Bonsa Girma Fufa^1*, Galana Mamo Ayana²

¹Department of statistics, Collage of Natural and Computational Sciences, Diredawa University, Diredawa, Ethiopia

²Department of Epidemiology and Biostatistics, School of Public Health, Haramaya University, Harar, Ethiopia

*Corresponding Author: Bonsa Girma Fufa, Department of statistics, Collage of Natural and Computational Sciences, Diredawa University, Diredawa, Ethiopia.

Abstract
Background: Diabetes mellitus is a chronic, progressive disease characterized by elevated blood glucose levels. There are three significant categories of diabetes based on etiology and cause of disease: type 1 diabetes (TIDM), type 2 diabetes(T2DM), and gestational diabetes. The primary purpose of this study was to assess the predictors of longitudinal fasting blood sugar and survival time-to-first recovery of adult diabetic patients in Jimma Hospital.

Methods: The retrospective data from all the admitted follow-up diabetic patients who have followed at least two visits from September 1, 2018, to August 30, 2020, in Jimma Hospital is used in this study. The study follows three stages, first fitting separate survival diabetic and longitudinal models, and the Joint Model is provided next.

Result: The mean baseline age of adult diabetic patients was 45.32 years. Out of 315 adult patients 111(35.2%) were females and 204(64.8% males) from 111 females and 204 males 88(79.3%) and 156(76.5%) were recovered from female and male respectively. Generally, 244(77.5%) adult diabetic patients recovered. The result from the joint model showed that longitudinal FBS is significantly associated with survival time. The estimated association parameters (α0= -0.089, α1 = -0.198) indicate that both outcome is negatively associated, and lower values of the FBS are associated with better survival time to recovery.

Conclusion: The results of both separate and joint models were consistent. This study revealed that education level, DBP, BMI, type of Diet, type of medication, frequency of meals, and age significantly affected the change in FBS. Furthermore, the presence of hypertension, type of medication, type of Diet, frequency of meals, educational level, BMI, comorbid condition, DBP, gender, and age were significant predictors of time to recovery of adult diabetic patients. Based on the findings of this study, maintaining a healthy body weight by eating a healthy diet along with a lower blood glucose level is essential to control blood sugar in the body and prevent long-term complications.

Keywords: DM, FBS, Longitudinal sub-model, Survival sub-model, Time-to-recovery, Joint model.

Introduction
Background
Diabetes mellitus is a severe and chronic disease that occurs either when the pancreas does not produce enough insulin (a hormone that regulates blood glucose) or when the body cannot effectively use the insulin it produces. DM is a chronic, progressive disease characterized by elevated levels of blood glucose [1]. Diabetes can lead to severe damage in many parts of the body, such as the heart, blood vessels, eyes, kidneys, and nerves, and can increase the overall risk of dying prematurely, but steps to control the disease and lower the risk of complications do exist. There are three significant categories of diabetes based on etiology and cause of disease, namely type 1 diabetes (TIDM), type 2 diabetes(T2DM), and gestational diabetes [2].

In 2019, one in two (50.1%), or 231.9 million of the 463 million adults living with diabetes (overwhelmingly type 2 diabetes, aged 20–79 years) are unaware that they have the condition. It is estimated that 79.4% live in low- and middle-income countries, most of whom are between 40 and 59 years of age [3]. Based on the 2019 estimates, by 2030 a projected 578.4 million, and by 2045, projected 700.2 million adults aged 20–79 will be living with diabetes.

According to a recent report on diabetes in Africa, an estimated 19.4 million adults aged 20-79 years were living with diabetes, representing a regional prevalence of 3.9%. In Africa, 45.9% of people with diabetes live in low-income countries and 54.1% in middle-income countries. The highest prevalence (8.8%) of diabetes in the Region is among adults aged between 65 and 69 years, and more than half (59.7%) of people living with diabetes are unaware of their condition [3].

In Ethiopia, the prevalence of diabetes is expected to be 5.1% in 2035 from 4.4% in 2013 [4]. According to the International Diabetes Federation Atlas (IDFA), in 2013, there were 1.9 million diabetes cases of 20–79-year-old people and 34,262 diabetes-related deaths, and 1.2 million of them were diabetic patients from rural areas. A few studies in Ethiopia report the prevalence of diabetes mellitus in a range from 0.5% to 6.5% [5]. Moreover, the prevalence of undiagnosed diabetes, those who are neither aware of raised blood sugar nor taking any anti-diabetic medications, was very high in Ethiopia [6]. If diabetes is not recognized early and treated, it may pose serious health problems as it progresses to affect other organs of the body [7].

