A semi-parametric mixed models for longitudinally measured fasting blood sugar level of adult diabetic patients
Abstract
Background
At the diabetic clinic of Jimma University Specialized Hospital, health professionals provide regular follow-up to help people with diabetes live long and relatively healthy lives. Based on patient condition, they also provide interventions in the form of counselling to promote a healthy diet and physical activity and prescribing medicines. The main purpose of this study is to estimate the rate of change of fasting blood sugar (FBS) profile experienced by patients over time. The change may help to assess the effectiveness of interventions taken by the clinic to regulate FBS level, where rates of change close to zero over time may indicate the interventions are good regulating the level.
Methods
In the analysis of longitudinal data, the mean profile is often estimated by parametric linear mixed effects model. However, the individual and mean profile plots of FBS level for diabetic patients are nonlinear and imposing parametric models may be too restrictive and yield unsatisfactory results. We propose a semi-parametric mixed model, in particular using spline smoothing to efficiently analyze a longitudinal measured fasting blood sugar level of adult diabetic patients accounting for correlation between observations through random effects.
Results
The semi-parametric mixed models had better fit than the linear mixed models for various variance structures of subject-specific random effects. The study revealed that the rate of change in FBS level in diabetic patients, due to the clinic interventions, does not continue as a steady pace but changes with time and weight of patients.
Conclusions
The proposed method can help a physician in clinical monitoring of diabetic patients and to assess the effect of intervention packages, such as healthy diet, physical activity and prescribed medicines, because individualized curve may be obtained to follow patient-specific FBS level trends.
Keywords
Diabetes mellitus Fasting blood sugar Linear mixed model Semi-parametric mixed modelAbbreviations
- AIC
Akaike’s information criteria
- BIC
Bayesian information criteria
- FBS
Fasting blood sugar level
- IDF
International diabetes federation
- JUSH
Jimma University specialized hospital
- P-IRLS
Penalized iteratively reweighted least squares
- REML
Restricted maximum likelihood
- RLRT
Restricted likelihood ratio tests
- SD
Standard deviation
Background
Diabetes mellitus is a metabolic disorder of multiple etiology characterized by chronic hyperglycaemia with disturbances of carbohydrate, fat and protein metabolism resulting from defects in insulin secretion, insulin resistance, or both [1]. The long-term effects of untreated diabetes mellitus might results in health complications, such as visual disability and nerve disease [2, 3, 4, 5], among others. A person is considered to be diabetic if he or she has fasting blood sugar (FBS) level value of greater than or equal to 7.0 mmol/L (126 mg/dL) or 2-h blood sugar level of greater than or equal to 11.1 mmol/L (200 mg/dL) or glycated hemoglobin (HbA_{1}) level of 6.5% or higher [6].
There are three main types of diabetes, namely type 1 diabetes, type 2 diabetes and gestational diabetes. The type 1 diabetes is caused by an auto-immune reaction, in which the patient body defense system attacks the insulin producing beta cells in the pancreas and hence the body can no longer produce the insulin it needs. Whereas in type 2 diabetes, the body is able to produce insulin, however it becomes resistant so that the insulin is ineffective. The type 2 diabetes is characterized by high levels of blood sugar or glucose resulting from defects in insulin production, insulin action, or both. The gestational diabetes is a form of diabetes that appears during pregnancy. It can lead to serious health risks for both the mother and child [7]. The risk factors that are associated with type 1 diabetes include family history of diabetes (diabetes history in one parent or both), infections and other environmental influences such as exposure to a viral illness, the presence of damaging immune system cells, i.e. autoantibodies and dietary factors low vitamin D consumption [8]. Whereas, for type 2 diabetes the risk factors are excess body weight, physical inactivity, poor nutrition, family history of diabetes, past history of gestational diabetes and older age [9]. The risk factors for increase or decrease in fasting blood sugar level of a patient include overweight, family history of diabetes, age, type of diabetes, blood pressure and gender [7]. The focus of this study however is on type 1 and type 2 diabetes.
