Determinants of Time-to-recovery from Hypertension using a Comparison of Various Parametric Shared Frailty Models: In Case of Felege Hiwot Refral Hospital.

Background : Hypertension is a major public health problem that is responsible for morbidity and mortality. In Ethiopia hypertension is becoming a double burden due to urbanization. The study aimed to identify factors that affect time-to-recovery from hypertension at Felege Hiwot Referral Hospital. Retrospective study design was used at FHRH. Methods : The data was collecte d in patient’s chart from September 2016 to January 2018. Kaplan Meier survival estimate and Log-Rank test were used to compare the survival time. The AFT and parametric shared frailty models were employed to identify factors associated with the recovery time of hypertension patients. All the fitted models were compared by using AIC and BIC. Results : Eighty one percent of sampled patients were recovered to normal condition and nineteen percent of patients were censored observations. The median survival time of hypertensive patients to attain normal condition was 13 months. Weibull- inverse Gaussian shared frailty model was found to be the best model for predicting recovery time of hypertension patients. The unobserved heterogeneity in residences as estimated by the Weibull-Inverse Gaussian shared frailty model was θ=0.385 (p -value=0.00). Conclusion : The final model showed that age, systolic blood pressure, related disease, creantine, blood urea nitrogen and the interaction between blood urea nitrogen and age were the determinants factors of recovery status of patients at 5% level of significance. The result showed that patients to diabetic patients and the interaction BUN and age were shorten recovery status of hypertension patients.


Background
Hypertension is one of chronic disease and the most commonly health problems resulting from high blood pressure during circulation. Hypertension is a silent public-health problem and a modifiable risk factor for non -communicable disease (NCD) and mortality. Globally, WHO report shows that 40 million deaths and more than 70% of mortality had been caused by NCDs in 2015 and 2016 respectively (Bayray et al., 2018). Annually, the number of worldwide deaths from NCDs is expected to increase to 52 million by 2030 (Organization, 2014).Globally, the prevalence of hypertension was 1.3 billion individuals which represents 31% of adults (Bloch, 2016). Among adults aged 18 years and above the prevalence of hypertension was nearby 22% in 2014 (Organization, 2014). And also hypertension is estimated to the risk factor for 7.5 million deaths and 57 million disability individuals in each year. In addition it is responsible for an estimated 45% and 51% of deaths due to heart disease and stroke respectively (Organization, 2009). Hypertension has been in the past regarded as a disease of affluence in economically higher class society but this has changed radically in the last two decades, that developing nations are not furnished to handle due to the health community's focus on infectious diseases and lack of awareness (Roger et al., 2011). In a sense, the prevalence is increasing amongst poor sections of society. The worldwide prevalence of hypertension in adults aged 25 years and above was 40%, with the highest is found in Africa (46%) while the lowest is found in America (35%), based on world health organization report. Generally, high income countries have a lower prevalence of hypertension (35%) than lower and middle income countries (46%) (Alwan, 2011).
Among total deaths due to cardiovascular disease, 80% occur in low and middle-income countries, with the highest death rate reported in Africa (Oo et al., 2017). Hypertension is one of the most commonly detected danger factor for cardiovascular disease in Sub-Saharan African countries (Hendriks et al., 2012). In this area, countries are experiencing an unexpected rise in the incidence of hypertension. There was an estimated 74.7 million hypertension patients in Sub-Saharan African countries and by the year 2025, the estimated number of hypertensive patients will be projected to increase by 68% to 125.5 million (Ogah and Rayner, 2013).Ethiopia is one of Sub-Saharan African countries in which, non-communicable diseases and their related risk factors are becoming a double burden due to urbanization and economic development.
Hypertension is also one of the most serious non-communicable chronic diseases in Ethiopia (Twagirumukiza et al., 2011). For many developing countries, the prevention and control of hypertension have given attention. Though, control of blood pressure and awareness about treatment is very little among under developing countries including Ethiopia (Organization).During the years 2011-2025, economic victims associated with NCDs like diabetes mellitus, hypertension in under developed countries are expected to be over $7 trillion these may causes millions of people below the poverty line. Ethiopia has not methodically studied the epidemiological situation of hypertension nor did they establish systematic programs for create awareness ,prevention and control hypertension yet (Bloom et al., 2011).
Hypertension control is very low; among ten patients only eight patients are aware of their hypertension not controlled. These control levels are attributable to poor compliance and health seeking behaviors which are affected by a number of factors. Strategies are very important to improve usage of time to attain respectable control of hypertension. Understanding controlled time for hypertension is important in designing programs for hypertension control time and enhancing quality standards in healthcare delivery (Musinguzi et al., 2018).Even if health professionals try to control blood pressure, the changes over time of blood pressure level on different factors that accelerate blood pressure are still not well understood. Hypertensive patients who have had a strong family history of hypertension, higher heart rate and a greater cardiovascular response controlled their BP compared to the normotensive family historynegative control population (Falkner et al., 1981).Baseline systolic BP and age were significantly associated with control of hypertension but not diastolic BP. Factors like increasing age, current smoking, higher initial systolic BP and serum creantine were associated with uncontrolled BP at the final visit that is do not control their BP in a short period of time. A larger drop in SBP with little increase in the number of antihypertensive drugs is important to control BP in a short period of time (Xiong et al., 2013).In this study, parametric share frailty models were used to analyze correlated hypertension by assuming that patients within the same cluster (residence) shares similar risk factors. Frailty model can be parametric or semi-parametric. The choice of distribution for the base line hazard is very important than the choice of frailty distribution (Fine et al., 2003). Unlike proportional hazards model, AFT models measured the direct effect of the explanatory variables on the survival time instead of hazard. They are also less affected by the choice of probability distribution (Lambert et al., 2004). Hence, in this research weibul, lognormal and log-logistic distributions were used and compared their efficiencies. And for the frailty distribution, gamma and inverse Gaussian frailty distributions were assumed. The aim of this study is to determine time to recovery from hypertension and identify significant predictors at Felege Hiwot referral hospital (FHRH).

