medical article

1. Scientific African
Longitudinal Analysis of intraocular pressure and its associated risk factors of
glaucoma patients Using Bayesian Linear Mixed Model: A data from Felege Hiwot
Hospital, Ethiopia
--Manuscript Draft--
Manuscript Number: SCIAF-D-21-01614R3
Article Type: Full Length Article
Section/Category: Life and Health Sciences
Keywords: Bayesian analysis; Deviance Information Criteria (DIC); Glaucoma patient Data;
Longitudinal Data Analysis
Corresponding Author: Denekew Bitew Belay, Ph.D
Bahir Dar University College of Science
Bahir Dar University, ETHIOPIA
First Author: Denekew Bitew Belay, Ph.D
Order of Authors: Denekew Bitew Belay, Ph.D
Minilik Derseh, MSc
Destaw Damtie, PhD
Yegnanew A. Shiferaw, PhD
Senait Cherie Adigeh, MSc
Abstract: Background : Glaucoma is a neurodegenerative condition that affects the eye and is
associated with increased intraocular pressure. Intraocular pressure is the fluid
pressure inside the eye and its disturbance often is implicated in the development of
pathologies such as glaucoma, uveitis and retinal detachment. The aim of the present
study was to identify factors that affect the longitudinal intraocular pressure of
glaucoma patients attending the ophthalmology clinic at Felege Hiwot Comprehensive
Specialized Hospital, Bahir Dar, Ethiopia, using a Bayesian linear mixed model
analysis.
Methods: In a longitudinal study with data obtained from glaucoma patients admitted
to Felege Hiwot Hospital, the measurement of intraocular pressure change was
applied. The study subjects were enrolled in the period between 1 st January 2016
and 1 st January 2020 and a total of 328 patients were selected for the study. Data
were explored using descriptive statistics and individual and mean profile plots
throughout study time. A Bayesian linear mixed model for the longitudinal data was
used along with their model comparison, model estimation, model diagnosis and
missing data analysis.
Results: The analysis included 328 individuals with 9 for maximum and 2 for
minimum repeated measurements of intraocular pressure change, including the
baseline. From the Bayesian linear mixed model variables, observation time, age,
place of residence, gender, the cup-disk ratio of patients, type of medication (like
Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox with Pilocarpin), and blood
pressure of the glaucoma patients significantly affected the intraocular pressure
changes over time. However, the type of medication (Diamox and Timolol with
Diamox) did not affect the intraocular pressure changes over time.
Conclusion: Based on the Bayesian linear mixed model analysis, we found that the
predictor variables of age, blood pressure, family history, residence, gender, diabetic
disease, treatment duration, stages of glaucoma, type of medication and cup-disk ratio
significantly affected the average intraocular pressure and had a positive association
with the responses of intraocular pressure of glaucoma patients. Furthermore, the type
of medication was statistically significant and negatively associated with the responses
to intraocular pressure.
Recommendation: We recommend the health professionals to give more attention to
the type of medication especially Timolol with Pilocarpin, Timolol with Diamox and
Timolol with Diamox with Pilocarpine. And taking the combination with the other type of
medication minimizes the risk of blindness and intraocular pressure.
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2. Suggested Reviewers: Awoke Seyoum Tegegne, PhD
Associate Professor, Bahir Dar University
bisrategebrail@yahoo.com
The suggested reviewer has many experience in mixed model applications
Aweke Abebaw Mitiku, PhD
Assistant Professor, Bahir Dar University
abebawaweke@gmail.com
he has an experience reviewing similar research papers
Muluwork Ayele Derebe, MSc
Assistant Professor, Bahir Dar University
muluwerkayele@gmail.com
the suggested reviewer has similar work experience in mixed model application
Ashenafi Abate Woya, MSc
Assistant Professor, Bahria University
ashu.abate@gmail.com
He has very good experience in reviewing similar research works.
Derbachew Asfaw Teni, PhD
Assistant Professor, Arba Minch University
dasfaw469@gmail.com
He has good experience in reviewing similar research works.
Lijalem Melie Tesfaw, MSc
Assistant Professor, Bahir Dar University
lijalemmelie@gmail.com
The suggested reviewer has good experience in review.
Opposed Reviewers:
Response to Reviewers:
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3. Date 08/03/2022
Response the editor’s Comments
Dear Editor,
We are pleased to submit a revised version of the manuscript entitled "Longitudinal Analysis
of intraocular pressure and its associated risk factors of glaucoma patients Using Bayesian
Linear Mixed Model: A data from Felege Hiwot Hospital, Ethiopia". All diagnostics plots
are moved to appendix. We are very grateful for the Editor for helping us to improve the
manuscript.
Denekew Bitew Belay
Corresponding author
Cover Letter

4. Date 08/03/2022
Response the editor’s Comments
Dear Editor,
We are pleased to submit a revised version of the manuscript entitled "Longitudinal Analysis
of intraocular pressure and its associated risk factors of glaucoma patients Using Bayesian
Linear Mixed Model: A data from Felege Hiwot Hospital, Ethiopia". All diagnostics figustre
are moved to appendix. We are very grateful for the Editor for helping us to improve the
manuscript.
Denekew Bitew, Belay
Corresponding author
Response to Reviewers

