Time to Blindness and its associated risk factors of glaucoma patients Using Bayesian Survival Model: A data from Felege Hiwot Specialized Hospital, Ethiopia

Time to Blindness and its associated risk factors of glaucoma patients Using Bayesian Survival Model: A data from Felege Hiwot Specialized Hospital, Ethiopia | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Time to Blindness and its associated risk factors of glaucoma patients Using Bayesian Survival Model: A data from Felege Hiwot Specialized Hospital, Ethiopia MINILIK DERSEH YISMAW This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3981132/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 3 You are reading this latest preprint version Abstract Background : Glaucoma is a neurodegenerative condition that affects the eye and is associated with increased intraocular pressure. Intraocular pressure is carefully regulated and disturbance is often involved in the development of pathologies such as glaucoma, uveitis, and retinal detachment. The study identified factors that have an association with longitudinal IOP and time in glaucoma patients attending an ophthalmology clinic at the Felege Hiwot Comprehensive Specialised Hospital, Ethiopia using Bayesian survival model analysis. Methods: A time-to-event study with data obtained from Felege Hiwot Comprehensive Specialized Hospital, glaucoma patients enrolled in an ophthalmology clinic, the measurement of IOP change approximately every six months and the time of an event occurring were taken. Study subjects were enrolled between the 1 January 2016 and 1 st January 2020 period. A total of 328 patients were selected for this study. The Bayesian Weibull proportional hazard model for the survival data analysis was used. Results: 328 patients were included in the analysis, with 2 being the minimum and 9 being the maximum for repeated measurements of IOP change, including the baseline. The hazard function of the Bayesian Weibull PH model is significantly determined by covariates such as age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration, and advanced stage of glaucoma in patients. Conclusion: Based on the Bayesian Weibull PH model, the predictors of age CI (0.2691, 0.4919), blood pressure (0.9148, 0.8017), diabetic disease (0.1015, 0.1943) , long treatment duration (0.2326, 0.4437), a dvanced stages of glaucoma ( 0.1099, 1.328) , and cup-disk ratio>0.7 (-0.159, -0.015) were significantly affect the average intraocular pressure (IOP). Also, the predictor type of medication was statistically significant and negatively associated with the responses to IOP. Recommendation: Health professionals give more attention to the type of medication especially (Timolol with Pilocarpin, Timolol with Diamox, and Timolol with Diamox with Pilocarpin), to minimize intraocular pressure when the patients are back again in the hospital. Bayesian Analysis Glaucoma patient Longitudinal Analysis Survival Analysis Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Introduction Glaucoma is a neurodegenerative condition that affects the eye and is associated with an increase in intraocular pressure (IOP). When left untreated, patients may gradually experience visual field loss, and even lose their sight completely. It is the second leading cause of blindness around the world. The most common cause of irreversible blindness in the world is glaucoma. Currently, 80 million people have glaucoma [ 1 ]. Intraocular pressure is the fluid pressure of the eye. As pressure is a measure of force per area, IOP is a measurement involving the magnitude of the force exerted by the aqueous humor on the internal surface area of the anterior eye. Intraocular pressure is carefully regulated, and disturbance is often implicated in the development of pathologies such as glaucoma, uveitis, and retinal detachment. Chronic evaluation of IOP has been infamously implicated in the pathogenesis of primary open-angle glaucoma (POAG) and vision-damaging problems. Intraocular pressure is traditionally measured by tonometry, which gives an estimate of the pressure inside the anterior eye based on the resistance to flattening of a small area of the cornea. IOP in the eye is measured by using a tonometer. During the tonometry process, eye drops are used to numb the eye. The doctor or technician uses a device called a tonometer to measure the inner pressure of the eye. A small amount of pressure is applied to the eye by a tiny device or by a warm puff of air. The range for normal pressure is 12 to 22 mm Hg. Eye pressure is unique to each person [ 2 ]. In the Ethiopian National Blindness and Low Vision Survey, which was conducted in 2005, glaucoma was found to be the fifth leading cause of blindness in Ethiopia (contributing 5.2% to the total blindness) [ 3 ]. The fact that the survey included individuals with visual acuity worse than 6/18 in either eye and the exclusion of patients with corneal opacity from intra-ocular measurement could have resulted in an underestimation of the prevalence of glaucoma [ 4 ]. The estimated number of people visually impaired in the world is 285 million, with 39 million blind and 246 million having low vision [ 1 ]. Blindness prevalence rates vary widely but the evidence suggests that approximately 21% of Africans are blind [ 5 ]. The prevalence of blindness and low vision in Ethiopia is 3.7% and 1.6% respectively [ 6 ]. This indicates that the burden of eye disease in Ethiopia poses huge economic and social impacts on individuals, society, and the nation at large. The prevalence is greater in the rural population [ 6 ]. Some studies related to glaucoma have been conducted to determine factors that affect longitudinal outcomes. For example, a study was conducted on the Influence of intraocular pressure reduction on the progression of normal-tension glaucoma and associated risk factors using the Cox proportional hazards model to identify factors of glaucoma progression [ 7 ]. The study conducted to determine the probability of an eye with normal tension glaucoma progressing to legal blindness under standard ophthalmic care using Kaplan-Meier life table analysis was used to estimate the probability of progressing to blindness in one or both eyes Sawada. Since, to our knowledge concerned, there is no studies have been documented on survival analysis of time to blindness of glaucoma patients in the Bayesian approach at Felege Hiwot Comprehensive and Specialized Felege Hiwot Hospital, Bahir Dar, Ethiopia. Therefore, this study focused on the Bayesian survival model of the longitudinal change in IOP in glaucoma patients. The aim is to investigate factors affecting the time to blindness of glaucoma patients using the Bayesian survival model application in the case of Felege Hiwot Comprehensive and Specialized Hospital, Bahir Dar, Ethiopia. Materials and Methods Data settings A retrospective study design was carried out to retrieve relevant information from glaucoma patient medical records to address the objectives of the study. The glaucoma patients were a source of population for this study. Data were collected from the medical chart of glaucoma patients in the ophthalmology clinic at FHSCH under the follow-up time of January 2016 and 1st January 2020 both the longitudinal and survival data were extracted from the patient's chart which contains socio-demographic and clinical information of all glaucoma patients under the follow-up. The survival endpoint of interest was the time to blindness in patients with glaucoma. To give an equal chance and select representative samples of recorded glaucoma patients under the follow-up period simple random sampling selection of attendants for every six months was employed. From the total of 2981 recently recorded glaucoma patients reported by the ophthalmology clinic at the hospital within the study period, only 328 were included in the study per six-month survey, which satisfied inclusion criteria. Data were collected using medical chart review from 131 patients with glaucoma at the time of patients visiting the hospital. Data were extracted from the patient's chart and self-administration questionnaire which contains sociodemographic data including family history of sampled glaucoma patients and behavioral factors. Study Variables Response variable The response variable of this study was the time to blindness of glaucoma patients or the time the study until the occurrence of blindness of glaucoma patients. The associated risk factors used in this study are broadly classified as sociodemographic and Clinical variables. Socio-demographic variables are: Sex, Age at baseline, Residence, Religion, Marital status, Occupation, and Education level. The clinical variables are: Stage of glaucoma, Type of medication, Duration of treatment, Cup-Disc ratio, Family history, Observation time, Presence of diabetes mellitus, Presence of Hypertension, Presence of pneumonia, and Chronic Kidney disease. Bayesian survival models Nonparametric and semi-parametric Bayesian methods in survival analysis have recently become quite popular due to recent advances in computing technology and the development of efficient computational algorithms for implementing these methods. Such methods have become quite common and well-accepted in practice since they offer a more general modeling strategy that contains fewer assumptions [ 8 ]. The survival outcome of the posterior distribution is proportional to the product of the likelihood and the prior distributions. Prior distributions play a very important role in Bayesian statistics [ 8 – 11 ]. In a parametric model, the distribution of outcomes (time to blindness) is specified in terms of a finite number of unknown parameters. The likelihood function of the set of unknown parameters \(\left(\theta \right)\) in the presence of right censoring can be written as: $$L\left(\theta \right)= \prod _{j=1}^{n}{f({t}_{j}/{X}_{j},\theta )}^{{\delta }_{j}}*{S({t}_{j}/{X}_{j},\theta )}^{{1-\delta }_{j}}$$ The log-likelihood form can be written as: $$\mathcal{l}\left(\theta \right)= \sum _{j=1}^{n}[log({f({t}_{j}/{X}_{j},\theta )}^{{\delta }_{j}})+{log\left(S\right({t}_{j}/{X}_{j},\theta )}^{{1-\delta }_{j}})]$$ Where \(\text{f}({\text{t}}_{\text{j}}/{\text{X}}_{\text{j}},{\theta })\) and \(\text{S}({\text{t}}_{\text{j}}/{\text{X}}_{\text{j}},{\theta })\) are the density and survival distributions, respectively. If the posterior distribution for the model specified above does not have closed-form solutions for the parameters. Markov Chain Monte Carlo (MCMC) techniques can be used to sample from the joint posterior distribution of these models. The full conditional likelihood distributions for the unknown parameters \(\theta\) models as: $$\pi \left(\theta ,t,X\right)\propto \prod _{j=1}^{n}{f\left(\frac{{t}_{j}}{{X}_{j}},\theta \right)}^{{\delta }_{j}}*{S\left(\frac{{t}_{j}}{{X}_{j}},\theta \right)}^{{1-\delta }_{j}}*\pi \left(\theta \right),$$ Where, \({\pi }\left({\theta }\right)\) is the prior distribution for every unknown parameter and \({\theta }\) is a generic label for the vector of all the unknown parameters in the assumed model. Bayesian Estimation In this section, prior distributions and the parameters are chosen for the parameters, and the general MCMC algorithm is outlined for estimating the posterior distributions of the parameters and latent variables [ 11 ]. The prior distributions are conjugate if the underlying variables are normal. Bayesian estimation of the model defense can be performed by a simple Gibbs sampler as long as all response components are joint. For discrete outcomes, auxiliary mixture sampling leads to an augmented joint model for which a Gibbs sampling scheme is available. The sampling of the additional mixture for continuous response is developed in [ 12 ]. Markov chain Monte Carlo (MCMC) methods use computer simulation of Markov chains in parameter space. The Markov chains are defined in such a way that the posterior distribution, in a given statistical inference problem, is the asymptotic distribution. One of the standard approaches to define such Markov chains is Gibbs sampling. We used MCMC techniques for the posterior computation. In the special case where all the underlying and latent variables have normal distribution the MCMC algorithm is Gibbs sampler that follows a simple from. Prior Distribution The prior distribution is a key part of Bayesian inference (Bayesian methods and modeling) and represents information about an uncertain parameter that is combined with the probability distribution of new data to obtain the posterior distribution, which in turn is used for future inferences and decisions involving [ 13 – 15 ]. The prior distribution is an intrinsic part of the Bayesian approach and the most obvious feature that distinguishes it from the classical approach. Much of the controversy about which inference paradigm is better has centered on the prior distribution. In Bayesian inference, a prior probability distribution, often called simply the prior, of an uncertain parameter ϴ or latent variable is a probability distribution that expresses uncertainty about ϴ before the data are taken into account [ 16 , 17 ]. The parameters of a prior distribution are called hyper-parameters, to distinguish them from the parameters ϴ of the model. When applying Bayes’ theorem, the prior is multiplied by the likelihood function and then normalized to estimate the posterior