In year 2015, there were an estimated 415 million adults aged 20–79 years living with diabetes worldwide [10], including 193 million who are undiagnosed. There were approximately 5 million people estimated to have died from diabetes worldwide in the same year, and a majority of these were the result of cardiovascular complications. In Africa Region, the number of adults living with diabetes estimated at 14.2 million whereas in Ethiopia the number is estimated 1 to 10 million in year 2015. The Region has the highest proportion of undiagnosed diabetes, 9.5 million (about 66.7%) of people with diabetes are unaware they have the disease and in Ethiopia there are 500 thousand to 5 million such cases [11, 12].
At the diabetic clinic of Jimma University Specialized Hospital (JUSH), health professionals provide regular follow-up to help people with diabetes live long and relatively healthy lives. Depending on patients conditions, e.g. FBS level, they also provide interventions in the form of counselling to promote a healthy diet and physical activity and prescribing medicines.
The main objective of the current study is to assess the factors that affect the FBS level of adult diabetic patients. In addition to assessing the factors that affect the FBS level over time, we are also interested to estimate the rate of change of FBS profile experienced by patients over time. The change may help to assess the effectiveness of interventions taken by the clinic to regulate FBS level, where rates of change close to zero over time may indicate the interventions are good regulating the level. These changes are determined using first derivatives of penalized regression splines [13, 14].
The rest of the paper is organized as follows. The data, some basic review of variance-covariance structure of the parametric linear mixed model, semi-parametric mixed models and inferences related them are introduced in “Methodology” section. The results from applying these methods on the the study data are discussed in “Results” section. Finally discussion, and conclusions and pointers for future study are given in “Discussion” and “Conclusion” sections respectively.
Methodology
Study data
The fasting blood sugar (FBS) level data used in this paper arises from a retrospective study conducted in Jimma University Specialized Hospital (JUSH) diabetic clinic. The hospital is located in Jimma town 352 km to the Southwest of Addis Ababa, the capital of Ethiopia. It is a teaching hospital and gives service to the southwestern part of Oromia region, some part of southern nations and nationalities and Gamella regions of Ethiopia. All diabetic patients aged 18 years or older, who were coming to JUSH diabetic clinic for their regular follow up during periods September 2011 and June 2014 were eligible for this study. During their follow up, patients FBS level along with other characteristics such as weight are measured and recorded in the individual follow up chart. The data in the chart include time (measured in months, where baseline or initial date was given a value 0), patient gender, age, type of diabetes (Type 1 diabetes or Type 2 diabetes) and family diabetes history. The duration between initial and the last recorded visits ranged from one to 36 months. Patients with at least two observations were included in the analyses leading to a total of 534 patients and 4390 observations. Permission of the study was obtained from Postgraduate research office of Jimma University, College of Natural Sciences and JUSH.
Variance-covariance structures and inference
Variance-covariance structures
Linear mixed models for selection of variance-covariance structure for FBS level, JUSH, September 2011 - June 2014
Subject-specific random effect | Linear mixed model |
---|---|
M_{1}: Random intercept | \(Y_{ij} = \beta _{0} + \beta _{1}\,t_{ij} + \beta _{2}\,t_{ij}^{2} + \sum _{l=1}^{L} \theta _{l}\,x_{ijl} + b_{0_{i}} + \varepsilon _{ij}\) |
M_{2}: Linear random effects | \(Y_{ij} = \beta _{0} + \beta _{1}\,t_{ij} + \beta _{2}\,t_{ij}^{2} + \sum _{l=1}^{L} \theta _{l}\,x_{ijl} + b_{0_{i}} + b_{1_{i}}\,t_{ij} + \varepsilon _{ij}\) |
M_{3}: Quadratic random effects | \(Y_{ij} \,=\, \beta _{0} \,+\, \beta _{1}\,t_{ij} \,+\, \beta _{2}\,t_{ij}^{2} \,+\, \sum _{l=1}^{L} \theta _{l}\,x_{ijl} \,+\, b_{0_{i}} \,+\, b_{1_{i}}\,t_{ij} \,+\, b_{2_{i}}\,t_{ij}^{2} \,+\,\varepsilon _{ij}\) |
In Table 1, for instance the subject-specific random intercept \(b_{0_{i}}\) in the quadratic random effects model (M_{3}) is considered to capture correlation of the FBS level measurements over time within the patient and it is assumed that subject-specific random slopes for linear as well as for quadratic time effects to capture different evolution of FBS level over time. Note that these subject-specific random structures are different for each patient.