5. Longitudinal Analysis of intraocular pressure and its associated risk factors of glaucoma patients Using Bayesian Linear Mixed Model:
A data from Felege Hiwot Hospital, Ethiopia
Denekew Bitew Belay1*
, Minilik Derseh2
, Destaw Damtie3
, Yegnanew A. Shiferaw4
and Senait Cherie Adigeh1
1
Bahir Dar University, Department of Statistics, Bahir Dar, Ethiopia
2
Debre Tabor University, Department of Statistics, Debre Tabor, Ethiopia
3
Bahir Dar University, Department of Biology, Bahir Dar, Ethiopia
4
University of Johannesburg, Department of Statistics, Johannesburg, South Africa
*Corresponding author: denekew.t.h@gmail.com
Abstract
Background: Glaucoma is a neurodegenerative condition that affects the eye and is associated with increased intraocular pressure. Intraocular
pressure is the fluid pressure inside the eye and its disturbance often is implicated in the development of pathologies such as glaucoma, uveitis
and retinal detachment. The aim of the present study was to identify factors that affect the longitudinal intraocular pressure of glaucoma
patients attending the ophthalmology clinic at Felege Hiwot Comprehensive Specialized Hospital, Bahir Dar, Ethiopia, using a Bayesian linear
mixed model analysis.
Revised Manuscript with Changes Marked

6. Methods: In a longitudinal study with data obtained from glaucoma patients admitted to Felege Hiwot Hospital, the measurement of intraocular
pressure change was applied. The study subjects were enrolled in the period between 1st
January 2016 and 1st
January 2020 and a total of 328
patients were selected for the study. Data were explored using descriptive statistics and individual and mean profile plots throughout study time.
A Bayesian linear mixed model for the longitudinal data was used along with their model comparison, model estimation, model diagnosis and
missing data analysis.
Results: The analysis included 328 individuals with 9 for maximum and 2 for minimum repeated measurements of intraocular pressure change,
including the baseline. From the Bayesian linear mixed model variables, observation time, age, place of residence, gender, the cup-disk ratio of
patients, type of medication (like Pilocarpin, Timolol with Pilocarpin, Timolol with Diamox with Pilocarpin), and blood pressure of the
glaucoma patients significantly affected the intraocular pressure changes over time. However, the type of medication (Diamox and Timolol with
Diamox) did not affect the intraocular pressure changes over time.
Conclusion: Based on the Bayesian linear mixed model analysis, we found that the predictor variables of age, blood pressure, family history,
residence, gender, diabetic disease, treatment duration, stages of glaucoma, type of medication and cup-disk ratio significantly affected the
average intraocular pressure and had a positive association with the responses of intraocular pressure of glaucoma patients. Furthermore, the
type of medication was statistically significant and negatively associated with the responses to intraocular pressure.
Recommendation: We recommend the health professionals to give more attention to the type of medication especially Timolol with Pilocarpin,
Timolol with Diamox and Timolol with Diamox with Pilocarpine. And taking the combination with the other type of medication minimizes the
risk of blindness and intraocular pressure.
Keywords: Bayesian Analysis, Intraocular pressure, Glaucoma, Longitudinal Data Analysis.
1. Introduction
Glaucoma is a neurodegenerative condition that affects the eye and increases 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 globe. Around

7. 80 million people are currently estimated to have glaucoma worldwide [1]. Intraocular pressure is a measurement involving the magnitude of
the force exerted by the aqueous humor on the internal surface area of the anterior eye. Disturbance in IOP leads to the development of
pathologies such as glaucoma, uveitis, and retinal detachment. Chronic IOP results in primary open-angle glaucoma (POAG) and vision-
damaging problems. It is traditionally measured by tonometry, which estimates the pressure inside the anterior eye based on the resistance to
flattening of a small area of the cornea. Normal-pressure ranges from 12 to 22 mm Hg, even though eye pressure is unique to each person[2].
According to the Ethiopian National Blindness and Low Vision Survey conducted in 2005, national glaucoma was the fifth leading cause of
blindness (contributing to 5.2% of the total blindness)[2] . The survey included individuals with visual acuity worse than 6/18 in either eye or the
exclusion of patients with corneal opacity from intraocular measurement could have resulted in underestimating the prevalence of glaucoma [3]
The estimated number of visually impaired people is globally 285 million, 39 million blind, and 246 million have low vision [1]. Blindness
prevalence rates vary widely, but evidence suggests that approximately 21% of Africans are blind [4]. In Ethiopia, the prevalence of blindness
and low vision are 3.7% and 1.6%, respectively[5]. It indicates that the burden of eye disease in Ethiopia poses enormous economic and social
impacts on individuals, society, and the nation at large and the prevalence is high among the rural population[3].
Some studies have been conducted on glaucoma to determine factors that affect the longitudinal outcomes. For example, [5-8] were studies
conducted to determine the factors that affect the longitudinal change of IOP using a classical linear mixed model approach. But there is no
documented study that used a linear mixed model of longitudinal IOP change of glaucoma patients in the Bayesian approach at Felege Hiwot
Comprehensive and Specialized Hospital, Bahir Dar, Ethiopia. Using the Bayesian approach gains the advantages of additional information by
adding a prior distribution for the parameters of interest. Allowing distributional assumption for the parameters of interest helps to more
accurately model and estimate the parameters with additional information which comes from the prior. Therefore, combining the two sources of
information using the Bayesian theorem will make the inference more reliable. This study thus, focused on the Bayesian linear mixed model of
longitudinal IOP change in glaucoma patients. This study aimed to investigate factors affecting the average longitudinal intraocular pressure of