probability distribution, which is the conditional distribution of ϴ given the data [ 12 , 17 ]. Prior Distribution for Weibull Proportional Hazards Model The baseline hazard can be assumed of a specific parametric form. For the priors of the model parameters, we make the standard assumptions following [ 18 – 23 ]. In particular, for the regression coefficients β of the longitudinal sub-model and the coefficients γ of the survival submodel we used multivariate normal priors. For variance-covariance matrices, we assumed an inverse Wishart distribution, and for the variance-covariance parameters we took as a prior an inverse gamma. For all parameters, vague priors have been chosen. To perform Bayesian inference with the Weibull model, we must specify the prior distribution for the parameters of the model for p-dimensional regression parameter β (for all regression coefficients) and the nonparametric baseline hazard rate \({h}_{0}\) (t) (constant hazard) that is assumed to be locally essential: To place a normal prior for β, β \(\sim\) N ( \({\mu }_{0}\) , \({\sigma }_{\beta }^{2}\) ) and \({h}_{0}\sim\) Gama (α, β) Some Bayesian approaches have also proposed a method based on approximating the posterior distribution of the parameters in the proportional hazard model by defining a Gaussian prior on regression coefficients. A loss function was then imposed to select a parsimonious model. A semiparametric Bayesian approach was utilized by [ 24 ] who used a discrete gamma process for the baseline hazard function and a multivariate Gaussian prior for the coefficient vector [ 25 , 26 ]. Markov chain Monte Carlo (MCMC) methods are used for posterior sampling. It involves sampling directly from the full conditional distribution, Metropolis-Hastings (MH) sampling [ 16 ], and adaptive rejection sampling (ARS) [ 15 , 23 ]. Posterior Distributions In a Bayesian approach, model parameters are treated as random variables and assign probability to each, which is the major difference to the likelihood approach. The assumed distributions for the parameters are called prior distributions. Bayesian estimation and inference are based on the posterior distribution, which is the conditional distribution of unobserved quantities given the observed data [ 12 , 27 ]. The posterior distribution for all unknown parameters θ is then given by: $$f\left(\theta /{Y}_{i},{T}_{i}\right)=\frac{f\left({Y}_{i},{T}_{i}/\theta \right)\pi \left(\theta \right)}{\int f\left({Y}_{i},{T}_{i}/\theta \right)\pi \left(\theta \right)d\theta }$$ Where, \(\text{f}\left({\theta }/{\text{Y}}_{\text{i}},{\text{T}}_{\text{i}}\right)\) is the posterior probability distribution \(\text{f}\left({\text{Y}}_{\text{i}},{\text{T}}_{\text{i}}/{\theta }\right)\) is the likelihood function, \(\pi \left(\theta \right)\) is the prior probability distribution and \(\int \text{f}\left({\text{Y}}_{\text{i}},{\text{T}}_{\text{i}}/{\theta },{\text{b}}_{\text{i}}\right){\pi }\left({\theta }\right)\text{d}{\theta }\) is the marginal constant. Thus, the posterior probability distribution becomes: $$\pi \left(\theta /{Y}_{i},{T}_{i}\right)\propto f\left({Y}_{i},{T}_{i}/\theta \right)\pi \left(\theta \right)$$ Where \(\text{f}\left({\theta }/\text{Y}, \text{T}\right)\) is the posterior probability distribution, \(\text{f}\left(\text{Y}, \text{T}/{\theta }\right)\) is the likelihood function and π (θ) is the prior probability distribution. In the Bayesian framework, inference follows from the posterior distribution. Bayesian survival model inference is then based on samples drawn from the posterior distribution using an MCMC algorithm such as the Gibbs sampler and Metropolis-Hastings. For example, the posterior means and variances of the parameters can be estimated based on these samples, and Bayesian inference can then be based on these estimated posterior means and variances. This sampling can be done using Win BUGS software. We selected very vague prior distributions in our Win BUGS analysis. That is, we chose priors and hyper-parameter values in such a way that the priors will have minimal impact relative to the data. Results The baseline socio-demographic and clinical characteristics of patients included in the analysis are presented in Table 1 . Of 328 patients, 113(30.1%) were women and the remaining 215(69.9%) were men. Similarly, 142(43.5%) were from rural areas and the rest 186(57.5%) were from urban areas. Similarly, 98 (22.9%), 50 (15.2%), 62 (18.9%), and 54 (16.5%) had blood pressure, and diabetic disease, respectively, the same for others. Table 2 shows that the mean of patients with a baseline age of glaucoma at enrollment in the ophthalmology clinic on January 1, 2016, up to January 1, 2020, was 55.86 years with a standard deviation (SD) of 17.35, the minimum and maximum age in years was 6 and 89 respectively. The average baseline IOP count is 30.99 mmHg with a standard deviation (SD) of 9.57 the minimum and maximum IOP in mmHg were 9.50 and 51.70, respectively, of 328 glaucoma patients. The results of the analysis showed that from 328 patients included in the study, 106 (32.6%) were non-blind while 222(67.8%) were blind. Table 1 Descriptive statistics of potential predictor variables of Glaucoma and time to Blindness in the data sample size 328. Covariates Category Frequency Percentage Sex of patient Female 113 30.1% Male 215 69.9% Residence Rural 142 44.8% Urban 186 55.2% Treatment duration Short 135 41.2% Medium 106 32.3% Long 87 26.5% Type of medicine Diamox 65 19.8% Timolol 22 6.7% Pilocarpin 51 15.5% Timolol with Pilocarpin. 60 18.3% Timolol with Diamox 61 18.6% Timolol with Diamox with Pilocarpin 69 21.0% Diabetic disease No 278 84.8% Yes 50 15.2% Blood pressure No 230 70.1% Yes 98 29.9% Cup-Disk-Ratio Less than and equal to 0.7 173 52.7% Greater than 0.7 155 47.3% Stage of Glaucoma Early 122 37.2% Moderate 53 16.2% Advanced 153 46.6% Family history No 228 69.1% Yes 102 30.9% Pneumonia disease No 266 81.1% Yes 62 18.9% Chronic kidney disease No 274 83.5% Yes 54 16.5% Table 2 Baseline characteristics of continuous variables of glaucoma patients Variables N Mean S.E. SD. Maximum Minimum Age 328 56 0.958 17.35 89 6 TOP 328 30.99 0.53 9.57 51.7 9.5 Non-parametric analysis for survival data Kaplan- Meier survival curves The Kaplan-Meier survival curves for each study variable provide an initial insight into the shape of the survival function. The Kaplan-Meier survival curves to see whether there is a difference in time to blindness between different categories of the covariates. In Fig. 2 we observe a difference between the survival curves. The plots indicate that patients who had less than or equal to the 0.7-cup disk ratio had a better survival experience than patients who had a greater survival experience than those who had a greater than 0.7-cup disk ratio. Figure 3 the plots indicate that male patients had higher survival experience than female patients. Results of the Weibull Model The survival data is analyzed with both Weibull and Exponential models in which results are presented in Table 3 . Because none of the covariates are time-varying, the regression equation for the log-relative hazard in the absence of random effects is: From Table 3 above, it is easy to observe that the parameter estimates of both the Weibull and the Exponential models differ significantly. The estimated Weibull shape parameter \(\widehat{\rho }\) is 2.0157 with a 95% CI (1.671, 2.305) which is significantly greater than one indicating that the rates of blindness increase over time. Lastly, the smaller DIC for the Bayesian Weibull model and the significance of the shape parameter ensure that it is better to use the Bayesian Weibull model than the Bayesian Exponential model. Thus, we have decided to use the Bayesian Weibull survival model for two reasons; firstly, its parameters are easily interpretable as compared to the other parametric models and secondly, it is the only model having both proportional hazards and accelerated failure time properties. Accordingly, subsequent analysis of the survival data is based on a Bayesian Weibull model. In this model, The estimated average regression coefficients of age, blood pressure, diabetic, types medication (like Pilocarpin, Timolol with Pilocarpin, Diamox, Timolol with Diamox, Timolol with Diamox with Pilocarpin), medium treatment duration, long treatment duration, moderate stage of glaucoma, advanced stage of glaucoma and cup-disk ratio of patients are 0.172, 0.147, 0.240, 0.095, 0.090, -0.413, -0.2530, -0.2406, 0.1617, 0.1908, 0.2547, 0.3205 and − 0.3398, respectively. Among those covariates, age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration and advanced stage of glaucoma of patients are statistically significant at the 0.05 significance level. However the type of medicine, cup-disk ratio, and the moderate stage of glaucoma are statistically insignificant at a 0.05 level of significance. These estimates show that an increases in age of the patients increase the risk of blindness and an increase cup-disk ratio (greater than 0.7) of the patients reduces the hazard of blindness as compared with patients who have less than or equal to 0.7. Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure. Table 3 Posterior Means and 95% Credible Intervals for Population Parameters of the Bayesian (Survival Model using both Weibull and Exponential Distributions) Parameters Weibull Distribution Model Exponential Distribution Model Posterior Mean 95% CI MC error Posterior Mean 95% CI MC error Intercept ( \({\beta }_{21}\) ) 10.56 (10.32, 11.323) 0.00179 7.016 (6.801.8.0369) 0.00275 Age ( \({\beta }_{22}\) ) 0.172 (0.125, 0.322) 6.114E-4 0.3447 (0.2691, 0.4919) 7.414E-4 BP (yes) ( \({\beta }_{23}\) ) 0.147 (0.099, 0.197) 6.805E-4 0.8517 (0.9148, 0.8017) 8.815E-4 Diabetic(yes) ( \({\beta }_{24}\) ) 0.240 (0.179, 0.300) 9.686E-4 0.1583 (0.1015, 0.1943) 8.266E-4 Types of medication Pilocarpin ( \({\beta }_{25}\) ) 0.095 (0.055, 0.135) 8.578E-4 -0.3398 (-0.7527, -0.135) 7.578E-4 Diamox ( \({\beta }_{26}\) ) 0.090 (-0.052, 0.130) 8.298E-4 -0.3494 (-0.3860, 0.0831) 8.298E-4 Timolol and Pilocarpin ( \({\beta }_{27}\) ) -0.413 (-0.713, -0.007) 6.018E-4 -0.3220 (-0.4478, -0.178) 6.018E-4 Timolol and Diamox ( \({\beta }_{28}\) ) -0.2530 (-0.313,-0.192) 9.631E-4 -0.077 (-0.6864, 0.2618) 9.631E-4 Timolol, Diamox and Pilocarpin ( \({\beta }_{29}\) ) -0.2406 ( -0.301,-0.179) 9.646E-4 -0.1425 (-0.1935,-0.0912) 9.646E-4 Treatment duration Medium ( \({\beta }_{\text{2,10}}\) ) 0.1617 (0.136, 0.189) 6.261E-4 0.1625 (0.0960, 0.2093) 6.261E-4 Long ( \({\beta }_{\text{2,11}}\) ) 0.1908 (0.190, 0.220) 5.942E-4 0.2789 (0.2326, 0.4437) 5.942E-4 Stage of Glaucoma Moderate ( \({\beta }_{\text{2,12}}\) ) 0.2547 (-0.1925,0.256) 6.224E-4 -0.2634 (-0.1428,0.302) 6.224E-4 Advanced ( \({\beta }_{\text{2,13}}\) ) 0.3205 (0.2481, 0.414) 2.945E-4 0.1214 (0.1099, 1.328) 2.945E-4 Cup-Disk ratio CDR > 0.7 ( \({\beta }_{\text{2,14}}\) ) -0.3398 (-0.392, 0.286) 3.459E-4 -0.0217 (-0.159, -0.015) 3.459E-4 \(\widehat{\rho }\) 2.0157 (1.671, 2.305) DIC 2196.173 2826.700 Assessing Convergence Diagnosis of the Bayesian Model In this study, we used three MCMC sampling chains, 75000 iterations each, and with three initial values was used. Applying three up to five initial values helps us to determine the convergence problem clearly and to give a solution for that [ 20 , 27 , 28 ]. The convergence assessment was also checked using output values of autocorrelation plot, density plot, and time series plots of estimated covariates. Under hierarchical data, rather than checking convergence for every element of a vector of random effects, one could take a random subset [ 11 , 17 , 20 ]. Figures 4 show the time series plot of observation time (obstime), covariate gender, and age respectively. This shows that the generated values of a parameter were generated for each iteration in a chain. Since the chains overlap each other and chains do not increase and decrease (parallel) it shows the model (posterior distribution) converged for the initial value. We can say that the used MCMC algorithm is a good convergence. The time series plot of the history of iterations of the Bayesian joint model shows a reasonable degree of randomness between iterations. The above Figs. 5 , displayed three Auto-correlation function plots (for Observation time, Gender, and Age), we can say that the sampled values of the Markov Chain are independent from the time when the Auto-correlation is diminishing before lag 20 and to diminishing completely after lag 20. Since the above three Figs. 5 show a lower Autocorrelation (because their histogram is low and completely diminishing after lag 20), then the distribution converges very well. As shown in Figs. 6 ; the density plot for the covariate observation time (obstime), gender, and age shows the curve of normal distribution, this implies the coefficient has a normal distribution and the simulated parameter values were converged. Therefore, the Gibbs sampler has been converged to the target density. Figures 7 , showed that the Gelman Rubin statistic plot and tests within and between chain variance for each parameter in the posterior density. From the above three graphs, the blue line indicates the within the individual chain variance, the green line indicates the between chain variance and the red line indicates R (ratio of within and between chain variance). From the above Figs. 7 , the given