Study area
The study was conducted at Felege Hiwot referral hospital which provides good services for the management of Hypertension in Bahir Dar city administration. Bahir Dar is found at a distance of 565 kilometers away from Addis Ababa and located on the Southern shore of Lake Tana, the source of the Blue Nile. According to the report of the chief executive officer of FHRH, the total population served by the hospital is about 12 million per year, and there are around 400 health care professionals.

Data type
The data was retrospective survival data and it is secondary data that were found at Felege Hiwot referral Hospital which was followed up by doctors from September 2016 to January 2018. The patients with incomplete recording of baseline data were excluded from the study.

Study design
A hospital based cross-sectional study design using secondary data from the retrospective record was conducted based on data from the chronic illness medication and follow up at FHRH. The source of population consists of all hypertensive patients who were joined for follow up in this hospital from September 2016 to January 2018. From the total registered patients at the hospital, only 299 of them were included in the study. The selected patients were picked-up by using systematic random sampling. Patients' id from review charts of hypertensive patients were used to find response and independent variables.

Sample size determination
In survival analysis (time to event data) sample size is determined by using the formula, Sample Size Calculations for Survival Analysis below (Collett, 2003)

Variables in the Study
The response variable of this study is the survival time of the hypertensive patients, that is the length of time from the start date of taking antihypertensive drugs until the date of recovery and the candidate predictors that included in this study were sex, age, residence, systolic blood pressure (mmhg), creantine(mg/dl), other related disease, blood urea nitrogen (mg/dl), diastolic blood pressure(mmhg) and number of medication.

Survival analysis
Survival analysis is a collection of statistical procedures that used to analysis data for outcome variable of interest in time until an event occurs. In a survival analysis, we usually refer to the time variable as survival time, because it gives the time that an individual has "survived" over some follow up period. We also typically refer to the event as a failure, because the event of interest usually is death, disease incidence, or some other negative individual experience. However, survival time may be "time to return to work after an elective surgical procedure," in which case failure is a positive event (Kleinbaum & Klein ,2005).The survival time of an individual is said to be censored when the end-point of interest has not been observed for that individual (Collett, 2015).