8. glaucoma patients using the Bayesian linear mixed model among patients visiting Felege Hiwot Comprehensive and Specialized Hospital
(FHSCH), Bahir Dar, Ethiopia.
2. Data and Methods
2.1 Data
A retrospective study design was employed to retrieve the relevant information from the medical records of glaucoma patients in the
ophthalmology clinic of FHSCH during the follow-up times in the period ranging from 1st
January 2016 to 1st
January 2020. The longitudinal
data were extracted from the patient’s charts and included all glaucoma patients' socio-demographic and clinical information under the follow-
up. The longitudinal response variable of IOP was measured in millimeters of mercury (mmHg). The IOP was measured approximately every six
months irrespective of the number of patient visits to the ophthalmology clinic of FHCSH and it contained nine visits including the baseline
visit.
A simple random sampling was employed for selecting a sample of attendants for every six months. From the total of 2981 recently recorded
glaucoma patients as reported by the ophthalmology clinic at the hospital, only 328 satisfied the inclusion criteria and were included in the study
per six months of the survey. Additional information about the patients was collected using self-administered questionnaires.
2.2 Study Variables
Response variable: The response variable of this study was intraocular pressure measured in mmHg for glaucoma patients.
The associated risk factors used in this study are broadly classified as socio-demographic and clinical variables. The socio-demographic
variables are sex, age at baseline, residence, religion, marital status, occupation, and education level. Clinical variables include the 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.

9. 2.3 Linear Mixed Model for Longitudinal Data
The linear mixed model is a widely used model in which random effects are introduced to incorporate the between-subjects variation and within-
subject correlation in the data. The random-effects not only determine the correlation structure between observations on the same subject, but
they also take account of heterogeneity among subjects, due to unobserved characteristics.
The general linear mixed-effects model is defined as [9].
𝑦𝑖 = 𝑋𝑖𝛽 + 𝑍𝑖𝑏𝑖 + 𝜀𝑖 (1).
In equation (1), 𝑦𝑖 is the (𝑛𝑖 x 1) vector of repeated measurements of intraocular pressure, 𝛽 is a (p x 1) vector of the fixed effects parameter,
𝑋𝑖is a (𝑛𝑖 x p) known design matrix corresponding to fixed effects, 𝑏𝑖 is a (q x 1) vector of random effects parameters, 𝑍𝑖 is a (𝑛𝑖 x q) known
design matrix corresponding to random effects and 𝜀𝑖is the (𝑛𝑖 x 1) vector of the error terms.
2.4 Bayesian longitudinal models
The estimated random effect for each change is done by changing the distribution of the random effects[10]. From the Bayesian perspective,
inferential interest focuses on the posterior distribution of the regression coefficients 𝛽 and random effect parameters. Allowing distributions for
the random effects may more accurately model our prior beliefs or allow us to express our uncertainty about the true distribution of the random
effects in a better way. The posterior distribution by applying “Bayes’ theorem” is given as;
𝑃(𝜃|𝑦) =
𝑃(𝑦|𝜃)𝑃(𝜃)
𝑃(𝑦)
=
𝐿(𝜃|𝑦)𝑃(𝜃)
𝑃(𝑦)
(2)
Where 𝑃(𝜃) and 𝑃(𝜃|𝑦) denote the prior and posterior probabilities of 𝜃 respectively; 𝜃 is a set of an unknown parameters and y is the observed
data. And

10. p(y) = ∫ L(θ/y)π(θ)dθ , which is a normalizing factor (constant) and this equation can be simplified as equation (3):
π(θ/y) ∝ L(θ/y)π(θ) (3).
2.5 Bayesian Estimation
In this section, prior distributions are chosen for the parameters, and a general MCMC algorithm is outlined for estimating the posterior
distributions of the parameters and the latent variables[11, 12]. The prior distributions are conjugated if the underlying variables are normal.
Bayesian estimation is performed by a simple Gibbs sampler as long as all response components are joint. For discrete outcomes, an auxiliary
mixture sampling leads to an augmented joint model for which a Gibbs sampling scheme is available. Auxiliary mixture sampling for continuous
response was developed in[13]. Markov Chain Monte Carlo (MCMC) method is used for simulation of Markov chain samples.
2.5.1 Prior Distribution
The prior distribution is a key part of Bayesian inference and represents the information about an uncertain parameter that is combined with the
probability distribution of new data to yield the posterior distribution, which in turn is used for future inferences and decisions involving [12-15].
The prior distribution is an intrinsic part of the Bayesian approach and it is the most obvious feature that distinguishes it from the classical
approach. Much of the controversy about which inference paradigm is better has centered on 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 considered[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[13, 17].
2.5.2 Prior Distribution of Longitudinal Mixed Effect Model