Tests for zero variance components
Adequate variance-covariance structure is essential to obtain valid model based inferences for the fixed effects or for parameters in the mean structure of the model [18]. Over-parametrization of the variance-covariance structure leads to inefficient estimation and potentially poor assessment of standard errors for the estimation of the mean structure, i.e. fixed effects, whereas a too restrictive specification invalidates inferences about the mean response profile when the assumed structure does not hold.
The likelihood ratio test for testing, for example \(H_{0}: \sigma ^{2}_{b_{0}} = 0\) versus \(H_{1}: \sigma ^{2}_{b_{0}} > 0\) for model M_{1}, has an asymptotic \(0.5\,\chi ^{2}_{0} + 0.5\,\chi ^{2}_{1}\) mixture distribution under H_{0} [19], if the vector of FBS level can be divided into a large number of independent and identically distributed sub-vectors both under H_{0} and H_{1}. However, this assumption usually does not hold, for example in linear mixed models or for unbalanced data [20, 21, 22]. Note that the FBS level data are unbalanced in the sense that all patients do not have equal number of measurements, hence the independent and identically distributed assumption can be violated in the linear mixed models used in this paper. Therefore, we used the exact finite sample null distribution of the restricted likelihood ratio test (RLRT) statistic derived by Crainiceanu and Ruppert [22] to test a zero random effect variance in M_{1}. However, since models M_{2} and M_{3} contain more than one random effect, the tests for a zero random effect variance in these models were done using the exact finite sample null distribution of the RLRT statistic derived by Greven et al. [21].
Semi-parametric mixed effects model
Estimation of parameters
where β=(β_{0},β_{1},…,β_{p},θ_{1},…,θ_{L})^{′} is a (p+L+1)×1 vector of fixed effects which is common to the n individuals, X_{i} is an m_{i}×(p+L+1) design matrix associating β to y_{i}, v=(b_{1},b_{2},…,b_{K}) is a K-dimensional vector of random coefficients in the summand in Eq. (2), Z_{i(f)} is the m_{i}×K matrix for the pth-degree spline basis functions, \(\textbf {u}_{i} = \left (b_{0_{i}}, b_{1_{i}}, b_{2_{i}}\right)'\) is subject-specific vector of random effects, Z_{i(u)} is an m_{i}×3 design matrix which relates u_{i} to the response y_{i} and \(\textbf {e}_{i} = \left (e_{1i}, e_{2i}, \ldots, e_{{im}_{i}}\right)'\) is an m_{i}-dimensional vector of within-individual errors. Furthermore, it is assumed that \(\textbf {v} \sim \mathcal {N}\left (\textbf {0}, \sigma _{b}^{2}\,\textbf {I}_{K}\right)\), \(\textbf {u}_{i} \sim \mathcal {N}(\textbf {0}, \textbf {G})\), \(\textbf {e}_{i} \sim \mathcal {N}\left (\textbf {0}, \textbf {R}_{i}\right)\), v, u_{i} and e_{i} are assumed to be pairwise independent with and between subjects for i=1,2,…,n. Note that G and R_{i} are 3×3 and m_{i}×m_{i} variance-covariance matrices, respectively.
and \(\phantom {\dot {i}\!}\textbf {b} = (b_{1}, b_{2}, \ldots, b_{k}, b_{0_{1}}, b_{1_{1}}, b_{2_{1}}, \ldots, b_{0_{n}}, b_{1_{n}}, b_{2_{n}})'\). Estimation of the coefficients of penalized and unpenalized terms in model (4) was done using a penalized iteratively reweighted least squares (P-IRLS) based on 20 equidistant knots in the range of FBS level and a smoothing parameter selection was done by REML [23].