Survivor function S (t)
The survivor function is the probability that the survival time of a randomly selected subject is greater than or equal to some specified time t(G. Rodr´ıguez , 2010). Let T be a random variable associated with the survival times, t be the specified value of the random variable T and f (t) be the underlying probability density function of the survival time T.The cumulative distribution function F (t), which represents the probability that a subject selected at random will have a survival time less than some stated value t, is given Using the above the survivor function, S (t), can be given as From equations (1) and (2) the relationship between f (t) and S (t) can be derived as Survivor functions have the following characteristics that:  They are non-increasing.

Hazard function h (t)
The hazard function h (t) gives the instantaneous potential for failing at time t, given that the individual has survived up to time t(G. Rodr´ıguez , 2010). The hazard function h (t) ≥ 0, is given By applying the theory of conditional probability and the relationship in equation, 3 the hazard function can be expressed in terms of the underlying probability density function and the survivor function becomes The corresponding cumulative hazard function H (t) is defined as (-H(t) and f(t)=h(t)s(t).from the above relation we can understand if Specifying one of the four functions f(t), S(t), h(t) or H(t) specifies the other three functions. Under the parametric approach, the baseline hazard function is defined as a parametric function and the vector of its parameters, say , is estimated together with the regression coefficients and the frailty parameter(s).

Estimation of survivorship function
In practice, when using actual data, we usually obtain estimated survivor function and obtain curves that are step functions, rather than smooth curves.

The Kaplan-Meier estimate of the survival function
The number of observed events at t (j), j = 1… r. Then the K-M estimator of S(t) is defined as the Kaplan-Meier (KM) estimator is the standard non-parametric estimator of the survival function used for estimating the survival probabilities from observed survival times both censored and uncensored (Kaplan and Meier ,1958).
Suppose that r individuals have failures in a group of individuals, let 0 ≤ t (1) ≤…<t(r) <∞ be the observed ordered death times. Let r(j) be the size of the risk set t(j), where risk set denotes the collection of individuals alive and uncensored just before t(j). Let d (j) be the number of observed events at ) ( j t =1… . Then the K-M estimator of ( ) is defined by This estimator is a step function that changes values only at the time of each birth interval. The cumulative hazard function of the KM estimator can be estimated as:

Comparison of Survivorship Functions
When comparing groups of subjects, it is always a good idea to begin with a graphical display of the data in each group. The figure in general shows if the pattern of one survivorship function lying above another which means the group defined by the upper curve had a more favorable survival experience than the group defined by the lower curve (Sudarno and A Prahutama ,2019).
The general form of this test statistic is given by

…………………………………………………………………… (7)
In this expression ni is the total number of individuals or risk before time t(i) di is the total number of deaths at t(i) wi is the total number of deaths at time t (i) The contribution to the test statistic depends on which of the various tests is used, but each may be expressed in the form of a ratio of weighted sums over the observed survival times.
Under the null hypothesis that the two survivorship functions are the same, and assuming that the censoring experience is independent of the group, and that the total number of observed events and the sum of the expected number of events is large, Q follows a chi-square distribution with one degree of freedom. We can also use the above test to compare k groups. In this study we used the log rank test which is special cases of Q.

Log-rank test
The log rank test, sometimes called the Cox-Mantel test, is the most well-known and widely used test statistic. This test is based on weights equal to one, i.e. wi=1 Therefore, the log rank test statistic becomes

2.8Accelerated failure time model
Parametric models are very applicable to analyze survival data; there are relatively few probability distributions for the survival time that can be used with these models. The

Weibul Accelerated Failure Time Model
The Weibul distribution is very flexible model for time-to-event data. It has a hazard rate which is monotone increasing, decreasing, or constant. The Weibul distribution (including the exponential distribution as a special case) can also be parameterized as an AFT model, and they are the only family of distributions to have this property. The results of fitting a Weibul model can therefore be interpreted in either framework (Klein & Moeschberger, 2003).
In terms of the log-linear representation of the model, if T i has a Weibull distribution, then ε i has a type of the extreme value distribution known as Gumbel distribution. This is an asymmetric distribution with survival function S ε i (ε) = exp(−e ε ) , −∞ < ε < ∞ Then the survivor function of T i = exp (μ + α ′ + σε i ) is And the hazard function is

log-logistic accelerated failure time model
Unlike Weibul hazard, Log-logistic distribution is not a monotonic function of time. However, situations in which the hazard function changes direction can arise. In cases where one comes across to censored data, using log-logistic distribution is mathematically more advantageous than other distributions.
According to the study of Gupta et al. (1999), the log-logistic distribution is proved to be suitable in analyzing survival data conducted by Cox (1972), Cox and Oakes (1984), Bennet (1983).The cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring (Bennett, 1983). The log-logistic distribution is very similar in shape to the log-normal distribution, but is more suitable for use in the analysis of survival data. The log-logistic model has two parameters  and  , where  the scale parameter is and  is the shape parameter. The probability density function of Log-logistic distribution given by The survival function of is then And the hazard function of for the ℎ individual is then