posterior density or distribution is converged because those red lines are very close to one. The above Figs. 8 displayed the trace plot of three betas (for obstime, gender, and age) is the plot of parameter value at a time t against the iteration. The plot shows that the model is converged because the trace plot moves around the model of the distribution. Therefore based on the above all figures, Assessing these plots indicates that the parameter traces look like straight hairy colorful caterpillars, with the three chains fluctuating rapidly around their equilibrium, and that there are no obvious upward or downward trends. Besides, the autocorrelation plots show little correlations kernel density plots show bell-like posterior distributions and the Gelman-Rubin statistic shows that the ratio of between to within variability is close to 1. All plots show that the model is converged very well. Convergence assessment by Markov chain Monte Carlo errors (MCMC error) Table 4 shows that the values of MC errors are very low (less than 0.05 for all parameters) in comparison to its posterior summaries of standard errors, thus the posterior density has converged to the target density. Therefore, from the above two ways of convergence assessment, we observe that posterior density converged very well to the target density. Table 4: Convergence assessment by Markov chain Monte Carlo errors node mean sd MC error 2.5% median 97.5% Beta[ 1 ] 12.890 0.166 6.767E-4 0.5755 10.246 0.1228 Beta[ 2 ] 2.233 0.1589 7.818E-4 0.6633 0.2807 0.001445 Beta[ 3 ] 1.519 0.159 6.531E-4 0.6189 0.2642 0.05068 Beta[ 4 ] 0.247 0.09777 5.004E-4 0.4437 0.25 -0.04812 Beta[ 5 ] 0.596 0.1464 0.001012 0.4339 0.2056 0.1556 Beta[ 6 ] -1.432 0.1623 6.409E-4 -0.6201 -0.2624 0.0635 Beta[ 7 ] -0.161 0.1162 0.001111 -0.6177 -0.3532 -0.1725 Beta[ 8 ] -0.422 0.1255 8.235E-4 -0.4225 -0.2121 0.08255 Beta[ 9 ] -0.375 0.1362 6.59E-4 -0.5904 -0.2779 -0.02603 Beta[ 10 ] -0.566 0.09064 5.665E-4 -0.4863 -0.2889 -0.1253 Beta[ 11 ] 0.971 0.1181 5.975E-4 0.4707 0.2431 0.01236 Beta[ 12 ] 0.678 0.1332 8.553E-4 0.4317 0.2119 0.1052 Beta[ 13 ] 0.772 0.1542 7.052E-4 0.6378 0.2767 0.006175 Discussion The main objective of this thesis is Bayesian analysis of time-to-event data. The Bayesian Weibull proportional hazard model was fitted and found that Among those covariates age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration and advanced stage of glaucoma of patients are statistically significant at 0.05 level of significance. But Diamox type of medicine, cup-disk ratio, and moderate stage of glaucoma are statistically insignificant at 0.05 level of significance. From the Bayesian Weibull PH model we have found that the parameter estimate for IOP change is negative. It shows that when the IOP change is increased the hazard of a patient on the contrary decreases (0.05 significant level). Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). This indicates blood pressure has a positive influence on the hazard of patients but a negative influence on the survival of patients. Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure. It is supported by [ 31 , 32 ] The parametric models: exponential, and weibull were compared using DIC information criteria, and maximum likelihood to select the appropriate survival time distribution. Then, found that the Bayesian Weibull model is more appropriate for the data under study, survival time distribution. Therefore; we have decided to use the Bayesian Weibull PH model for survival data. From the fitted Bayesian weibull model the estimated Weibull shape parameter \(\widehat{\rho }\) is 2.0157 with a 95% CI (1.671, 2.305), which is significantly greater than one indicating that the rates of blindness increase over time. It supports [ 33 – 35 ] baseline IOP, age and time were significant factors of IOP change of glaucoma patients, but contradicts the stage of glaucoma patients, which were not significant but which were significant in our case. The significance of sex is also supported by [ 35 ] and a study conducted in Ghana (‘Glaucoma patients in Ghana’, 2002; Publishing and All, 2005), but not supported by (Kebede et. al.) and a study conducted in Ghana (Burton, 2016). Conclusion In this study, Bayesian survival analysis for the time-to-blindness of glaucoma patients under follow-up is demonstrated. The Bayesian Weibull model and the significance of the shape parameter assures that it is better to use the Bayesian Weibull model than the Bayesian Exponential model. Thus, we have decided to use the Bayesian Weibull survival model for two reasons; firstly, its parameters are easily interpretable as compared to the other parametric models and secondly, it is the only model having both proportional hazards and accelerated failure time properties. Accordingly, subsequent analysis of the survival data is based on a Bayesian Weibull proportional hazard model. Bayesian Weibull proportional hazard model shows that covariates age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration, and advanced stage of glaucoma of patients were positively associated with time to the blindness of glaucoma patients. These estimates show that an increase in the age of the patients increases the hazard of blindness and an increase cup-disk ratio (greater than 0.7) of the patients reduces the hazard of blindness as compared with patients who have less than or equal to 0.7. Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure. Based on the findings of this study, the researcher recommended the following points for researchers, Health professionals and those concerned should focus on controlling glaucoma cases special patients who have high blood pressure and patients who have a cup-disk ratio greater than 0.7 during the follow-up time to reduce intraocular pressure of glaucoma patients and also give more attention for type of medication specially (Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin), to reduce progression of glaucoma when the patients back again in the hospital. Abbreviations AIC = Akaki information criteria, BIC= Bayesian information criteria, DIC= Deviance Information Criteria, FHCSH = Felege Hiwot Comprehensive Specialized Hospital, IOP = Intraocular Pressure, MCMC = Markov Chain Monte Carlo, POAG= Primary Open Angle Glaucoma Declarations Ethical consideration and consent to participation Ethical clearance was obtained from the ethical review board of the College of Sciences, Bahir Dar University, and permission for data collection was obtained from the Felege Hiwot Comprehensive Specialized Hospital management. Informed consent was also obtained for each patient before the data collection. Consent to publication This manuscript is not published in another journal and is not under consideration for publication in any other journals. The authors have agreed for the manuscript to be submitted to this journal. Availability of data and materials : The data, which is available with the correspondence author, will be made available upon request. Competing interests : Authors declared that there is no conflict of interest. Funding: Not applicable Authors’ Contributions MD was involved in the study design, performed the data extraction, analyzed the data, and produced the manuscript; MD was involved in the study design, counseled at each stage, and read the paper and also contributed to the manuscript’s development. The final paper was critically evaluated and approved by authors. Acknowledgments Bahir Dar Specialized and Referral Hospital and all health staff are gratefully acknowledged for the data they supplied for our health research. 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(2004). https://doi.org/10.1590/S0042-96862004001100009 Han, X., Niu, Y., Guo, X., Hu, Y., Yan, W., He, M.: Age-Related Changes of Intraocular Pressure in Elderly People in Southern China : Lingtou Eye Cohort Study. 2010, 2–11 (2016). https://doi.org/10.1371/journal.pone.0151766 Rudnicka, A.R., Mt-Isa, S., Owen, C.G., Cook, D.G., Ashby, D.: Variations in Primary Open-Angle Glaucoma Prevalence by Age, Gender, and Race : A Bayesian Meta-Analysis. 4254–4261 (2006). https://doi.org/10.1167/iovs.06-0299 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editor assigned by journal 21 Mar, 2024 Submission checks completed at journal 19 Mar, 2024 First submitted to journal 23 Feb, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3981132","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":281937491,"identity":"bc5eb74a-145b-4cf3-9fe4-a75f1e9483fd","order_by":0,"name":"MINILIK DERSEH YISMAW","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4UlEQVRIiWNgGAWjYDACZgaGAw8MEhgbQJwPQMzGToyWBKgWxhkgLczE2JTAANHCzAM1BC/Qbec9eCChIE22X/r4w8c2v7bJ8zEzMH74mINbi9lhvgSgw3KMZ/blGBvn9t02bGNmYJacuQ2fFh4DoJaKxA1neNikc3tuMwK1sDHzEqNl/xn2578te27bE6slJ3EDD4MZM8OP24nEakkznnGGx1iyt+F2chszYzN+v5w/Y/zhw59k2f4e9ocffvy5bTu/vfngh494tKACxjYw2UCsehD4Q4riUTAKRsEoGCkAANKeUl7xCcY2AAAAAElFTkSuQmCC","orcid":"","institution":"Debre Tabor University","correspondingAuthor":true,"prefix":"","firstName":"MINILIK","middleName":"DERSEH","lastName":"YISMAW","suffix":""}],"badges":[],"createdAt":"2024-02-23 08:15:27","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3981132/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3981132/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":53197644,"identity":"632aa68e-1c33-49e7-b87d-77f8b97c9961","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":9727,"visible":true,"origin":"","legend":"\u003cp\u003eThe overall estimate of Kaplan-Meier survivor function plot of glaucoma patients\u003c/p\u003e","description":"","filename":"floatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/f37377ad4b9255d6994ffc63.png"},{"id":53197643,"identity":"408673b2-5b43-4eb8-96b8-0a528cb68dcc","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":18195,"visible":true,"origin":"","legend":"\u003cp\u003eKaplan-Meier survival plot for cup-disc ratio for glaucoma patients\u003c/p\u003e","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/b23408d03228986af75e882f.png"},{"id":53197642,"identity":"74b673c3-4a54-44b8-81da-4e1753ec0bb5","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":17008,"visible":true,"origin":"","legend":"\u003cp\u003eKaplan-Meier survival plot for sex of glaucoma patients\u003c/p\u003e","description":"","filename":"floatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/c611889aa75e319e06e35b55.png"},{"id":53197645,"identity":"cb7409df-9b61-41d4-81e5-55cdb22e4d21","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":381345,"visible":true,"origin":"","legend":"\u003cp\u003eTime series plot for observation time of intercept, gender, and age\u003c/p\u003e","description":"","filename":"floatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/ec6cca32a8a70ab28674b3c6.png"},{"id":53197647,"identity":"ecac75df-6b03-4d8a-9840-05e869d43434","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":72292,"visible":true,"origin":"","legend":"\u003cp\u003eAuto-correlation function plot for observation time of intercept, gender, and age\u003c/p\u003e","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/144e04245298a1b0bf567c08.png"},{"id":53198794,"identity":"1fb21d22-894b-4454-9e5b-c67233a437a8","added_by":"auto","created_at":"2024-03-21 18:48:30","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":162744,"visible":true,"origin":"","legend":"\u003cp\u003eDensity plot for Observation time of intercept, gender, and age\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/cdb24e5332b1e2da2ed1faa1.png"},{"id":53197648,"identity":"1bc335e5-2137-4cf4-912e-200edeec7275","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":273386,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eGelman Rubin statistic plot for Observation time of intercept, gender, and age\u003c/em\u003e\u003c/p\u003e","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/30f3324474e8889f6a6f889f.png"},{"id":53197649,"identity":"db55eb31-f04c-4a30-868d-191027643c1c","added_by":"auto","created_at":"2024-03-21 18:40:30","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":391780,"visible":true,"origin":"","legend":"\u003cp\u003eGelman Rubin statistic plot for Obstime of intercept, gender, and age\u003c/p\u003e","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/dfc47e04daa6658574f13a42.png"},{"id":53199408,"identity":"e4d0739e-ef73-4798-bb82-1c6c9211e05a","added_by":"auto","created_at":"2024-03-21 18:56:32","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1157451,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3981132/v1/1966d684-dc8a-4838-9a0c-96e16dfad117.