The correspondence between the penalized spline smoother and the optimal predictor in a mixed model framework enables us to take advantage of the existing methodology for mixed model analysis and the use of mixed model software, such as the function gamm in mgcv R package, for fitting the penalized spline model and the MIXED and GLIMMIX procedures in SAS [24]. This implementation of penalized smoothing in the linear mixed model framework also provides an automated approach to obtain a smoothing parameter and flexibility to extend the models [17].
In this paper, parameters in the fitted models are estimated by restricted maximum likelihood (REML) method because the statistical hypotheses that were considered have the same mean structures between models under the null and alternative hypotheses. Furthermore, maximum likelihood estimators of variance components are biased downward as they do not take into account the degrees of freedom lost in the estimation of fixed effects (e.g. see Ruppert et al. [16]).
Model selection and inference
Testing whether the inclusion of spline effects in the parametric model improves model fit or not is equivalent to testing \(H_{0}: \sigma ^{2}_{b} = 0\) versus \(H_{1}: \sigma ^{2}_{b} > 0\). In this paper, due to the second objective of the study, a quadratic penalized spline was added in Eq. (1), therefore neither of the two methods discussed in “Variance-covariance structures and inference” section can be used to test \(H_{0}: \sigma ^{2}_{b} = 0\) [27] instead an approximate F-test of Hastie and Tibshi [28] was applied. For Hastie and Tibshi approximate F-test, residual degrees of freedom for the null and alternative model fits are used in the place of the number of parameters in each model.
Rate of change over time and simultaneous confidence bands
The quantile h_{(1−α)} can be approximated using simulations. First we simulate from realization of (5) and computation of (7) can be repeated for a large number of times, say N times, to obtain \(\tilde {h}^{1}_{1-\alpha }, \tilde {h}^{2}_{1-\alpha }, \ldots, \tilde {h}^{N}_{1-\alpha }\). The value with rank N×(1−α) is used as h_{1−α}.
The proposed semi-parametric mixed models were fitted with the the gamm function available in R package mgcv [29] and the linear mixed models using the lme function available in R package nlme.
Results
Patients baseline characteristics
Baseline characteristics of adult diabetic patients in JUSH, September 2011 - June 2014
Type of diabetes | |||||
---|---|---|---|---|---|
Characteristics | Type 1 | Type 2 | p-value | Overall | |
Gender | Male, N (%) | 87 (16.29%) | 255 (47.75) | 0.9935 | 342 (64.04%) |
Female, N (%) | 48 (8.99%) | 144 (26.97%) | 192 (35.96%) | ||
Family history | No, N (%) | 37 (6.93%) | 380 (71.16%) | <0.0001 | 417 (78.09%) |
Yes, N (%) | 98 (18.35%) | 19 (3.56%) | 117 (21.91%) | ||
Age, mean (SD) | 34.55 (11.92) | 48.63 (13.78) | <0.0001 | 45.4 (14.62) | |
Weight, mean (SD) | 58.83 (11.10) | 64.02 (13.74) | <0.0001 | 62.83 (13.36) | |
FBS, mean (SD) | 171.38 (102.39) | 162.73 (80.66) | 0.0139 | 164.72 (86.20) |
Parametric mixed models
Mean structure
where Type and F.History represent diabetes type and family history of diabetes, respectively.