Log-normal accelerated failure time
The lognormal distribution is also defined for random variables that take positive values and so may be used as a model for survival data. If the survival times are assumed to have a log-normal distribution, the baseline survival function and hazard function respectively are given by.

Parameter estimation
Parameters of AFT models can be estimated by maximum likelihood method. The likelihood of n observed survival time, t1, t2, t3… tn, the likelihood function for right censored data is given Z= {zji} is a vector of covariates for the th j subject. The Maximum likelihood parameter estimates are found by using a Newton-Raphson procedure which can be done by software.

Shared frailty model
Multivariate or shared frailty model is a conditional independence model in which frailty is common to all subjects in a cluster. The concept of frailty provides a suitable way to introduce random effects in the model to account for association and unobserved heterogeneity. In its simplest form, a frailty is an unobserved random factor that modifies multiplicatively the hazard function of an individual or cluster of individuals. (Vaupel&Manton et al., 1979) The frailties i u are assumed to be distributed to be identically and independently with a mean of zero and a variance of unity. If the number of subjects is 1 for all groups, the univariate frailty model is obtained (Wienke, 2011); otherwise the model is called the shared frailty model (Hougaard, 2000) because all subjects in the same cluster share the same value of frailty.
The main assumption of a shared frailty model is that all individuals in cluster i share the same value of frailty Zi (i = 1, 2, 3 ..., n), and this is why the model is called the shared frailty model.
The lifetimes are assumed to be conditionally independent with respect to the shared frailty. This shared frailty is the cause of dependence between lifetimes within the clusters.
Extensive research has been devoted to the frailty issue in survival analysis and generalized linear model (GLIM  Vaupel et al.(1979) assumed that frailty is distributed across individuals as a gamma distribution.
In this study we used gamma distributions. These two distributions are the main frailty distributions widely used in the literature because of its simplicity and mathematical tractability.
From an analytical and computational view gamma is a very convenient distribution. Arguably, this is the most popular frailty model due to its mathematical tractability, by (De Sherbinin et al., 2008).

The Gamma shared Frailty Distribution
The gamma distribution has been widely applied as a mixture distribution (Vaupel et al., 1979).From a computational and analytical point of view, it fits very well to failure data. It is widely used due to mathematical tractability (Wienke, 2010).
The density of a gamma-distributed random variable with parameter θ is given bY  is the gamma function, it corresponds to a Gamma distribution Gam (μ, θ) with μ fixed to 1 for identifiability. Its variance is then θ, with Laplace transform.
The conditional survival function of the gamma frailty distribution is given by (Gutierrez ,2002) And the conditional hazard function is given by: Where S (t) and h (t) are the survival and the hazard functions of the baseline distributions. For the Gamma distribution, the Kendall's Tau (Hougaard,2000), which measures the association between any two event times from the same cluster in the multivariate case, can be compute by:

Parameter estimation
For right-censored clustered survival data, the observation for subject where Xij denote the vector of covariates for the ij-th observation.
In the parametric setting, estimation is based on the marginal likelihood in which the frailties have been integrated out by averaging the conditional likelihood with respect to the frailty distribution. Under assumptions of non-informative right-censoring and of independence between the censoring time and the survival time random variables, given the covariate information, the marginal log-likelihood of the observed data can be written as.