11. From the likelihood function of longitudinal mixed effects model (LMEM), the unknown population parameters are θ where θ= {𝛽𝑖, 𝜎ℇ
2
, 𝑏𝑖, Ʃ𝑣
} which represents the coefficient of the risk factors, variance of the error terms, vector of random effects parameters and covariance matrix of
random effects, respectively. Then the parameters are assumed to be independent of one another. Under the Bayesian framework, we also need
to specify prior distributions for unknown parameters as follows. The prior distribution of the parameters of the model for p-dimensional
regression parameter β is β~N (𝜇𝑜, 𝜎𝑜
2
), and the prior of the variance component is 𝜎ℇ
2
~IG (α, β) (fixed part of mixed effect model).
The priors’ distribution of the parameters of the model for q-dimensional parameter 𝑏𝑖 is 𝑏𝑖~Exp (𝜐𝑖) I(υ> 2) and covariance matrix of random
effects is Ʃ𝑣~IW(Ω, ν) (random part of mixed effects model).
But the mutually independent Normal (N), Inverse Gamma (IG), Exponential truncated model truncated at 2 (Exp) and Inverse Wishart (IW)
prior distributions are chosen to facilitate posterior computations[17]. The super-parameter matrices 𝜎𝑜
2
and Ω can be assumed to be diagonal for
convenient implementation.
2.6 Posterior Distributions
In a Bayesian approach, model parameters are treated as random variables and assigned a probability to each, which is the main difference from
the likelihood approach. The assumed distributions for the parameters are called prior distributions. Bayesian estimation and inference are based
on the posterior distribution are conditional distribution of unobserved quantities given the observed data[13]. The joint posterior distribution for
all unknown parameters θ and random effects 𝑏𝑖 is then given by:
𝑓 (𝜃/𝑌𝑖, i
b ) =
𝑓(𝑌𝑖/𝜃,𝑏𝑖)𝜋(𝜃)
∫ 𝑓(𝑌𝑖/𝜃,𝑏𝑖)𝜋(𝜃) 𝑑𝜃
(4).
Where, f(θ/Yi ,bi) is the posterior probability distribution, f(Yi/θ, bi) is the likelihood function, 𝜋(𝜃) is the prior probability distribution, and

12. 




 d
b
Y
f i
i )
(
)
,
/
( is the marginal constant. Thus, the posterior joint probability distribution becomes:
𝜋 (𝜃/𝑌𝑖, i
b ) ∝ 𝑓(𝑌𝑖/𝜃, 𝑏𝑖)𝜋(𝜃) (5).
Where f (θ/Y, i
b ) is the posterior probability distribution, f(Y/θ, b) is the likelihood function, and π (θ) is the prior probability distribution.
In the Bayesian framework, inference follows from the full posterior distribution. The Bayesian linear mixed 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.
3. Results
The baseline socio-demographic and clinical characteristics of the respondents are shown in Table 1. Out of 328 patients, 113(30.1%) were
females and the remaining 215(69.9%) were males. On the other hand, 142(43.5%) of these sample patients were rural dwellers, while the rest
186(57.5%) were urban dwellers. Likewise, 98(29.9%), 50(15.2%), 62(18.9%), and 54(16.5%) of the respondents had blood pressure, diabetic
disease, pneumonia, and chronic kidney disease, respectively.

13. Table 1: Descriptive statistics of potential predictor variables of glaucoma and time to blindness (n = 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%

14. In Table 2, the mean age of glaucoma patients enrolled in ophthalmology clinic from 1st January 2016 to 1st January 2020 was 55.86 years with
a standard deviation (SD) of 17.35 and an age range of 6 to 89. IOP was measured at baseline at an average of 30.99 mmHg and a standard
deviation (SD) of 9.57 with a range of 9.50 to 51.70. Among the 328 patients included in the study, 106 (32.6%) of them were not blind, whereas
222 (70.8%) of them were blind.
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
IOP 328 30.99 0.53 9.57 51.7 9.5
3.1 Exploring the Longitudinal Data
We conducted exploratory data analysis to investigate various associations and patterns exhibited in the data. Additionally, individual profile
plots, mean structure plots, and variance plots were obtained to gain some insights into the data.
Profile Plots of Glaucoma Patients
The individual profile plot was obtained to gain some insights into the data or to show the pattern of the data over time.