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Time to Blindness and its associated risk factors of glaucoma patients Using Bayesian Survival Model: A data from Felege Hiwot Specialized Hospital, Ethiopia","fulltext":[{"header":"Introduction","content":"\u003cp\u003eGlaucoma is a neurodegenerative condition that affects the eye and is associated with an increase in intraocular pressure (IOP). When left untreated, patients may gradually experience visual field loss, and even lose their sight completely. It is the second leading cause of blindness around the world. The most common cause of irreversible blindness in the world is glaucoma. Currently, 80\u0026nbsp;million people have glaucoma [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIntraocular pressure is the fluid pressure of the eye. As pressure is a measure of force per area, IOP is a measurement involving the magnitude of the force exerted by the aqueous humor on the internal surface area of the anterior eye. Intraocular pressure is carefully regulated, and disturbance is often implicated in the development of pathologies such as glaucoma, uveitis, and retinal detachment. Chronic evaluation of IOP has been infamously implicated in the pathogenesis of primary open-angle glaucoma (POAG) and vision-damaging problems. Intraocular pressure is traditionally measured by tonometry, which gives an estimate of the pressure inside the anterior eye based on the resistance to flattening of a small area of the cornea. IOP in the eye is measured by using a tonometer. During the tonometry process, eye drops are used to numb the eye. The doctor or technician uses a device called a tonometer to measure the inner pressure of the eye. A small amount of pressure is applied to the eye by a tiny device or by a warm puff of air. The range for normal pressure is 12 to 22 mm Hg. Eye pressure is unique to each person [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn the Ethiopian National Blindness and Low Vision Survey, which was conducted in 2005, glaucoma was found to be the fifth leading cause of blindness in Ethiopia (contributing 5.2% to the total blindness) [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The fact that the survey included individuals with visual acuity worse than 6/18 in either eye and the exclusion of patients with corneal opacity from intra-ocular measurement could have resulted in an underestimation of the prevalence of glaucoma [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. The estimated number of people visually impaired in the world is 285\u0026nbsp;million, with 39\u0026nbsp;million blind and 246\u0026nbsp;million having low vision [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Blindness prevalence rates vary widely but the evidence suggests that approximately 21% of Africans are blind [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. The prevalence of blindness and low vision in Ethiopia is 3.7% and 1.6% respectively [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. This indicates that the burden of eye disease in Ethiopia poses huge economic and social impacts on individuals, society, and the nation at large. The prevalence is greater in the rural population [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eSome studies related to glaucoma have been conducted to determine factors that affect longitudinal outcomes. For example, a study was conducted on the Influence of intraocular pressure reduction on the progression of normal-tension glaucoma and associated risk factors using the Cox proportional hazards model to identify factors of glaucoma progression [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. The study conducted to determine the probability of an eye with normal tension glaucoma progressing to legal blindness under standard ophthalmic care using Kaplan-Meier life table analysis was used to estimate the probability of progressing to blindness in one or both eyes Sawada. Since, to our knowledge concerned, there is no studies have been documented on survival analysis of time to blindness of glaucoma patients in the Bayesian approach at Felege Hiwot Comprehensive and Specialized Felege Hiwot Hospital, Bahir Dar, Ethiopia. Therefore, this study focused on the Bayesian survival model of the longitudinal change in IOP in glaucoma patients. The aim is to investigate factors affecting the time to blindness of glaucoma patients using the Bayesian survival model application in the case of Felege Hiwot Comprehensive and Specialized Hospital, Bahir Dar, Ethiopia.\u003c/p\u003e"},{"header":"Materials and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003eData settings\u003c/h2\u003e \u003cp\u003eA retrospective study design was carried out to retrieve relevant information from glaucoma patient medical records to address the objectives of the study. The glaucoma patients were a source of population for this study. Data were collected from the medical chart of glaucoma patients in the ophthalmology clinic at FHSCH under the follow-up time of January 2016 and 1st January 2020 both the longitudinal and survival data were extracted from the patient's chart which contains socio-demographic and clinical information of all glaucoma patients under the follow-up. The survival endpoint of interest was the time to blindness in patients with glaucoma.\u003c/p\u003e \u003cp\u003eTo give an equal chance and select representative samples of recorded glaucoma patients under the follow-up period simple random sampling selection of attendants for every six months was employed. From the total of 2981 recently recorded glaucoma patients reported by the ophthalmology clinic at the hospital within the study period, only 328 were included in the study per six-month survey, which satisfied inclusion criteria.\u003c/p\u003e \u003cp\u003eData were collected using medical chart review from 131 patients with glaucoma at the time of patients visiting the hospital. Data were extracted from the patient's chart and self-administration questionnaire which contains sociodemographic data including family history of sampled glaucoma patients and behavioral factors.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003eStudy Variables\u003c/h2\u003e \u003cp\u003e \u003cstrong\u003eResponse variable\u003c/strong\u003e \u003cp\u003eThe response variable of this study was the time to blindness of glaucoma patients or the time the study until the occurrence of blindness of glaucoma patients.\u003c/p\u003e \u003c/p\u003e \u003cp\u003eThe associated risk factors used in this study are broadly classified as sociodemographic and Clinical variables. Socio-demographic variables are: Sex, Age at baseline, Residence, Religion, Marital status, Occupation, and Education level. The clinical variables are: Stage of glaucoma, Type of medication, Duration of treatment, Cup-Disc ratio, Family history, Observation time, Presence of diabetes mellitus, Presence of Hypertension, Presence of pneumonia, and Chronic Kidney disease.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003eBayesian survival models\u003c/h2\u003e \u003cp\u003eNonparametric and semi-parametric Bayesian methods in survival analysis have recently become quite popular due to recent advances in computing technology and the development of efficient computational algorithms for implementing these methods. Such methods have become quite common and well-accepted in practice since they offer a more general modeling strategy that contains fewer assumptions [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe survival outcome of the posterior distribution is proportional to the product of the likelihood and the prior distributions. Prior distributions play a very important role in Bayesian statistics [\u003cspan additionalcitationids=\"CR9 CR10\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. In a parametric model, the distribution of outcomes (time to blindness) is specified in terms of a finite number of unknown parameters. The likelihood function of the set of unknown parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\theta \\right)\\)\u003c/span\u003e\u003c/span\u003e in the presence of right censoring can be written as:\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$L\\left(\\theta \\right)= \\prod _{j=1}^{n}{f({t}_{j}/{X}_{j},\\theta )}^{{\\delta }_{j}}*{S({t}_{j}/{X}_{j},\\theta )}^{{1-\\delta }_{j}}$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe log-likelihood form can be written as:\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\mathcal{l}\\left(\\theta \\right)= \\sum _{j=1}^{n}[log({f({t}_{j}/{X}_{j},\\theta )}^{{\\delta }_{j}})+{log\\left(S\\right({t}_{j}/{X}_{j},\\theta )}^{{1-\\delta }_{j}})]$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{f}({\\text{t}}_{\\text{j}}/{\\text{X}}_{\\text{j}},{\\theta })\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{S}({\\text{t}}_{\\text{j}}/{\\text{X}}_{\\text{j}},{\\theta })\\)\u003c/span\u003e\u003c/span\u003eare the density and survival distributions, respectively.\u003c/p\u003e \u003cp\u003eIf the posterior distribution for the model specified above does not have closed-form solutions for the parameters. Markov Chain Monte Carlo (MCMC) techniques can be used to sample from the joint posterior distribution of these models.\u003c/p\u003e \u003cp\u003eThe full conditional likelihood distributions for the unknown parameters \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\theta\\)\u003c/span\u003e\u003c/span\u003e models as:\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\pi \\left(\\theta ,t,X\\right)\\propto \\prod _{j=1}^{n}{f\\left(\\frac{{t}_{j}}{{X}_{j}},\\theta \\right)}^{{\\delta }_{j}}*{S\\left(\\frac{{t}_{j}}{{X}_{j}},\\theta \\right)}^{{1-\\delta }_{j}}*\\pi \\left(\\theta \\right),$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\pi }\\left({\\theta }\\right)\\)\u003c/span\u003e\u003c/span\u003e is the prior distribution for every unknown parameter and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\theta }\\)\u003c/span\u003e\u003c/span\u003e is a generic label for the vector of all the unknown parameters in the assumed model.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003eBayesian Estimation\u003c/h2\u003e \u003cp\u003eIn this section, prior distributions and the parameters are chosen for the parameters, and the general MCMC algorithm is outlined for estimating the posterior distributions of the parameters and latent variables [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. The prior distributions are conjugate if the underlying variables are normal. Bayesian estimation of the model defense can be performed by a simple Gibbs sampler as long as all response components are joint. For discrete outcomes, auxiliary mixture sampling leads to an augmented joint model for which a Gibbs sampling scheme is available. The sampling of the additional mixture for continuous response is developed in [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Markov chain Monte Carlo (MCMC) methods use computer simulation of Markov chains in parameter space. The Markov chains are defined in such a way that the posterior distribution, in a given statistical inference problem, is the asymptotic distribution. One of the standard approaches to define such Markov chains is Gibbs sampling. We used MCMC techniques for the posterior computation. In the special case where all the underlying and latent variables have normal distribution the MCMC algorithm is Gibbs sampler that follows a simple from.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003ePrior Distribution\u003c/h2\u003e \u003cp\u003eThe prior distribution is a key part of Bayesian inference (Bayesian methods and modeling) and represents information about an uncertain parameter that is combined with the probability distribution of new data to obtain the posterior distribution, which in turn is used for future inferences and decisions involving [\u003cspan additionalcitationids=\"CR14\" citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. The prior distribution is an intrinsic part of the Bayesian approach and the most obvious feature that distinguishes it from the classical approach. Much of the controversy about which inference paradigm is better has centered on the prior distribution. In Bayesian inference, a prior probability distribution, often called simply the prior, of an uncertain parameter ϴ or latent variable is a probability distribution that expresses uncertainty about ϴ before the data are taken into account [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. The parameters of a prior distribution are called hyper-parameters, to distinguish them from the parameters ϴ of the model. When applying Bayes\u0026rsquo; theorem, the prior is multiplied by the likelihood function and then normalized to estimate the posterior probability distribution, which is the conditional distribution of ϴ given the data [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003ePrior Distribution for Weibull Proportional Hazards Model\u003c/h2\u003e \u003cp\u003eThe baseline hazard can be assumed of a specific parametric form. For the priors of the model parameters, we make the standard assumptions following [\u003cspan additionalcitationids=\"CR19 CR20 CR21 CR22\" citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. In particular, for the regression coefficients \u003cem\u003eβ\u003c/em\u003e of the longitudinal sub-model and the coefficients γ of the survival submodel we used multivariate normal priors. For variance-covariance matrices, we assumed an inverse Wishart distribution, and for the variance-covariance parameters we took as a prior an inverse gamma. For all parameters, vague priors have been chosen. To perform Bayesian inference with the Weibull model, we must specify the prior distribution for the parameters of the model for p-dimensional regression parameter β (for all regression coefficients) and the nonparametric baseline hazard rate \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{0}\\)\u003c/span\u003e\u003c/span\u003e(t) (constant hazard) that is assumed to be locally essential:\u003c/p\u003e \u003cp\u003eTo place a normal prior for β, β\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sim\\)\u003c/span\u003e\u003c/span\u003eN (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{0}\\)\u003c/span\u003e\u003c/span\u003e,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{\\beta }^{2}\\)\u003c/span\u003e\u003c/span\u003e) and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({h}_{0}\\sim\\)\u003c/span\u003e\u003c/span\u003eGama (α, β)\u003c/p\u003e \u003cp\u003eSome Bayesian approaches have also proposed a method based on approximating the posterior distribution of the parameters in the proportional hazard model by defining a Gaussian prior on regression coefficients. A loss function was then imposed to select a parsimonious model. A semiparametric Bayesian approach was utilized by [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] who used a discrete gamma process for the baseline hazard function and a multivariate Gaussian prior for the coefficient vector [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e26\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eMarkov chain Monte Carlo (MCMC) methods are used for posterior sampling. It involves sampling directly from the full conditional distribution, Metropolis-Hastings (MH) sampling [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], and adaptive rejection sampling (ARS) [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e].