Comparison of models
Model selection and comparison are the most common problems of statistical practice, with numerous procedures for selecting among a set of candidate models (Kadane and Lazar ,2004).There are several methods of model selection like AIC, BIC and LRT. However, in some circumstances, it might be useful to easily obtain AIC value for a series of candidate models (Munda et al., 2012). In this study, AIC criteria were used to compare various candidate models and the model with the smallest AIC value is considered as a better fit. AIC can be obtained by Where L is the maximum likelihood value, K is the number of covariates and C the number of model specific distributional parameters. In addition to this criteria, also BIC and likelihood ratio test are important in order to compare models, LRT is a best model selection technique, when models are nested (one model contains a subset of the explanatory variables in another), however, BIC is available for non-nested model.

Model checking and diagnostics
After a model has been fitted, the adequacy of the fitted model needs to be assessed. The methods for assessment of model checking for this study were used evaluation of the Parametric Baselines and the Cox Snell Residuals.

Evaluation of the parametric baselines
The appropriateness of model with the weibul baseline has a property that the log (-log(S (t)) is Therefore the log-failure odds can be written as: ,therefore the appropriateness of model with the log logistic baseline can graphically be evaluated by plotting log time log ) ( Kaplan-Meier survival estimate (Duchateau & Janssen, 2008 (t) is linear, the lo-normal distribution is appropriate for the given data set.

Cox-Snell residuals
The residual that is most widely used in the analysis of survival data is the Cox-Snell residual, the Cox-Snell residual is given by (Cox and SNELL). For the parametric regression problem, analogs of the semi parametric residual plots can be made with a redefinition of the various residuals to incorporate the parametric form of the baseline hazard rates (Klein, 2012). The Cox-Snell residual for the th k individual with observed survival time k t is defined Where Ĥ is refer to the cumulative hazard function of the fitted model. If the model fits the data, then the k r 's should have a standard (  =1) exponential distribution, so that a hazard plot k r versus the Nelson-Aalen estimator of the cumulative hazard of the k r 's should be a straight line with intercept zero and slope unity.
Note: the Cox-Snell residuals will not be symmetrically distributed about zero and cannot be Negative.   The mean, median, minimum, maximum and standard deviation of continuous variables included in this study is given table2 below.

The Kaplan-Meier Survival Estimate for Recovery Time of hypertension Patients
Non-parametric survival analysis is very important to visualize the survival of time-to-recovery of patients from hypertension at FHRH under different levels of covariates. Moreover, it gives information on the shape of the survival and hazard functions of hypertension data set given figure 1 below. Survival time of distributions for time-to-recovery is estimated for each group using the K-M method and in order to compare the survival curves of two or more groups, logrank and Wilcoxon tests.    BUN is ≤35 (mg/dL) (i.e. patients whose blood urea nitrogen ≤35 (mg/dL) have shorter recovery time compared to those have blood urea nitrogen >35(mg/dL).

Comparison of Survival Experiences of Recovery Time
It is vital to do some statistical tests that will be used as initiation to our subsequent findings.
Here we start with the test of equality of probabilities across different groups of categorical variables using Log Rank and Breslow tests. The results of both log-rank and Breslow test for survival difference were highly significant showed table3 below. By using both tests there were significant differences in survival experience among groups of sex, residence, related disease, creantine, blood urea nitrogen and number of medications. These showed that all categorical variables had statistically significant differences in survival probabilities.

Univariable Analysis
A univariable analysis was performed in order to see the effect of each covariate on recovery time of hypertension patients and to select variables to be included in the multivariable analysis.
The result of univariable analysis indicated that age, sex, residence, SBP, DBP, creantine, blood urea nitrogen and number of medications were significant for Cox proportional hazards model at 25% level of significance. Therefore all predictor variables were included in the multivariable analysis.

Multivariable analysis
The multivariable analysis of recovery time of hypertension patients using Cox proportional hazards model was fitted by including all the covariates significant in the univariable analysis at 25% level of significance. Covariates which become insignificant in the multivariable analysis were then removed one by one from the model starting from the largest p-value by using purposeful variable selection technique. Accordingly, the covariates age, sex, residence, systolic blood pressure, related disease, creantine, blood urea nitrogen and the interaction between age and sex were significant at 5% level of significance (Appendix B).

Assumption checking for Cox proportional hazard model
While the Goodness of fit testing approach is employed for Cox PH model, it provides a test statistic and p-value for assessing the PH assumption for a given predictor of interest. Then the PH assumption was checked by using graphical method, adding time dependent covariates in the Cox model and tests based on the Schoenfeld residuals.