Joint analysis is an elegant approach to modeling the association between time-dependent covariates and the event of interest when the covariate trajectory is not entirely observed and is subject to measurement error and biological variation. When the primary interest is in the association between such endogenous time- dependent covariates and survival, an alternative modeling framework, known as the joint modeling framework for longitudinal and time-to-event data [8], has been introduced in the literature. The design followed by package JMbayes requires first separately fitting a generalized linear mixed model for the longitudinal part and a Cox- PH model for the survival part.

Results
Descriptive Analysis
The data consists of 315 adult diabetic patients aged equal to or above 18 years old and placed under treatments that have followed between September 1, 2018, and August 30, 2020, in Jimma Hospital. Descriptive statistics of baseline covariates for adult diabetic patients were illustrated in (Table 4.1).

Among the 315 patients eligible for the study, 204 (64.8%) were males, and the remaining 111(35.2%) were females. We observed 244 (77.5%) recovery and 71(22.5%) censored observations. A large proportion of these patients, 250 (79.4%) were non-alcoholic users, while 65(20.6%) were alcohol users. Looking at the educational level of patients, 125 (39.7%) had attended secondary school. Most of the patients, 170(54.0%), were from urban residences. Most subjects were exposed to type II diabetes, and 232 (73.7%) subjects were included in the study. Regarding the family history of DM, 214(67.9%) had not, whereas 101 (32.1%) had a family history of diabetes. Of the total patients, the majority of patients, 223 (70.8%), had no history of hypertension, while 92 (29.2%) had. In addition, 134 (42.5%) patients were insulin users, 70 (22.2%) were OHA, and 111 (35.3%) patients used a combination of the two drugs. Regarding dietary type, 169 (53.7%) were patients who frequently used fruit and vegetables, 106(33.6%) were meat eaters, and the rest were patients from other types of Diets. Considering comorbidity of another disease, 202 (64.1%) patients had no comorbidity at their baseline, and the rest, 113 (35.9%), had co-infected by another disease at their baseline. Among study participants, 50 (15.9%) were those who ate less than three/per day,92(29.2%) was consuming 4-to 6 meals/per day, and most of them, 173 (54.9%), were still just eating three meals per day. Similarly, among a total number of patients, 177 (55.8%) were married, 65(19.2%) were single, 43 (15%) were divorced, and the rest of the patients were windowed.

Table 4.1: Frequencies and percentages of categorical covariates of DM patients who were enrolled in Jimma Hospital.
Status of DM patients

Separate Analysis of the Longitudinal Data
Exploring Individual Profiles and Mean Structure
(Figure 4.1) is an individual profile plot of longitudinal fasting blood sugar of adult diabetic patients by follow-up time. As indicated on the plots, the personal profile plots suggested that there were within and between variations of change in FBS measurements over time. The particular trajectory of FBS for patients shows that patients had considerably different FBS at baseline and over time. This suggests that a random intercept and slope model is a good start. However, the red line, which shows the mean structure of FBS measurement over time with the loess smoothing technique, suggested the linear change of the mean of FBS measurement over time.

To understand the possible relationships among the FBS means over time, a plot of a line connecting the average values computed at each time point is shown below (Figure 4.2). The mean structure plot suggests that the mean of the FBS profiles has a linear relationship over time. Since the data is not balanced, the loess smoothing technique is appropriate to offer the mean structure. Also, since the locally weighted least square (loess) smoothing curve of the individual profile plot also suggests the linear effect mean structure, the linear time effects were included as random effects in the model.

Figure 4.1: Individual profile plot and the evolution of mean structure of FBS overtime

Figure 4.2: The mean of FBS measurements evolution structure over time

Linear Mixed Effect Model
In this analysis, we considered sixteen variables. Among these variables, those significant at 25% modest significance level in the univariate analysis were used as a candidate for the multivariate analysis. The result in the appendix (Table 4.2) revealed that the variables age, BMI, DBP, SBP, type of Diet, type of medication, educational level, hypertension, frequency of meals per day, and comorbid condition were found to be significant at 25% level of significance and they are considered in the multivariate analysis. However, gender, place of residence. Type of diabetes, marital status, alcohol consumption, and family history of diabetes were not significant variables for the change in fasting blood sugar. After we chose the important covariates in the uni variable linear mixed model, we incorporated them into a multivariable linear mixed model for fasting blood sugar level data and fitted them with the estimated value of significant covariates.

In Table 4.3, the computed AIC and BIC are reported. The random intercept and random slope models have lower AIC and BIC values than the random intercept and quadratic slope models. Therefore, random intercept and random slope models were used in the linear mixed effect model that appropriately predicts the mean change of the fasting blood sugar over time.