\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003ePosterior Distributions\u003c/h2\u003e \u003cp\u003eIn a Bayesian approach, model parameters are treated as random variables and assign probability to each, which is the major difference to the likelihood approach. The assumed distributions for the parameters are called prior distributions. Bayesian estimation and inference are based on the posterior distribution, which is the conditional distribution of unobserved quantities given the observed data [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e]. The posterior distribution for all unknown parameters θ is then given by:\u003cdiv id=\"Equd\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equd\" name=\"EquationSource\"\u003e\n$$f\\left(\\theta /{Y}_{i},{T}_{i}\\right)=\\frac{f\\left({Y}_{i},{T}_{i}/\\theta \\right)\\pi \\left(\\theta \\right)}{\\int f\\left({Y}_{i},{T}_{i}/\\theta \\right)\\pi \\left(\\theta \\right)d\\theta }$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{f}\\left({\\theta }/{\\text{Y}}_{\\text{i}},{\\text{T}}_{\\text{i}}\\right)\\)\u003c/span\u003e\u003c/span\u003e is the posterior probability distribution \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{f}\\left({\\text{Y}}_{\\text{i}},{\\text{T}}_{\\text{i}}/{\\theta }\\right)\\)\u003c/span\u003e\u003c/span\u003e is the likelihood function, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\pi \\left(\\theta \\right)\\)\u003c/span\u003e\u003c/span\u003e is the prior probability distribution and\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\int \\text{f}\\left({\\text{Y}}_{\\text{i}},{\\text{T}}_{\\text{i}}/{\\theta },{\\text{b}}_{\\text{i}}\\right){\\pi }\\left({\\theta }\\right)\\text{d}{\\theta }\\)\u003c/span\u003e\u003c/span\u003e is the marginal constant. Thus, the posterior probability distribution becomes:\u003cdiv id=\"Eque\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Eque\" name=\"EquationSource\"\u003e\n$$\\pi \\left(\\theta /{Y}_{i},{T}_{i}\\right)\\propto f\\left({Y}_{i},{T}_{i}/\\theta \\right)\\pi \\left(\\theta \\right)$$\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{f}\\left({\\theta }/\\text{Y}, \\text{T}\\right)\\)\u003c/span\u003e\u003c/span\u003e is the posterior probability distribution, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\text{f}\\left(\\text{Y}, \\text{T}/{\\theta }\\right)\\)\u003c/span\u003e\u003c/span\u003e is the likelihood function and π (θ) is the prior probability distribution.\u003c/p\u003e \u003cp\u003eIn the Bayesian framework, inference follows from the posterior distribution. Bayesian survival model inference is then based on samples drawn from the posterior distribution using an MCMC algorithm such as the Gibbs sampler and Metropolis-Hastings. For example, the posterior means and variances of the parameters can be estimated based on these samples, and Bayesian inference can then be based on these estimated posterior means and variances. This sampling can be done using Win BUGS software. We selected very vague prior distributions in our Win BUGS analysis. That is, we chose priors and hyper-parameter values in such a way that the priors will have minimal impact relative to the data.\u003c/p\u003e \u003c/div\u003e"},{"header":"Results","content":"\u003cp\u003eThe baseline socio-demographic and clinical characteristics of patients included in the analysis are presented in Table \u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. Of 328 patients, 113(30.1%) were women and the remaining 215(69.9%) were men. Similarly, 142(43.5%) were from rural areas and the rest 186(57.5%) were from urban areas. Similarly, 98 (22.9%), 50 (15.2%), 62 (18.9%), and 54 (16.5%) had blood pressure, and diabetic disease, respectively, the same for others.\u003c/p\u003e\n\u003cp\u003eTable \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e shows that the mean of patients with a baseline age of glaucoma at enrollment in the ophthalmology clinic on January 1, 2016, up to January 1, 2020, was 55.86 years with a standard deviation (SD) of 17.35, the minimum and maximum age in years was 6 and 89 respectively. The average baseline IOP count is 30.99 mmHg with a standard deviation (SD) of 9.57 the minimum and maximum IOP in mmHg were 9.50 and 51.70, respectively, of 328 glaucoma patients. The results of the analysis showed that from 328 patients included in the study, 106 (32.6%) were non-blind while 222(67.8%) were blind.\u003c/p\u003e\n\u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab1\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eDescriptive statistics of potential predictor variables of Glaucoma and time to Blindness in the data sample size 328.\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCovariates\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eCategory\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eFrequency\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePercentage\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eSex of patient\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eFemale\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e113\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e30.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMale\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e215\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e69.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eResidence\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eRural\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e142\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e44.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eUrban\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e186\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e55.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"3\"\u003e\n \u003cp\u003eTreatment duration\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eShort\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e135\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e41.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMedium\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e106\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e32.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLong\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e87\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e26.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"6\"\u003e\n \u003cp\u003eType of medicine\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDiamox\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e65\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e19.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e22\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e6.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePilocarpin\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e51\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol with Pilocarpin.\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e18.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol with Diamox\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e61\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e18.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol with Diamox with Pilocarpin\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e69\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e21.0%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eDiabetic disease\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNo\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e278\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e84.8%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eYes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e15.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eBlood pressure\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNo\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e230\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e70.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eYes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e98\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e29.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eCup-Disk-Ratio\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLess than and equal to 0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e173\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e52.7%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eGreater than 0.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e155\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e47.3%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"3\"\u003e\n \u003cp\u003eStage of Glaucoma\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eEarly\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e122\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e37.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModerate\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e16.2%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAdvanced\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e153\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e46.6%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eFamily history\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNo\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e228\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e69.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eYes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e102\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e30.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003ePneumonia disease\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNo\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e266\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e81.1%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eYes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e62\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e18.9%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eChronic kidney disease\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eNo\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e274\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e83.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eYes\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e54\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e16.5%\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab2\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003eBaseline characteristics of continuous variables of glaucoma patients\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eVariables\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eN\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eS.E.\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eSD.\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMaximum\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMinimum\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAge\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e328\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.958\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e17.35\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e89\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTOP\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e328\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e30.99\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e0.53\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"char\"\u003e\n \u003cp\u003e9.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e51.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.5\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003c/p\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003eNon-parametric analysis for survival data\u003c/h2\u003e\n \u003cdiv id=\"Sec12\" class=\"Section3\"\u003e\n \u003ch2\u003eKaplan- Meier survival curves\u003c/h2\u003e\n \u003cp\u003eThe Kaplan-Meier survival curves for each study variable provide an initial insight into the shape of the survival function. The Kaplan-Meier survival curves to see whether there is a difference in time to blindness between different categories of the covariates.