Test of proportional hazard assumption by generating time varying covariates
Generating time varying covariates by creating interactions of the predictors and a function of survival time is another method of checking proportional hazard assumption. The output in appendix B Table 5 indicates that the p-value of the covariates sex, related disease, creantine, and blood urea nitrogen was less than 0.05, indicating that the proportional hazard assumption was not satisfied.

Test of proportional hazard assumption by schoenfeld residual
The Schoenfeld residual is one of the methods used to check the PH assumption. In this study the p-value was checked for testing the assumption is fulfilled or not. The global test result in (Appendix B Table 6) shows that p-value is significant (p-value=0.021). This implied that there is evidence to contradict the proportionality assumption. So the proportionality assumption is not fulfilled.  implying that the proportional hazards assumption has been violated.

Accelerated Failure Time Model
Since proportional hazards assumption was not satisfied, the accelerated failure time model is an alternative model for the analysis of this data. We fitted the data using accelerated failure time model with Weibull, Lognormal and Log-logistic as a baseline distribution. For each model fit, we did both univariable and multivariable analysis.

Univariable Analysis
A univariable analysis was performed to select variables to be included in the multivariable analysis. The univariable analysis was fitted for each covariate by AFT model using different baseline distributions. From the outputs in univariable analysis (Appendix A), age, sex, systolic blood pressure, diastolic blood pressure, related disease, creantine, blood urea nitrogen and number of medication were significant at 25% level of significance in all base line distributions.
Hence, based on the univariable analysis these variables were included in the multivariable analysis.

Multivariable analysis and Model Comparison
Multivariable analysis of Weibull, lognormal, and log-logistic models was done by using all significant covariates at univariable analysis. The output of multivariable analysis in appendix B indicates that age, related disease, creantine, blood urea nitrogen and the interaction between age and blood urea nitrogen were significant at 5% level of significant in all AFT models. The covariate systolic blood pressure was significant in Weibull and lognormal AFT models. While, test of ph assumption diastolic blood pressure and the interaction between blood urea nitrogen and systolic blood pressure were significant in Log logistic model. Model comparison was done using the covariates that are significant in multivariable analysis. AIC and BIC were used to compare models. The best model is the one with the lowest value of AIC and BIC. From Table 4 below the AIC and BIC values of Weibull were the smallest (170.58 and 205.46) implying that Weibull accelerated failure time model was the best model to describe the hypertension data.

AIC= (Akaike's Information Criteria) and BIC= (Bayesian Information Criteria)
An acceleration factor greater than one (positive coefficient) indicates extending the recovery time while an acceleration factor less than one (negative coefficient) indicates shortened recovery time. Based on Table 5 the predictors: age, systolic blood pressure, blood urea nitrogen>1.5 mg/dL and creantine>35 mg/dL had positive coefficient indicating that these factors extend recovery time. Patients having kidney disease, other disease and have no disease had acceleration factor less than one (negative coefficient) indicating that they have shorter recovery time as compared to diabetic patients (ref). The interaction between blood urea nitrogen and age had acceleration factor less than one indicate that this factor shorten the recovery time.
A statistical test for H0: ) log( p = 0 yields a p-value of 0.000.We would reject the null hypothesis and decide p is not equal to 1, suggesting that the exponential model is not appropriate.

Parametric Shared Frailty Model Results
In the previous section, three AFT models were fitted and compared to analyze the survival time of hypertension patients. To identify baseline distribution this study used Akaike Information Criteria and Bayesian Information Criteria. As a resultshowed in   Table below indicates a chi-square value of 21.48 with one degree of freedom resulting a highly significant with p-value of 0.000. This implied that the frailty component had significant contribution to the model. The Kendall's tau (τ) is used to measure the association within the residence. From the results of this study the values of Kendall's tau (τ) for the Weibull-Inverse Gaussian frailty was 0.141. The estimated value of shape parameter of Weibull-Inverse Gaussian frailty was (p=3.803). This indicates that the shape of hazard functions is increases up as time increase since its value is greater than one.

