommendations

Based on the findings of this study, it is recommended for the Ghana Health Service and other healthcare agencies within the Muncipality to institute advocacy programmes on; the need for lifestyle modifications, raising awareness about TB infections and the need for early screening and treatment. This will get the public well informed and help in the prevention of TB infections. Education on the need for lifestyle modifications can help prevent conditions such as diabetes and contribute in reducing the risk of TB infection in the Municipality. As a high risk group, it is also recommended for males to consider frequent screening for early detection and treatment of TB in the Bolgatanga Municipality.

4.3 Study Limitations

The use of secondary data limited the choice of variables in the study. Consequently, important variables such as HIV, hypertension and alcohol usage which could influence TB infection could not be captured in the current study. Nonetheless, the findings of this study are useful and have policy implications in the Bolgatanga Municipality.

Corresponding author: Ernest Zamanah, Department of Biometry, School of Mathematical Sciences, C. K. Tedam University of Technology and Applied Sciences, Navrongo, Ghana, E-mail: ezamanah@yahoo.com

Appendix

See Appendix Tables 1–5.

Table 1:

Parameter estimates of Bayesian skew normal regression on the tuberculosis data.

Parameter		Estimates	Standard error	95 % Credible interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		48.26	3.18	(41.82, 54.36)	1.00	5,926	4,413
Sex	Male	−4.86	2.08	(−9.20, −1.10)	1.00	5,104	4,127
Sex	Female (Ref)
Marital status	Married	−3.18	2.95	(−8.86, 2.76)	1.00	4,583	4,431
	Single	−9.24	23.11	(−15.17, −3.03)	1.00	4,775	4,254
	Divorced (Ref)
Employment status	Unemployed	−1.97	1.51	(−4.92, 1.06)	1.00	8,459	3,633
Employment status	Employed (Ref)
Religion	Muslim	0.91	2.31	(−3.73, 5.50)	1.00	7,303	4,457
	Traditional	5.21	1.48	(2.31, 8.13)	1.00	7,299	4,561
	Christian (Ref)
Educational level	Primary	8.00	1.98	(4.09, 11.86)	1.00	7,062	5,058
	Secondary	6.57	1.64	(3.46, 9.87)	1.00	5,717	5,034
	Tertiary	11.29	2.53	(6.35, 16.22)	1.00	6,601	4,805
	None (Ref)
Diabetes status	Yes	−5.76	1.54	(−8.87, −2.78)	1.00	8,111	4,629
Diabetes status	No (Ref)

Table 2:

Parameter estimates of Bayesian lognormal regression on the tuberculosis data.

Parameter		Estimates	Standard error	95 % Credible interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		3.84	0.08	(3.69, 4.00)	1.00	5,883	4,308
Sex	Male	−0.16	0.04	(−0.23, −0.09)	1.00	6,930	4,579
Sex	Female (Ref)
Marital status	Married	−0.05	0.07	(−0.19, 0.09)	1.00	5,292	4,698
	Single	−0.21	0.07	(−0.36, −0.07)	1.00	5,170	4,516
	Divorced (Ref)
Employment status	Unemployed	−0.06	0.04	(−0.13, 0.01)	1.00	7,516	4,548
Employment status	Employed (Ref)
Religion	Muslim	0.03	0.05	(−0.08, 0.13)	1.00	9,238	4,763
	Traditional	0.11	0.03	(0.04, 0.18)	1.00	7,812	4,288
	Christian (Ref)
Educational level	Primary	0.20	0.04	(0.12, 0.29)	1.00	7,861	5,012
	Secondary	0.18	0.04	(0.10, 0.25)	1.00	7,332	4,405
	Tertiary	0.26	0.06	(0.14, 0.28)	1.00	7,766	4,266
	None (Ref)
Diabetes status	Yes	−0.15	0.04	(−0.22, −0.07)	1.00	7,914	3,966
Diabetes status	No (Ref)

Table 3:

Parameter estimates of Bayesian shifted-lognormal regression on the tuberculosis data.