\u003c/p\u003e\n \u003cp\u003eIn Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e we observe a difference between the survival curves. The plots indicate that patients who had less than or equal to the 0.7-cup disk ratio had a better survival experience than patients who had a greater survival experience than those who had a greater than 0.7-cup disk ratio.\u003c/p\u003e\n \u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e the plots indicate that male patients had higher survival experience than female patients.\u003c/p\u003e\n \u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\n \u003ch2\u003eResults of the Weibull Model\u003c/h2\u003e\n \u003cp\u003eThe survival data is analyzed with both Weibull and Exponential models in which results are presented in Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. Because none of the covariates are time-varying, the regression equation for the log-relative hazard in the absence of random effects is:\u003c/p\u003e\n \u003cp\u003eFrom Table \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e above, it is easy to observe that the parameter estimates of both the Weibull and the Exponential models differ significantly. The estimated Weibull shape parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\rho }\\)\u003c/span\u003e\u003c/span\u003e is 2.0157 with a 95% CI (1.671, 2.305) which is significantly greater than one indicating that the rates of blindness increase over time.\u003c/p\u003e\n \u003cp\u003eLastly, the smaller DIC for the Bayesian Weibull model and the significance of the shape parameter ensure that it is better to use the Bayesian Weibull model than the Bayesian Exponential model. Thus, we have decided to use the Bayesian Weibull survival model for two reasons; firstly, its parameters are easily interpretable as compared to the other parametric models and secondly, it is the only model having both proportional hazards and accelerated failure time properties. Accordingly, subsequent analysis of the survival data is based on a Bayesian Weibull model. In this model, The estimated average regression coefficients of age, blood pressure, diabetic, types medication (like Pilocarpin, Timolol with Pilocarpin, Diamox, Timolol with Diamox, Timolol with Diamox with Pilocarpin), medium treatment duration, long treatment duration, moderate stage of glaucoma, advanced stage of glaucoma and cup-disk ratio of patients are 0.172, 0.147, 0.240, 0.095, 0.090, -0.413, -0.2530, -0.2406, 0.1617, 0.1908, 0.2547, 0.3205 and \u0026minus;\u0026thinsp;0.3398, respectively. Among those covariates, age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration and advanced stage of glaucoma of patients are statistically significant at the 0.05 significance level. However the type of medicine, cup-disk ratio, and the moderate stage of glaucoma are statistically insignificant at a 0.05 level of significance.\u003c/p\u003e\n \u003cp\u003eThese estimates show that an increases in age of the patients increase the risk of blindness and an increase cup-disk ratio (greater than 0.7) of the patients reduces the hazard of blindness as compared with patients who have less than or equal to 0.7. Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure.\u003c/p\u003e\n \u003cp\u003e\u003c/p\u003e\u0026nbsp;\u003ctable id=\"Tab3\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\n \u003cp\u003ePosterior Means and 95% Credible Intervals for Population Parameters of the Bayesian (Survival Model using both Weibull and Exponential Distributions)\u003c/p\u003e\n \u003c/div\u003e\n \u003c/caption\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\" rowspan=\"2\"\u003e\n \u003cp\u003eParameters\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eWeibull Distribution Model\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003eExponential Distribution Model\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePosterior Mean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e95% CI\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMC error\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003ePosterior Mean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e95% CI\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMC error\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eIntercept (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{21}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(10.32, 11.323)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.00179\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.016\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(6.801.8.0369)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.00275\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAge (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{22}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.172\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.125, 0.322)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.114E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3447\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.2691, 0.4919)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.414E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBP (yes) (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{23}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.147\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.099, 0.197)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.805E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.8517\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.9148, 0.8017)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.815E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDiabetic(yes) (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{24}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.240\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.179, 0.300)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.686E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1583\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.1015, 0.1943)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.266E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTypes of medication\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003ePilocarpin (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{25}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.095\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.055, 0.135)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.578E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.3398\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.7527, -0.135)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.578E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDiamox (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{26}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.090\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.052, 0.130)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.298E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.3494\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.3860, 0.0831)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.298E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol and Pilocarpin (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{27}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.413\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.713, -0.007)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.018E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.3220\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.4478, -0.178)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.018E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol and Diamox (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{28}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2530\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.313,-0.192)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.631E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.077\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.6864, 0.2618)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.631E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTimolol, Diamox and Pilocarpin (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{29}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2406\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e( -0.301,-0.179)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.646E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.1425\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.1935,-0.0912)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e9.646E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eTreatment duration\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eMedium (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{\\text{2,10}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1617\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.136, 0.189)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.261E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1625\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.0960, 0.2093)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.261E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eLong (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{\\text{2,11}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1908\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.190, 0.220)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.942E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2789\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.2326, 0.4437)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.942E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eStage of Glaucoma\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eModerate (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{\\text{2,12}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2547\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.1925,0.256)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.224E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2634\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.1428,0.302)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.224E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eAdvanced (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{\\text{2,13}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.3205\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.2481, 0.414)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.945E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1214\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(0.1099, 1.328)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.945E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCup-Disk ratio\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eCDR\u0026thinsp;\u0026gt;\u0026thinsp;0.7 (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{\\text{2,14}}\\)\u003c/span\u003e\u003c/span\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.3398\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.392, 0.286)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.459E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.0217\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e(-0.159, -0.015)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e3.459E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\rho }\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.0157\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e\n \u003cp\u003e(1.671, 2.305)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"2\"\u003e\u0026nbsp;\u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eDIC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003e2196.173\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\" colspan=\"3\"\u003e\n \u003cp\u003e2826.700\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003cp\u003e\u003c/p\u003e\n \u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\n \u003ch2\u003eAssessing Convergence Diagnosis of the Bayesian Model\u003c/h2\u003e\n \u003cp\u003eIn this study, we used three MCMC sampling chains, 75000 iterations each, and with three initial values was used. Applying three up to five initial values helps us to determine the convergence problem clearly and to give a solution for that [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e]. The convergence assessment was also checked using output values of autocorrelation plot, density plot, and time series plots of estimated covariates. Under hierarchical data, rather than checking convergence for every element of a vector of random effects, one could take a random subset [\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e17\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e show the time series plot of observation time (obstime), covariate gender, and age respectively. This shows that the generated values of a parameter were generated for each iteration in a chain. Since the chains overlap each other and chains do not increase and decrease (parallel) it shows the model (posterior distribution) converged for the initial value. We can say that the used MCMC algorithm is a good convergence. The time series plot of the history of iterations of the Bayesian joint model shows a reasonable degree of randomness between iterations.