Comparison of Weibull AFT and Weibull-Inverse Gaussian shred frailty models
In this study, in order to compare the efficiency of the models AIC and BIC were used. As showed  BIC=205.46). These indicate that the Weibull-Inverse Gaussian shared frailty model is best model for hypertension dataset. Weibull-Inverse Gaussian shared frailty model take in to account the clustering effect (θ is significant at 5% level of significant), for this data Weibull-Inverse Gaussian shared frailty is better fit than Weibull accelerated failure time model.

Model diagnosis
After the model has been fitted, it is important to determine whether a fitted parametric model adequately describes the data or not. The study used two different methods to diagnose the parametric baselines.

Diagnostic Plots of the Parametric Baselines
To check the adequacy of the baseline accelerated failure time models with baseline Weibull, baseline log logistic and baseline log-normal can be graphically evaluated by plotting log(−log against log (t) respectively. If the plot of parametric model is approximately linear, the given baseline distribution is appropriate for the given dataset. Then based on the plots given in figure 6 below, the plot for the Weibull baseline distribution is approximately a straight line compare to that of log-logistic and lognormal baseline distribution. This evidence also support the decision made comparing baseline AFT models by using AIC value that Weibull baseline distribution is appropriate for the given dataset.

Cox-Snell residual plots
The Cox-Snell residuals method can be applied for parametric models. The Cox-Snell residuals are important to investigate how well the model fits the data. In this study we used the Cox-Snell residuals to check the overall goodness of fit for various parametric models. The Cox-Snell plot of Figure7 given below indicates that the Weibull AFT model fit the data better compare to log normal and log logistic AFT models. Nevertheless, the results of Cox-Snell were consistent with the results based on AIC and BIC. The plot makes approximately straight line through the origin for Weibull baseline distribution suggesting that it is appropriate for hypertension dataset.

Interpretation of Weibull-Inverse Gaussian shared frailty model results
From Weibull-Inverse Gaussian shared frailty model results in Table 7 above, controlling for other variables, Acceleration factor, 95% CI and p-value of the variable systolic blood pressure were 1.002, (1.00, 1.004) and .022 respectively. These indicated that systolic blood pressure was significant factors for the recovery time and for a one mmHg increase in systolic blood pressure the recovery time of hypertensive patients were prolonged by 1.002 times in month.
Acceleration factor and its 95% CI for related disease category of kidney, others and none were (ϕ =.859, 95% CI=0.760, 0.971), (ϕ= .628, 95% CI=0.550, 0.718) and (ϕ=.557, 95% CI=0.478, 0.649) respectively. This showed that patients having kidney disease, other and have no related disease shortened the recovery time by a factor of .859, .628 and .557 respectively compared to diabetes patients (Reference).The 95% CI did not include one showed that related disease was significantly important factor for the recovery time at 5% level of significance.
Acceleration factor and its 95% CI for the variable creantine were 1.142 and (1.051, 1.241) respectively. These indicated that creantine was significant factors for the recovery time and Showed that the recovery time of patients, who have had creantine level >1.5 mg/dL, were prolonged by 1.142 times as compared to patients those who have had creantine level ≤1.5 mg/dL.
The interaction between blood urea nitrogen and age had statistically significant effect on the recovery time, suggesting that as age increased by one year, the acceleration factor for patients who have had blood urea nitrogen >35 mg/dL was prolonged by exp (.013-.012) =1.001 with p value of .000 as compared to those patients who have had blood urea nitrogen ≤35 mg/dL.