Parameter		Estimates	Standard Error	95 % Credible Interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		3.80	0.09	(3.62, 3.97)	1.00	2,832	2,517
Sex	Male	0.17	0.04	(−0.25, −0.09)	1.00	4,457	3,423
Sex	Female (Ref)
Marital status	Married	−0.05	0.07	(−0.20, 0.09)	1.00	3,314	3,185
	Single	−0.22	0.08	(−0.38, −0.07)	1.00	3,235	3,191
	Divorced (Ref)
Employment status	Unemployed	−0.07	0.04	(−0.15, 0.00)	1.00	4,303	3,780
Employment status	Employed (Ref)
Religion	Muslim	0.03	0.06	(−0.09, 0.14)	1.00	5,434	3,524
	Traditional	0.12	0.04	(0.05, 0.19)	1.00	4,423	3,822
	Christian (Ref)
Educational level	Primary	0.21	0.05	(0.12, 0.31)	1.00	3,577	3,488
	Secondary	0.18	0.04	(0.11, 0.26)	1.00	4,159	4,039
	Tertiary	0.27	0.06	(0.14, 0.39)	1.00	4,359	3,051
	None (Ref)
Diabetes status	Yes	−0.15	0.04	(−0.23, −0.08)	1.00	4,166	3,820
Diabetes status	No (Ref)

Table 4:

Parameter estimates of Bayesian exponential regression on the tuberculosis data.

Parameter		Estimates	Standard error	95 % Credible interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		3.92	0.28	(3.39, 4.48)	1.00	5,451	4,711
Sex	Male	−0.20	0.14	(−0.47, 0.07)	1.00	7,490	4,901
Sex	Female (Ref)
Marital status	Married	−0.05	0.25	(−0.57, 0.43)	1.00	4,842	4,504
	Single	−0.20	0.27	(−0.74, 0.31)	1.00	4,788	4,323
	Divorced (Ref)
Employment status	Unemployed	−0.06	0.13	(−0.30, 0.19)	1.00	6,833	4,775
Employment status	Employed (Ref)
Religion	Muslim	0.06	0.20	(−0.32, 0.47)	1.00	7,489	4,084
	Traditional	0.10	0.12	(−0.14, 0.33)	1.00	7,684	4,736
	Christian (Ref)
Educational level	Primary	0.21	0.16	(−0.10, 0.52)	1.00	6,639	3,989
	Secondary	0.19	0.13	(−0.07, 0.45)	1.00	7,253	4,681
	Tertiary	0.27	0.21	(−0.13, 0.69)	1.00	7,585	4,936
	None (Ref)
Diabetes status	Yes	−0.15	0.13	(−0.39, 0.11)	1.00	7,835	4,484
Diabetes status	No (Ref)

Table 5:

Parameter estimates of Bayesian Fréchet regression on the tuberculosis data.

Parameter		Estimates	Standard error	95 % Credible interval	Rhat	Bulk_ESS	Tail_ESS
Intercept		3.90	0.08	(3.74, 4.06)	1.00	5,663	4,698
Sex	Male	−0.02	0.04	(−0.09, 0.05)	1.00	7,263	4,795
Sex	Female (Ref)
Marital status	Married	−0.09	0.07	(−0.23, 0.06)	1.00	4,887	4,239
	Single	−0.29	0.08	(−0.43, −0.13)	1.00	5,006	4,626
	Divorced (Ref)
Employment status	Unemployed	−0.11	0.04	(−0.19, −0.04)	1.00	7,734	4,437
Employment status	Employed (Ref)
Religion	Muslim	−0.03	0.06	(−0.15, 0.08)	1.00	7,707	4,204
	Traditional	0.15	0.04	(0.08, 0.23)	1.00	6,925	4,831
	Christian (Ref)
Educational level	Primary	0.23	0.05	(0.13, 0.32)	1.00	6,868	4,636
	Secondary	0.12	0.04	(0.05, 0.20)	1.00	6,933	4,606
	Tertiary	0.29	0.06	(0.16, 0.41)	1.00	6,654	5,012
	None (Ref)
Diabetes status	Yes	−0.11	0.04	(−0.19, −0.04)	1.00	7,041	4,219
Diabetes status	No (Ref)

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Received: 2025-04-06

Accepted: 2025-08-20

Published Online: 2025-09-19

Published in Print: 2025-11-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

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Editors Note
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Bringing Emotions into the Study of Responsiveness: The Case of Protective Policies
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https://doi.org/10.1515/spp-2025-0023

Keywords for this article

Bayesian framework; posterior distribution; widely applicable information criterion; Weibull model

Creative Commons

BY 4.0

Articles in the same Issue

Frontmatter
Editors Note
Editors’ Note
Special Issue: Emotional Dynamics of Politics and Policymaking; Guest Editors: Georg Wenzelburger and Beatriz Carbone
Bringing Emotions into the Study of Responsiveness: The Case of Protective Policies
Emotional Reactions to Protective Policies on the Political Spectrum
Regular Articles
Determinants of Political Instability in ECOWAS (1991–2022)
The Impact of International Remittances on Public Debt Sustainability in Kerala: Evidence from the FMOLS Approach
Bayesian Analysis of Tuberculosis Cases in Bolgatanga Municipality, Ghana–West Africa