(Table 4.4) showed the result of the linear mixed effect model and found the variables age, observation time, body mass index, diastolic blood pressure, type of Diet, educational level, type of medication, frequency of meals, and time by type of Diet and comorbid interaction were significantly associated with mean change in the fasting blood sugar at 5% level of significance

Table 4.2: Summary statistic of the longitudinal PBS of adult diabetic patients at different follow up months.

Table 4.3: Selection of random effect models to be included in linear mixed effect model.

Table 4.4: Parameter estimate of linear mixed effect model with random intercept and random slope model.

Separate Survival Data Analysis
Descriptive Analysis for Survival Time
The patients were followed up for a total of 36 months. The mean survival time of adult diabetic patients aged 18 years or older based on a cohort of 315 patients was 28.3 months with median and standard deviation of 9 and 3.3 months, respectively. The graph of the Kaplan- Meier estimate of total survival functions is shown in (Figure 4.3), indicating a decreasing pattern of survivorship function as we expected. The plots (Figure 4.3) show that there are relatively similar patterns of survival, with rapidly descending estimated survival functions. The line for female, Frequency of meals(four-six/day), education level (no formal education), type of medication (OHA), kind of Diet (Meat), Hypertension (Yes), and Comorbid (Yes) lies above their respective(belonging) categories, which indicates that they are prolonged longer to recover from the disease.

Figure 4.3: Kaplan-Meier survival plots of some selected categories of predictor variables.

Analysis of Multivariate Cox-PH Model
The result of multivariate analysis of the cox-PH model in (Table 4.5) indicates the variables gender, age, hypertension, comorbidity condition, body mass index, frequency of meals, type of Diet, type of medication, educational level, and diastolic blood pressure were significantly contributed to survival probability of DM Patients under treatment at 5% level of significance. The results reveal that after accounting for other confounders in the data, recovery takes longer with a unit increase of diastolic blood pressure and body mass index at baseline in diabetic patients. This implies that the lower the age, DBP, and BMI at baseline, the faster the rate of recovery (blood sugar level reaching the normal range) of diabetic patients.

Proportionality Hazard Assumption
Table (Table 4.6) shows a test of the proportionality assumption of the Cox proportional hazard model. P-values of the coefficient terms are not significant at a 5% significance level. This ascertains the validity of the assumption of the proportional hazards. The overall proportionality test (Global test) is also insignificant. Therefore, the proportionality assumption is not violated.

Table 4.6: Global test for proportionality assumption

Result based on the joint model.
After having appropriate separate models for the mean of the fasting blood sugar and time to recovery of adult diabetic patients, the next step is to explore a proper joint model that associates the longitudinally measured fasting blood sugar and time to recovery of adult diabetic patients. This study employed the collaborative modeling approach following the variable in the Cox proportional hazard model and linear mixed effect model, shown in the following. The estimate of the association parameter due to the random intercept and random linear slope of fasting blood sugar measurement (α0 = - 0.089, α1 = -0.198), which was negative, means that subject-specific baseline and linear slope of individual fasting blood sugar measurement marker was negatively associated with the hazard of recovery. The association parameter (α) was significantly different from zero (p-value< 0.05 ), indicating that there is a strong association between the fasting blood sugar and time to recovery of DM patients. The negative value of the association parameter showed that the intercept and slope of fasting blood sugar were negatively associated with the time to recovery of DM patients, and with a unit increase in the fasting blood sugar, the time to recovery was decreased. The fitted joint model is displayed in (Table 4.7)

 Table 4.7: Parameter estimate of joint modeling of longitudinal and survival outcomes.

Model Diagnosis
This study used standardized marginal residuals plot to assess the assumption of the linear mixed effect model and Schoenfeld residual to assess the proportionality assumption of the Cox proportional hazard model. To check these model assumptions, the study used standard residual plots to validate the assumptions behind linear mixed effect and Cox proportional hazard models when they are separately fitted. To validate the Cox proportional hazards model assumption of the survival sub-model, a graph of the Schoenfeld residuals was displayed to check the overall goodness of fit for survival sub-models. (Figure 4.4) the scaled Schoenfeld residuals are randomly distributed, and a loess-smoothed curve does not exhibit many departures from the horizontal line, suggesting that the proportional hazards assumption is not violated.

The plot shows that the longitudinal sub-model (FBS) response variables are typically distributed because the points are scattered on the line. From the exact Figure, the bottom plot shows a marginal residual versus fitted values plot of the standardized residuals for longitudinal Fasting blood sugar, and the fitted LOWESS curve is close to 0. Hence, it indicates validation of the assumption of normality of the error term in the longitudinal sub-model. The Marginal survival plot and marginal cumulative hazard plot showed that the survival probability (not developing recovery) came down and the likelihood of developing recovery status came up to one, respectively, when the follow-up time was increased.