15. Figure 1: Individual Profile Plots for IOP of glaucoma patients
Figure 1 displays the pattern of the overall individual plots of IOP measurements of a patient overtime and demonstrates the variability within
and between patients in IOP over time. Since the measurements were not equally spaced across the different subjects and data was not balanced,
the loess smoothing technique was used instead.

16. Figure 2: Loess smoothing plot with the average trend line of intraocular pressure (IOP) of glaucoma patients
Figure 2 displays that the red line loess smoothing technique suggests that the mean structure of the IOP is nearly linear overtime (i.e., the
relationship between IOP and follow-up time seems to be linear) and that the mean IOP decreased over time. This means the intraocular pressure
is decreasing when the patients get a series of treatments and hence the IOP decreases over time.
Exploring the mean structure of IOP for categorical variables
To explore the mean of IOP with each categorical variable over time, the following plots were considered.

17. Figure 3: The mean profile plots of IOP by place of residence
The mean IOP of glaucoma patients who lived in rural areas was higher at baseline follow-up time up to 54 months, but by the end of the study
period (at 60 months), they were similar for both groups (Figure 3). This signifies that the patients who were receiving the treatment at the center
in the hospital were getting better and better although urban dwellers were much better than the rural dwellers.

18. Figure 4: The Mean profile plots of IOP by blood pressure
Figure 4 displays that the mean IOP of the patient who had blood pressure was higher compared to the patient who had no blood pressure over
time, but the magnitudes of the differences decreased at the end of the follow-up time (at 60 months).
3.2 Bayesian linear mixed model results
For the Bayesian linear mixed-effects model to be valid, covariance among repeated measures must be modeled properly. To identify the
appropriate covariance structure, there are four commonly used covariance structures which include are compound symmetry (CS), first order
autoregressive (AR (1)), unstructured (UN), and Toeplitz (TOEP) were considered.

19. Table 3: Comparison of covariance structures for Bayesian linear mixed-effects model
Covariance structure AIC BIC LogLik
CS 15272.80 15337.34 -6497.13
AR (1) 15127.40 15192.74 -6357.50
UN 15171.20 15277.54 -6485.69
TOEP 15211.37 15248.14 -6486.80
According to Table 3 which presents the results of the comparison, we chose the model with the smallest AIC and BIC values of covariance
structure. Therefore, the first order autoregressive (AR (1)) covariance structure was selected due to the smallest AIC and BIC as compared to
the remaining covariance structures [19].

20. Table 4: Posterior means and 95% credible intervals (CI) for parameters of the Bayesian LMEM without and with patient-specific variances.
Parameters Without Patient-Specific
Variances
With Patient-Specific Variances
Posterior Mean 95% CI Posterior Mean 95% CI
Fixed Effects - - - -
Intercept(𝛽1,1) 12.630 (12.161,13.01) 12.980 (12.358,13.25)
Obstime (𝛽1,2) 2.572 (1.845, 2.835) 1.5395 (1.036,1.834)
Gender(female)(𝛽1,3) 0.5641 ( 0.031, 0.818) 1.8025 (1.003,2.024)
Age( 𝛽1,4) 0.3144 (0.174, 0.458) 0.3347 (0.314, 0.595)
Residence (urban) (𝛽1,5) 0.5077 ( 0.223, 0.850) 0.2174 (0.173, 0.268)
Pilocarpin (𝛽1,6) -0.8470 (-1.334,-0.359) -1.4442 (-1.752, -1.135)
Diamox (𝛽1,7) -0.1154 (-0.595, 0.364) -0.1347 (-0.186,0.083)
Timolol and Pilocarpin
(𝛽1,8)
-0.4801 (-1.443,-0.408) -0.2435 (-0.307,-0.178)
Timolol and Diamox
(𝛽1,9)
-0.3261 (-0.733, 0.373) -0.5513 (-0.686, 0.261)
𝑇𝑖𝑚𝑜𝑙𝑜𝑙, 𝐷𝑖𝑎𝑚𝑜𝑥 𝑎𝑛𝑑
𝑃𝑖𝑙𝑜𝑐𝑎𝑟𝑝𝑖𝑛 (𝛽1,10)
-0.5541 (-1.062, -0.046) -2.8943 (-3.336, -2.235)
BP (yes) (𝛽1,11) 0.9978 (0.464, 1.529) 1.1050 (0.609, 1.596)
CDR (>0.7) (𝛽1,12) 0.6271 (0.435, 0.704) 0.3312 (0.202, 0.543)
Family his (yes)(𝛽1,12) 0.7628 (0.535, 0.804) 0.7528 (0.552, 0.803)
𝜎2
ℰ 7.8314 (7.014, 8.456) - -
Random Effect - - - -
Intercept𝑣𝑎𝑟 (b0) 10.8300 (9.593,12.230) 10.7047 (9.536, 10.845)
Obstime 𝑣𝑎𝑟(b1) 0.8190 (0.575, 0.877) 0.3591 (0.335, 0.495)
µ𝑣 - - 2.2910 (2.201, 2.381)
𝜎2
𝑣 - - 0.6859 (0.526, 0.780)
DIC 26723.3 24537.7