Discussions
Hypertension remains a major public health problem that is associated with morbidity and mortality and the prevalence is still increasing (Chobanian, 2015). Therefore, this study attempted to identify factors that affect time-to-recovery from hypertension patients at FHRH, Bahir Dar using survival models. Factors concerned for this study were age, sex, residence, systolic blood pressure, diastolic blood pressure, related disease, creantine, blood urea nitrogen and number of medication. Lognormal, Weibull and log-logistic AFT distributions and gamma and inverse Gaussian frailty distributions were used to analyze this survival data.
Both univariable and multivariable analysis was performed to examine factors that affect recovery time of hypertension patients. The univariable analysis given in Appendix A revealed that variables age, sex, SBP, DBP, related disease, creantine, BUN and number of medication were significant at 25% level of significance. All significant covariates in univariable analysis were included in multivariable analysis.
This research showed that there was a clustering (frailty) effect on recovery time of hypertension patients which might be due to the heterogeneity in residence. Then, patients coming from the same residence share similar risk factors related to recovery time, and indicating that it was important to consider the cluster effect.
Comparison of models for different multivariable analysis was done by using AIC and BIC criteria. In this study, Weibull-inverse Gaussian shared frailty model had the smallest AIC and BIC i.e., 150.08 and 189.47 respectively, which is appropriate model for describing recovery time of patients with hypertension. The obtained results were discussed as follows: The finding of this study showed that systolic blood pressure had positive significant effect on recovery time that is as systolic blood pressure increase the recovery time of hypertension patients also prolonged. This finding was consistent with ( Gesese, 2017;Xiong et al., 2013).
Related disease is another prognostic factor that significantly predicts the recovery time of patients with hypertension. And showed that patients having kidney disease, other and have no related disease shortened the recovery time by a factor of .859, .628 and .557 respectively compared to diabetes patients (Reference). This study is supported by (Workie et al., 2017).
Another potential risk factor that accelerates recovery status of patients from hypertension was creantine and showed that the recovery time of hypertensive patients, who have had creantine >1.5 mg/dL, were prolonged by 1.142 times compared to patients those who have had creantine ≤1.5 mg/dL. This study was consist with (Zhang et al., 2018;Xiong et al., 2013).
The results of this study also suggested that the interaction between blood urea nitrogen and age had statistically significant effect on the recovery time, suggesting that as age increase by one year, the acceleration factor for patients who have had blood urea nitrogen level >35 mg/dL was prolonged by 1.001 as compared to those who have had blood urea nitrogen ≤35 mg/dL.
The related literatures have not been done by using interaction effect of variables in the area of hypertension. So, this study is unique that interaction effect of variables was assumed to see whether the effect one variable is different or not on the acceleration factor of another variable.
In this study, the time of recovery for patients with hypertension did not depend on sex which is similar with the study of (Belachew et al., 2018;Angaw et al., 2015;Mungati et al., 2014) suggested that sex was not statistically associated with hypertension using logistic regression model in Northwest Ethiopia , Addis Ababa and Zimbabwe respectively.

Conclusions
This study was based on hypertension patient dataset which is obtained from FHRH, Bahir Dar with an aim to identify predictors of recovery time of patients by applying survival models.
Parametric shared frailty models with different parametric baseline distributions and AFT models were applied. Among this using AIC and BIC, Weibull-inverse Gaussian shared frailty model is the best fitted model for the recovery time of patients with hypertension. There was a frailty (clustering) effect on time-to-recovery from hypertension that arises due to differences in distribution of timing of recovery among residence in Felege Hiwot referral hospital. This indicates the presence of heterogeneity and necessitates for the frailty models.
The result of Weibull-inverse Gaussian shared frailty model showed systolic blood pressure, related disease, creantine and the interaction between blood urea nitrogen and age were found significant predictors for the recovery time of patients among patients with hypertension in Felege Hiwot referral hospital. Among these significant predictors, systolic blood pressure and creantine prolong timing of recovery while, related disease and the interaction between blood urea nitrogen and age shorten timing of recovery for patients with hypertension. On the other hand, sex, diastolic blood pressure and number of medication were not significant predictors for the recovery time of patients with hypertension.

Ethics approval and consent to participate
The authors got an ethical approval certificate from Bahir Dar University Ethical approval committee, Bahir Dar, Ethiopia with Ref≠ BDU/54/2018 to use the secondary data related to patients. The Ethical approval certificate obtained in this committee can be attached up on request. Since the data used in the current investigation was secondary, there was no verbal or written consent from the participants.

Consent for publish
Not applicable.

Availability of data and materials
All data supporting the findings are contained in the manuscript. Anyhow, data are available from the corresponding author on rational request.

Competing interests
The author declares that they have no competing interests

Funding
No funding was received to provision the writing of this paper.

Authors' contribution
Nigist mulu and Yeshambel kindu wrote the proposal, analysis the data and draft the paper.
Yeshambel Kindu prepared the first draft of the manuscript. Yeshambel Kindu and Abay Kassie reviewed the manuscript .all authors read and approved the final manuscript.