\u003c/p\u003e\n \u003cp\u003eThe above Figs. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e, displayed three Auto-correlation function plots (for Observation time, Gender, and Age), we can say that the sampled values of the Markov Chain are independent from the time when the Auto-correlation is diminishing before lag 20 and to diminishing completely after lag 20. Since the above three Figs. \u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e show a lower Autocorrelation (because their histogram is low and completely diminishing after lag 20), then the distribution converges very well.\u003c/p\u003e\n \u003cp\u003eAs shown in Figs. \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e; the density plot for the covariate observation time (obstime), gender, and age shows the curve of normal distribution, this implies the coefficient has a normal distribution and the simulated parameter values were converged. Therefore, the Gibbs sampler has been converged to the target density.\u003c/p\u003e\n \u003cp\u003eFigures \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e, showed that the Gelman Rubin statistic plot and tests within and between chain variance for each parameter in the posterior density. From the above three graphs, the blue line indicates the within the individual chain variance, the green line indicates the between chain variance and the red line indicates R (ratio of within and between chain variance). From the above Figs. \u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e, the given posterior density or distribution is converged because those red lines are very close to one.\u003c/p\u003e\n \u003cp\u003eThe above Figs. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e displayed the trace plot of three betas (for obstime, gender, and age) is the plot of parameter value at a time t against the iteration. The plot shows that the model is converged because the trace plot moves around the model of the distribution.\u003c/p\u003e\n \u003cp\u003eTherefore based on the above all figures, Assessing these plots indicates that the parameter traces look like straight hairy colorful caterpillars, with the three chains fluctuating rapidly around their equilibrium, and that there are no obvious upward or downward trends. Besides, the autocorrelation plots show little correlations kernel density plots show bell-like posterior distributions and the Gelman-Rubin statistic shows that the ratio of between to within variability is close to 1. All plots show that the model is converged very well.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\n \u003ch2\u003eConvergence assessment by Markov chain Monte Carlo errors (MCMC error)\u003c/h2\u003e\n \u003cp\u003eTable 4 shows that the values of MC errors are very low (less than 0.05 for all parameters) in comparison to its posterior summaries of standard errors, thus the posterior density has converged to the target density. Therefore, from the above two ways of convergence assessment, we observe that posterior density converged very well to the target density.\u003c/p\u003e\n \u003cp\u003eTable 4: Convergence assessment by Markov chain Monte Carlo errors\u003c/p\u003e\n \u003cdiv class=\"gridtable\"\u003e\u0026nbsp;\u003ctable id=\"Tab4\" border=\"1\"\u003e\n \u003ccaption language=\"En\"\u003e\n \u003cdiv class=\"CaptionContent\"\u003e\u003cbr\u003e\u003c/div\u003e\n \u003c/caption\u003e\n \u003ccolgroup cols=\"7\"\u003e\u003c/colgroup\u003e\n \u003cthead\u003e\n \u003ctr\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003enode\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003emean\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003esd\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003eMC error\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e2.5%\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003emedian\u003c/p\u003e\n \u003c/th\u003e\n \u003cth align=\"left\"\u003e\n \u003cp\u003e97.5%\u003c/p\u003e\n \u003c/th\u003e\n \u003c/tr\u003e\n \u003c/thead\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e12.890\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.166\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.767E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.5755\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e10.246\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1228\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e2.233\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1589\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.818E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.6633\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2807\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.001445\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e1.519\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.159\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.531E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.6189\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2642\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.05068\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.247\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.09777\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.004E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4437\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.25\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.04812\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.596\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1464\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.001012\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4339\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2056\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1556\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-1.432\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1623\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e6.409E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.6201\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2624\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.0635\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.161\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1162\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.001111\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.6177\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.3532\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.1725\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.422\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1255\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.235E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.4225\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2121\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.08255\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e]\u003c/p\u003e\n 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align=\"left\"\u003e\n \u003cp\u003e-0.4863\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.2889\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e-0.1253\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.971\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1181\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e5.975E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4707\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2431\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.01236\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.678\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1332\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e8.553E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.4317\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2119\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1052\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003eBeta[\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e]\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.772\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.1542\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e7.052E-4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.6378\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.2767\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd align=\"left\"\u003e\n \u003cp\u003e0.006175\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n \u003c/div\u003e\n\u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eThe main objective of this thesis is Bayesian analysis of time-to-event data. The Bayesian Weibull proportional hazard model was fitted and found that Among those covariates age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration and advanced stage of glaucoma of patients are statistically significant at 0.05 level of significance. But Diamox type of medicine, cup-disk ratio, and moderate stage of glaucoma are statistically insignificant at 0.05 level of significance. From the Bayesian Weibull PH model we have found that the parameter estimate for IOP change is negative. It shows that when the IOP change is increased the hazard of a patient on the contrary decreases (0.05 significant level). Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). This indicates blood pressure has a positive influence on the hazard of patients but a negative influence on the survival of patients. Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure. It is supported by [\u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e32\u003c/span\u003e]\u003c/p\u003e \u003cp\u003eThe parametric models: exponential, and weibull were compared using DIC information criteria, and maximum likelihood to select the appropriate survival time distribution. Then, found that the Bayesian Weibull model is more appropriate for the data under study, survival time distribution. Therefore; we have decided to use the Bayesian Weibull PH model for survival data. From the fitted Bayesian weibull model the estimated Weibull shape parameter \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\widehat{\\rho }\\)\u003c/span\u003e\u003c/span\u003e is 2.0157 with a 95% CI (1.671, 2.305), which is significantly greater than one indicating that the rates of blindness increase over time.\u003c/p\u003e \u003cp\u003eIt supports [\u003cspan additionalcitationids=\"CR34\" citationid=\"CR33\" class=\"CitationRef\"\u003e33\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] baseline IOP, age and time were significant factors of IOP change of glaucoma patients, but contradicts the stage of glaucoma patients, which were not significant but which were significant in our case. The significance of sex is also supported by [\u003cspan citationid=\"CR35\" class=\"CitationRef\"\u003e35\u003c/span\u003e] and a study conducted in Ghana (\u0026lsquo;Glaucoma patients in Ghana\u0026rsquo;, 2002; Publishing and All, 2005), but not supported by (Kebede et. al.) and a study conducted in Ghana (Burton, 2016).\u003c/p\u003e"},{"header":"Conclusion","content":"\u003cp\u003eIn this study, Bayesian survival analysis for the time-to-blindness of glaucoma patients under follow-up is demonstrated. The Bayesian Weibull model and the significance of the shape parameter assures that it is better to use the Bayesian Weibull model than the Bayesian Exponential model. Thus, we have decided to use the Bayesian Weibull survival model for two reasons; firstly, its parameters are easily interpretable as compared to the other parametric models and secondly, it is the only model having both proportional hazards and accelerated failure time properties. Accordingly, subsequent analysis of the survival data is based on a Bayesian Weibull proportional hazard model.\u003c/p\u003e \u003cp\u003eBayesian Weibull proportional hazard model shows that covariates age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration, and advanced stage of glaucoma of patients were positively associated with time to the blindness of glaucoma patients.\u003c/p\u003e \u003cp\u003eThese estimates show that an increase in the age of the patients increases the hazard of blindness and an increase cup-disk ratio (greater than 0.7) of the patients reduces the hazard of blindness as compared with patients who have less than or equal to 0.7. Since the parameter of the covariate blood pressure has a positive sign implies the hazard increase (recovery time decrease). Those patients who have blood pressure have larger hazards but, lower survival than patients who have no blood pressure.\u003c/p\u003e \u003cp\u003eBased on the findings of this study, the researcher recommended the following points for researchers, Health professionals and those concerned should focus on controlling glaucoma cases special patients who have high blood pressure and patients who have a cup-disk ratio greater than 0.7 during the follow-up time to reduce intraocular pressure of glaucoma patients and also give more attention for type of medication specially (Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin), to reduce progression of glaucoma when the patients back again in the hospital.