21. Table 4 presents the posterior means and 95% credible intervals for the parameters for the conventional linear mixed effects model and for the
model incorporating patient-specific IOP variances. In both models, the observation time, age, place of residence, gender, cup-disk ratio of
patients, type of medicine (like Pilocarpine, Timolol with Pilocarpine, Timolol with Diamox with Pilocarpine), family history and blood pressure
of the glaucoma patients are statistically significant at 0.05 level of significance, since the 95% posterior credible interval excludes zero.
However, the type of medicine (Diamox and Timolol with Diamox) is statistically insignificant, since the interval includes zero. The table also
shows that the estimated subject- specific variance is 𝜎2
𝑣= 0.6859 with 95% credible interval (0.5264, 0.7800). Thus, it supports the assumption
of heterogeneous variance for repeated IOP measures. Furthermore, the reduction in the DIC for the model, including subject-specific variances,
indicates the need to account for subject-specific IOP variations in the analysis. The Bayesian LMEM incorporating subject-specific variances
has a smaller DIC than the conventional model, so we used it for Bayesian LMEM estimation in the current study.
The estimated average regression coefficients of observation time, age, place of residence, gender, the cup-disk ratio of patients, type of
medicine (like Pilocarpine, Timolol with Pilocarpine, Timolol with Diamox with Pilocarpine), family history, and blood pressure are 1.5395,
1.8025, 0.3347, 0.2174, -1.4442, -0.1347, -0.2435, -0.5513, -2. 8943, 1.1050, 0.7528 and 0.3312, respectively. These estimates show that, on
average, the longitudinal IOP measure significantly increases with an increase in time, cup-disk ratio, gender, age, residence, blood pressure and
family history of glaucoma, but it decreases with an increase of the type of medicine used as compared with their reference category.
3.3 Convergence Diagnostics
In this study, we used three MCMC sampling chains, 75000 iterations of each and three initial values. Applying from three up to five initial
values helps us to determine the convergence problem clearly and similar studies have been conducted in[18, 19].
The convergence assessment was also checked using output values of autocorrelation plots, density plots, and time series plots of estimated
covariates. Under hierarchical data, one could take a random subset instead of checking convergence for every element of a vector of random
effects [20-23].

22. Figures in the appendix (SM1-SM3) show the time series plots of observation time (obstime), covariates of gender and age, respectively, and
indicate the generated values of parameters for the iterations in the chain. Since the chains have, overlapped and mixed each other it shows that
the model (posterior distribution) converges for the targeted value. The time series plot of the history of each parameter’s iterations shows a
reasonable degree of randomness between iterations.
The Figures in the appendix (SM4-SM6) depicted the three autocorrelation function plots for observation time, gender and age. We can say that
the sampled values of the Markov Chain are independent of the time when the autocorrelation is decreasing before lag 20 and when the
autocorrelation is diminishing ultimately after lag 20. Those with the highest autocorrelation have lower convergence, while those with the
lowest autocorrelation have higher convergence [24]. Since the above three Figures 8-10 show lower autocorrelation (because their histogram is
low and ultimately diminishing after lag 20), the distribution is converged very well.
As shown in Figures from the appendix (SM7-SM9), the density plots for the predictor variables of time (obstime), gender and age reveal the
curve of normal distribution. Therefore, the coefficients have a normal distribution, and the simulated parameter values are converged. The
Gibbs sampler has thus been converged to the target density.
Figures (SM10-SM12) showed in the appendix that the Gelman Rubin statistic plots test within-the-individual chain variance for each parameter
in the posterior density. From the above three figures, the blue line indicates the within the individual chain variance and the green line indicates
the between-chain variance, and the red line indicates R (ratio of within- and between- chain variance). As shown in the above Figures 14-16, the
given posterior density or distribution is converged because those red lines are close to one another.
The figures shown in the in appendix (SM 11 –SM15) represent the trace plots of the three parameters (obstime, gender, and age). These plots
represent the parameter values at a given time t against an iteration. As the trace plots move around the distribution model, the model is
converged. Therefore, based on the above figures, assessing these plots indicates that the parameter traces look like straight hairy colorful

23. caterpillars, with the three chains fluctuating rapidly around their equilibrium. There are no obvious upward or downward trends. Besides, the
autocorrelation plots show minor correlations, and the kernel density plots show bell-like posterior distributions. The Gelman-Rubin statistic
shows that the ratio of between-to-within variability is close to 1. All the plots show that the MCMC samples are converged very well.
Table 5: Convergence assessment using Markov Chain Monte Carlo errors
As shown in Table 5, the MC errors are very low (less than 0.05 for all parameters) compared to the posterior summaries of standard errors;
therefore, the posterior density has converged to the target density. Based on the above two methods of convergence assessment, we can observe
that the posterior density is very close to the target density.
Parameters
Posterior
mean
SD MC error 2.5% median 97.5%
Beta[1] 12.9801 0.166 0.00068 0.5755 10.246 0.1228
Beta[2] 1.5395 0.1589 0.00078 0.6633 0.2807 0.001445
Beta[3] 1.8025 0.159 0.00065 0.6189 0.2642 0.05068
Beta[4] 0.3347 0.0977 0.00050 0.4437 0.25 -0.04812
Beta[5] 0.2174 0.1464 0.00101 0.4339 0.2056 0.1556
Beta[6] -1.4442 0.1623 0.00064 -0.6201 -0.2624 0.0635
Beta[7] -0.1347 0.1162 0.00111 -0.6177 -0.3532 -0.1725
Beta[8] -0.2435 0.1255 0.00082 -0.4225 -0.2121 0.08255
Beta[9] -0.5513 0.1362 0.00066 -0.5904 -0.2779 -0.02603
Beta[10] -2.8943 0.0906 0.00057 -0.4863 -0.2889 -0.1253
Beta[11] 1.1050 0.1181 0.00060 0.4707 0.2431 0.01236
Beta[12] 0.3312 0.1332 0.00086 0.4317 0.2119 0.1052
Beta[13] 0.7528 0.1542 0.00071 0.6378 0.2767 0.006175

24. 4. Discussion
This study uses the Bayesian linear mixed modeling to analyze longitudinal intraocular pressure changes in glaucoma patients at Felege Hiwot
Comprehensive Specialized Hospital, Bahir Dar, Ethiopia. As part of the Bayesian linear mixed model analysis, the assumption of normality was
checked by plotting actual, and square root transformed data from glaucoma patients and by comparing them to the QQ plot, histogram, and box
plots. Based on all plots of the actual data, normality is assumed and no transformation is required. To quantify subject-specific variation, the
IOP data were analyzed using the Bayesian linear mixed effects modeling. The first order autoregressive (AR (1)) covariance structure and
random intercept and slope models had a smaller AIC and BIC compared to other models.
The glaucoma patients data understudy was analyzed using different plots (individual profile and exploratory data analysis), followed by model-
based outputs. The exploratory analysis results for the mean structure also suggested that IOP increases in a linear pattern over time on average.
This is consistent with the results of [8, 25-27], in which time caused a decrease in IOP for the patients. It follows that the progression of the
disease will increase as the IOP rises. Additionally, the mean IOP of female patients is higher than that of males up to 60 months, and also it
proves to be significant over time. It confirms [27-34] baseline IOP, age, and time were important factors in glaucoma patient IOP change, but
contradicts the stages of glaucoma, which was not significant, but for our case significant. In addition a study in Ghana[35] supports the
significance of sex, although the results of other studies[36, 37] and [33, 38, 39] do not have similar conclusion.
A recent study conducted among a black population in the UK found no association between intraocular pressure and hypertension [29, 40]. In
this study, hypertension is positively correlated with intraocular pressure change, as confirmed by [41]. A population-based study of patients of
African descent with intraocular pressure[33, 42] did not reveal a relationship of IOP with diabetes mellitus, and a cross-sectional study in
London did not reveal a similar association[26, 35, 40, 42-47]. Despite this, this study indicates a positive association between intraocular
pressure and it is consistent with the findings of [32, 39, 48-52].
Gender, ethnicity [48, 52-55], long duration of disease, hypertension[33, 56], and cup-disk ratio [41, 49] have all been associated with higher
risk regarding intraocular pressure changes.

25. Generally, we have concluded that applying the Bayesian linear mixed model for longitudinal data analysis answers the questions that cannot be
answered by classical analysis for the given glaucoma patient dataset.
Further work is needed to understand better the benefits of using the Bayesian linear mixed models and how to choose the appropriate Bayesian
linear mixed model for a dataset.
5. Conclusion
The study involved a sample of 328 glaucoma patients who attended their follow-up in the ophthalmology clinic at Felege Hiwot Comprehensive
Specialized Hospital, Bahir Dar, over a period of four years (1st January 2016 and 1st January 2020). The study aimed to identify the
determinant factors of IOP change and the effects of the factors on the patients' IOP change. We chose the Bayesian linear mixed model
analysis of longitudinal IOP measurements, including the subject-specific variability, based on the minimum DIC values. The longitudinal
measured IOP change shows variability through time evolution and glaucoma patients attending the ophthalmology clinic improved their IOP
change. The Bayesian linear mixed model analysis shows that the predictor variables of age, residence, blood pressure, cup-disk ratio, family
history, and follow-up time affected the average IOP and had a positive association with the responses to IOP. Likewise, the type of medication
was statistically significant and negatively associated with responses to IOP. That means, on average, the longitudinal IOP measure increases
with an increase in time, cup-disk ratio, gender, age, residence, blood pressure and family history of glaucoma but decreases with a rise in the
type of medicine used. In light of the results of this study, the researchers suggested that health professionals and concerned bodies should focus
on controlling glaucoma cases with patients who have blood pressure and a cup-disk ratio greater than 0.7 during the follow-up time to reduce
intraocular pressure of glaucoma patients. Further, we recommended to pay more attention to the type of medication (Pilocarpine, Timolol with
Pilocarpine, Diamox with Pilocarpine) to reduce glaucoma progression when patients come back to the hospitals.