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cp\u003eAIC = Akaki information criteria, BIC= Bayesian information criteria, DIC= Deviance Information Criteria, FHCSH = Felege Hiwot Comprehensive Specialized Hospital, IOP = Intraocular Pressure, MCMC = Markov Chain Monte Carlo, POAG= Primary Open Angle Glaucoma\u0026nbsp;\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthical consideration and consent to participation\u003c/strong\u003e\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eEthical clearance was obtained from the ethical review board of the College of Sciences, Bahir Dar University, and permission for data collection was obtained from the Felege Hiwot Comprehensive Specialized Hospital management. Informed consent was also obtained for each patient before the data collection.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003eConsent to publication\u003c/strong\u003e\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eThis manuscript is not published in another journal and is not under consideration for publication in any other journals. The authors have agreed for the manuscript to be submitted to this journal.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAvailability of data and materials\u003c/strong\u003e: The data, which is available with the correspondence author, will be made available upon request.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e: Authors declared that there is no conflict of interest.\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003e\u0026nbsp;Funding:\u003c/strong\u003e\u003c/em\u003e\u003cem\u003e\u0026nbsp;Not applicable\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors\u0026rsquo; Contributions\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMD was involved in the study design, performed the data extraction, analyzed the data, and produced the manuscript; MD was involved in the study design, counseled at each stage, and read the paper and also contributed to the manuscript\u0026rsquo;s development. The final paper was critically evaluated and approved by authors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u003cem\u003e\u0026nbsp;\u003c/em\u003e\u003c/strong\u003eBahir Dar Specialized and Referral Hospital and all health staff are gratefully acknowledged for the data they supplied for our health research.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eQuigley, H.: and 2020. 262\u0026ndash;268 (2020). https://doi.org/10.1136/bjo.2005.081224\u003c/li\u003e\n\u003cli\u003eFarandos, N.M., Yetisen, A.K., Monteiro, M.J., Lowe, C.R.: Contact Lens Sensors in Ocular Diagnostics. 1\u0026ndash;19 (2014). https://doi.org/10.1002/adhm.201400504\u003c/li\u003e\n\u003cli\u003eMulusew, A., Yilikal, A.: Prevalence of congenital color vision defects among school children in five schools of Abeshge District, Central Ethiopia. 10\u0026ndash;14 (2013)\u003c/li\u003e\n\u003cli\u003eAdmassu, F., Prof, A., Assefa, Y.: A thesis submitted to the department of Optometry, college of Medicine and health sciences, university of Gondar in partial fulfillment of the requirements for the degree of master\u0026apos;s in clinical optometry. (2015)\u003c/li\u003e\n\u003cli\u003eSimunovic, M.P.: Colour vision deficiency. Eye. 24, 747\u0026ndash;755 (2009). https://doi.org/10.1038/eye.2009.251\u003c/li\u003e\n\u003cli\u003eBirhane, E., Teketay, D., Barklund, P.: East African Journal of Sciences ( 2007 ) Enclosures to Enhance Woody Species Diversity in The Dry Lands of Eastern Tigray, 1, 136\u0026ndash;147 (2007)\u003c/li\u003e\n\u003cli\u003eWise, L.A., Rosenberg, L., Radin, R.G., Mattox, C., Yang, E.B., Palmer, J.R., Seddon, J.M.: A Prospective Study of Diabetes, Lifestyle Factors, and Glaucoma Among African-American Women. Ann. Epidemiol. 21, 430\u0026ndash;439 (2011). https://doi.org/10.1016/j.annepidem.2011.03.006\u003c/li\u003e\n\u003cli\u003eAvci, E.: Bayesian survival analysis : Comparison of survival probability of hormone receptor status for breast cancer data Bayesian survival analysis : comparison of survival probability of hormone receptor status for breast cancer data Esin Avcı. (2017). https://doi.org/10.1504/IJDATS.2017.083061\u003c/li\u003e\n\u003cli\u003eMansourian, M., Reza, S.M.: Bayesian analysis of joint modeling of longitudinal and time-to-event data using some skew-elliptical distributions. 5, (2017). https://doi.org/10.15406/bbij.2017.05.00153\u003c/li\u003e\n\u003cli\u003eErango, M.A.: Bayesian Joint Modeling of Longitudinal and Survival Time Measurement of Hypertension Patients. (2020)\u003c/li\u003e\n\u003cli\u003eTaylor, B.M., Rowlingson, B.S.: An R Package for Bayesian Inference with Spatial Survival Models. 77, (2017). https://doi.org/10.18637/jss.v077.i04\u003c/li\u003e\n\u003cli\u003eLe, L., Li, J., Chi, E., Ibrahim, J.G.: A Bayesian Joint model for Longitudinal DAS28 Scores and Competing Risk Informative Drop Out in a Rheumatoid Arthritis Clinical Trial. 1\u0026ndash;16 (2018)\u003c/li\u003e\n\u003cli\u003eAlemayehu, E.: Bayesian Hierarchical Approach in Latent Gaussian Modeling for Tuberculosis Cases in Jimma Zone : Using INLA Method By Bayesian Hierarchical Approach in Latent Gaussian Modeling for Tuberculosis Cases in Jimma Zone : Using INLA Method. By : Endale Alemayehu. (2018)\u003c/li\u003e\n\u003cli\u003eAllenby, G.M., Rossi, P.E.: Chapter 20. 1\u0026ndash;59 (1995)\u003c/li\u003e\n\u003cli\u003eLesaffre, E.: Bayesian Biostatistics. (2014)\u003c/li\u003e\n\u003cli\u003eApproach, A.B., Nonlinear, T.O., Variable, L., The, U., Sampler, G., Metropolis-hastings, T.H.E., Arminger, G., Muthen, B.O., Angeles, L.O.S.: A BAYESIAN APPROACH TO NONLINEAR LATENT VARIABLE MODELS USING THE GIBBS SAMPLER AND THE METROPOLIS-HASTINGS. 63, 271\u0026ndash;300 (1998)\u003c/li\u003e\n\u003cli\u003eChen, R.: Bayesian Inference on Mixed-effects Models with Skewed Distributions for HIV longitudinal Data. (2012)\u003c/li\u003e\n\u003cli\u003eBayesian, I.: Bayesian hierarchical models 9.1.\u003c/li\u003e\n\u003cli\u003eData, T.: JM : An R Package for the Joint Modelling of. 35, (2010)\u003c/li\u003e\n\u003cli\u003eHuang, Y.: Joint Models and Their Applications.\u003c/li\u003e\n\u003cli\u003eRizopoulos, D.: Joint Models for Longitudinal and Survival Data. (2019)\u003c/li\u003e\n\u003cli\u003eRobert, C.P., Marin, J.M.: Bayesian Essentials with R : The Complete Solution Manual. 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(2017). https://doi.org/10.3390/ijgi6010016\u003c/li\u003e\n\u003cli\u003ePaddock, S.M.: Journal of Statistical Software. 57, (2014)\u003c/li\u003e\n\u003cli\u003eTaylor, P., Baghfalaki, T., Ganjali, M., Hashemi, R.: Bayesian Joint Modeling of Longitudinal Measurements and Time-to-Event Data Using Robust Distributions ROBUST DISTRIBUTIONS. 37\u0026ndash;41. https://doi.org/10.1080/10543406.2014.903657\u003c/li\u003e\n\u003cli\u003eButa, G.B., Goshu, A.T.: Bayesian Joint Modelling of Disease Progression Marker and Time to Death Event of HIV / AIDS Patients under ART Methods. 5, 1034\u0026ndash;1043 (2015). https://doi.org/10.9734/BJMMR/2015/12907\u003c/li\u003e\n\u003cli\u003eIntegration, M.C., In, I., Inference, B.: Markov chain Monte Carlo algorithms in Bayesian inference. (2006)\u003c/li\u003e\n\u003cli\u003eMuthukumarana, S., Lanka, S., Science, A.: Bayesian methods and applications using winbugs. (2010)\u003c/li\u003e\n\u003cli\u003eOmoti, A.E., Enock, M.E., Okeigbemen, V.W., Akpe, B.A., Fuh, U.C.: African Eyes. 16, 146\u0026ndash;151 (2009). https://doi.org/10.4103/0974-9233.56229\u003c/li\u003e\n\u003cli\u003eTimothy, C.O., Nneli, R.O.: THE EFFECTS OF CIGARETTE SMOKING ON INTRAOCULAR PRESSURE AND ARTERIAL BLOOD PRESSURE OF NORMOTENSIVE YOUNG NIGERIAN MALE ADULTS. 22, 31\u0026ndash;35 (2007)\u003c/li\u003e\n\u003cli\u003eResnikoff, S., Pascolini, D., Etya, D., Kocur, I., Pararajasegaram, R.: Policy and Practice Global data on visual impairment in the year 2002. (2004). https://doi.org/10.1590/S0042-96862004001100009\u003c/li\u003e\n\u003cli\u003eHan, X., Niu, Y., Guo, X., Hu, Y., Yan, W., He, M.: Age-Related Changes of Intraocular Pressure in Elderly People in Southern China : Lingtou Eye Cohort Study. 2010, 2\u0026ndash;11 (2016). https://doi.org/10.1371/journal.pone.0151766\u003c/li\u003e\n\u003cli\u003eRudnicka, A.R., Mt-Isa, S., Owen, C.G., Cook, D.G., Ashby, D.: Variations in Primary Open-Angle Glaucoma Prevalence by Age, Gender, and Race : A Bayesian Meta-Analysis. 4254\u0026ndash;4261 (2006). https://doi.org/10.1167/iovs.06-0299\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"bmc-public-health","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"pubh","sideBox":"Learn more about [BMC Public Health](http://bmcpublichealth.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/pubh/default.aspx","title":"BMC Public Health","twitterHandle":"@BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Bayesian Analysis, Glaucoma patient, Longitudinal Analysis, Survival Analysis","lastPublishedDoi":"10.21203/rs.3.rs-3981132/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3981132/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cem\u003e\u003cstrong\u003eBackground\u003c/strong\u003e\u003c/em\u003e\u003cem\u003e: Glaucoma is a neurodegenerative condition that affects the eye and is associated with increased intraocular pressure. Intraocular pressure is carefully regulated and disturbance is often involved in the development of pathologies such as glaucoma, uveitis, and retinal detachment. The study identified factors that have an association with longitudinal IOP and time in glaucoma patients attending an ophthalmology clinic at the Felege Hiwot Comprehensive Specialised Hospital, Ethiopia using Bayesian survival model analysis.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003eMethods:\u003c/strong\u003e\u003c/em\u003e\u003cem\u003e A time-to-event study with data obtained from Felege Hiwot Comprehensive Specialized Hospital, glaucoma patients enrolled in\u003c/em\u003e an \u003cem\u003eophthalmology clinic, the measurement of IOP change approximately every six months and the time of an event occurring were taken. Study subjects were enrolled between the 1\u003c/em\u003e\u003csup\u003e\u003cem\u003e \u003c/em\u003e\u003c/sup\u003e\u003cem\u003eJanuary 2016 and 1\u003c/em\u003e\u003csup\u003e\u003cem\u003est \u003c/em\u003e\u003c/sup\u003e\u003cem\u003eJanuary 2020 period. A total of 328 patients were selected for this study. The Bayesian Weibull proportional hazard model for the survival data analysis was used.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003eResults: \u003c/strong\u003e\u003c/em\u003e\u003cem\u003e328 patients were included in the analysis, with 2 being the minimum and 9 being the maximum for repeated measurements of IOP change, including the baseline. The hazard function of the Bayesian Weibull PH model is significantly determined by covariates such as age, blood pressure, diabetes, Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox, Timolol with Diamox with Pilocarpin, medium treatment duration, long treatment duration, and advanced stage of glaucoma in patients.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003eConclusion:\u003c/strong\u003e\u003c/em\u003e \u003cem\u003eBased on the Bayesian Weibull PH model, the predictors of age CI (0.2691, 0.4919), blood pressure (0.9148, 0.8017), diabetic disease \u003c/em\u003e(0.1015, 0.1943)\u003cem\u003e, long treatment duration (0.2326, 0.4437), a\u003c/em\u003edvanced \u003cem\u003estages of glaucoma (\u003c/em\u003e0.1099, 1.328)\u003cem\u003e, and cup-disk ratio\u0026gt;0.7 \u003c/em\u003e(-0.159, -0.015)\u003cem\u003e were significantly affect the average \u003c/em\u003eintraocular pressure \u003cem\u003e(IOP). Also, the predictor type of medication was statistically significant and negatively associated with the responses to IOP.\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u003cstrong\u003eRecommendation:\u003c/strong\u003e\u003c/em\u003e\u003cem\u003e Health professionals give more attention to the type of medication especially (Timolol with Pilocarpin, Timolol with Diamox, and Timolol with Diamox with Pilocarpin), to minimize intraocular pressure when the patients are back again in the hospital.\u003c/em\u003e\u003c/p\u003e","manuscriptTitle":"Time to Blindness and its associated risk factors of glaucoma patients Using Bayesian Survival Model: A data from Felege Hiwot Specialized Hospital, Ethiopia","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-21 18:40:25","doi":"10.21203/rs.3.rs-3981132/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorAssigned","content":"","date":"2024-03-21T11:26:25+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-03-19T11:41:04+00:00","index":"","fulltext":""},{"type":"submitted","content":"BMC Public Health","date":"2024-02-23T08:08:12+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"bmc-public-health","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"pubh","sideBox":"Learn more about [BMC Public Health](http://bmcpublichealth.biomedcentral.com/)","snPcode":"","submissionUrl":"https://www.editorialmanager.com/pubh/default.aspx","title":"BMC Public Health","twitterHandle":"@BMC_series","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"em","reportingPortfolio":"BMC Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1f96a437-6a76-415f-956e-43593d4826a2","owner":[],"postedDate":"March 21st, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2024-03-21T18:40:25+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-21 18:40:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3981132","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3981132","identity":"rs-3981132","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}