26. Abbreviations
AIC = Akaki information criteria
BIC= Bayesian information criteria
FHCSH =Felege Hiwot Comprehensive Specialized Hospital
IOP = Intraocular Pressure
LMEM= Linear Mixed Effects Model
MCMC = Markov Chain Monte Carlo
POAG= Primary Open Angle Glaucoma
Declarations
Ethical consideration and consent to participation
We obtained ethical clearance from the ethical review board of the College of Sciences at Bahir Dar University and permission for data
collection from the Felege Hiwot Comprehensive Specialized Hospital. Prior to data collection, informed consent was also obtained from each
patient.
Consent to publication
The manuscript has not been published in another journal, and it is not under consideration for publication in another journal. The authors have
agreed that the manuscript should be submitted to this journal.
Availability of data and materials: The data, which is available with the corresponding author, will be made available upon request.
Competing interests: All authors declared that there is no conflict of interest.
Consent for publication: Not applicable

27. Funding: Not applicable
Authors’ contributions:
DBB and MD were involved in the study design, performed the data extraction, analyzed data and drafted the manuscript; YAS, DD and SCA
were involved in study design, advising through each stage, and reviewed the manuscript. All authors have read critically and approved the final
manuscript.
Acknowledgements
We are grateful to Felege Hiwot Specialized Hospital and all health staff for their data for our health research. The manuscript was edited and
proofread for language by Dr. Ayenew from Department of English language and literature, Bahir Dar University.
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31. Beta[1] chains 1:3
iteration
1 25000 50000 75000
-2.0
-1.0
0.0
1.0
2.0
Figure SM1: Time series plots for observation time (Beta1)
Beta[2] chains 1:3
iteration
1 25000 50000 75000
-2.0
-1.0
0.0
1.0
Figure SM2: Time series plots for gender (Beta2)
Beta[3] chains 1:3
iteration
1 25000 50000 75000
-3.0
-2.0
-1.0
0.0
1.0
Figure SM3: Time series plots for age (Beta3)
Supplementary material

32. Beta[1] chains 1:3
lag
0 20 40
-1.0
-0.5
0.0
0.5
1.0
Figure SM4: Auto-correlation function plot for observation time (Beta1)
Beta[2] chains 1:3
lag
0 20 40
-1.0
-0.5
0.0
0.5
1.0
Figure SM5: Auto-correlation function plot for gender (Beta2)
Beta[3] chains 1:3
lag
0 20 40
-1.0
-0.5
0.0
0.5
1.0
Figure SM6: Auto-correlation function plot for age (Beta3)
Beta[1] chains 1:3 sample: 60000
-2.0 -1.0 0.0 1.0
0.0
1.0
2.0
3.0
4.0
Figure SM7: Density plot for observation time (Beta1)

33. Beta[2] chains 1:3 sample: 60000
-2.0 -1.0 0.0
0.0
1.0
2.0
3.0
4.0
Figure SM8: Density plot for gender (Beta2)
Beta[3] chains 1:3 sample: 60000
-2.0 -1.0 0.0
0.0
1.0
2.0
3.0
4.0
Figure SM9: Density plot for age (Beta3)
Beta[1] chains 1:3
start-iteration
903 10000 20000 30000
0.0
0.5
1.0
1.5
Figure SM10: Gelman- Rubin statistic plot for observation time (Beta1)

34. Beta[2] chains 1:3
start-iteration
903 10000 20000 30000
0.0
0.5
1.0
1.5
Figure SM11: Gelman -Rubin statistic plot for gender (Beta2)
Beta[3] chains 1:3
start-iteration
903 10000 20000 30000
0.0
0.5
1.0
1.5
Figure SM12: Gelman- Rubin statistic plot for age (Beta3)
Beta[1] chains 3:1
iteration
80950
80900
80850
-2.0
-1.0
0.0
1.0
Figure SM13: Gelman-Rubin statistic plot for obstime (Beta1)

35. Beta[1] chains 3:1
iteration
80950
80900
80850
-2.0
-1.0
0.0
1.0
Figure SM14: Gelman-Rubin statistic plot for gender (Beta2)
Beta[3] chains 3:1
iteration
80950
80900
80850
-1.5
-1.0
-0.5
0.0
0.5
Figure SM15: Gelman- Rubin statistic plot for age (Beta3)

36. Scanned by CamScanner
Declaration of Interest Statement

37. Date 06/09/2021
Authors’ contributions and authors statement
Denekew Bitew Belay and Minilik Derseh involved in the study design, performed the data
extraction, analyzing and drafted the manuscript; Destaw Damte, Yegnanew A. Shiferaw and
Senait Cherie Adigeh involved in study design and reviewed the manuscript. All authors have
read critically and approved the final manuscript.
With our best regards
Authors.
CREDIT author statement