Variance-covariance structure for random effects
Parameter estimates (standard errors, s.e.), p-values for associated t-tests and model fit criteria, FBS level of diabetes patients in JUSH, September 2011 - June 2014
Variance-components | ||||||
---|---|---|---|---|---|---|
Effects | Random intercept | Linear random effects | Quadratic random effects | |||
Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | |
Fixed effects | ||||||
Intercept | 304.362 (14.616) | <0.0001 | 306.756 (15.743) | <0.0001 | 303.139 (15.678) | <0.0001 |
Age | 0.252 (0.183) | 0.1693 | 0.212 (0.179) | 0.2362 | 0.197 (0.179) | 0.2699 |
Gender, Male | -2.605 (5.487) | 0.6352 | -1.968 (5.983) | 0.7424 | -2.609 (5.933) | 0.6603 |
Diabetes type, Type 2 | -9.758 (8.697) | 0.2624 | -10.553 (8.814) | 0.2317 | -10.581 (8.852) | 0.2325 |
Family history, Yes | -12.763 (8.478) | 0.1328 | -12.335 (8.606) | 0.1523 | -12.593 (8.643) | 0.1457 |
Time | -4.462 (0.870) | <0.0001 | -5.614 (1.071) | <0.0001 | -5.549 (1.116) | <0.0001 |
Time^{2} | 0.123 (0.018) | <0.0001 | 0.135 (0.020) | <0.0001 | 0.153 (0.025) | <0.0001 |
Weight | -1.981 (0.196) | <0.0001 | -1.991 (0.216) | <0.0001 | -1.906 (0.215) | <0.0001 |
Time × Weight | 0.016 (0.013) | 0.2139 | 0.032 (0.016) | 0.0439 | 0.025 (0.016) | 0.1162 |
Gender, Male × Time | -0.412 (o.363) | 0.2563 | -0.482 (0.443) | 0.2761 | -0.425 (0.444) | 0.3390 |
Variance components | ||||||
var(b_{0}) | 2135.023 | 2797.766 | 3352.606 | |||
var(b_{1}) | 4.575 | 40.343 | ||||
var(b_{2}) | 0.048 | |||||
Residual | 5023.386 | 4873.227 | 4723.609 |
The fixed effect estimates were consistent in sign but have slight differences in magnitude across the three different variance-covariance structures. The variables age, gender, diabetes type, family history, and time by weight and gender by time interactions were statistically nonsignificant in all models, except for time by weight interaction where its p-value marginally significant for subject-specific random intercept and slope model (i.e. linear random effects model). The covariates that were statistically significant at 5% level, i.e. Time, Time^{2} and weight and the time by weight interaction were retained for the subsequent analysis.
Parameter estimates (standard errors, s.e.) and p-values for associated t-tests for model M_{4}, FBS level of diabetes patients in JUSH, September 2011 - June 2014
Effects | Estimate (s.e.) | p-value |
---|---|---|
Fixed effects | ||
Intercept | 302.931 (13.330) | <0.0001 |
Time | -5.815 (1.061) | <0.0001 |
Weight | -1.968 (0.212) | <0.0001 |
Time × Weight | 0.031 (0.016) | 0.0509 |
Time^{2} | 0.134 (0.020) | <0.0001 |
Variance components | ||
var(b_{0}) | 2797.887 | |
var(b_{1}) | 4.601 | |
Residual | 4877.259 |
Semi-parametric mixed model
Parameter estimates (standard errors, s.e.), p-values for associated t-tests and variance components estimates of semi-parametric models under various variance structures, FBS level of diabetes patients in JUSH, September 2011 - June 2014
Variance structures | ||||
---|---|---|---|---|
Effects | Random intercept | Linear random effects | ||
Estimate (s.e.) | p-value | Estimate (s.e.) | p-value | |
Fixed effects | ||||
Weight | -1.908 (0.191) | <0.0001 | -1.899 (0.212) | <0.0001 |
Time | 28.264 (6.087) | <0.0001 | 26.742 (6.359) | <0.0001 |
Time × Weight | 0.017 (0.013) | 0.1837 | 0.031 (0.016) | 0.0536 |
Time^{2} | 0.408 (0.402) | 0.3095 | 0.448 (0.421) | 0.2875 |
s(Time)Fx1 | -2971.649 (551.992) | <0.0001 | -3014.737 (579.734) | <0.0001 |
Variance components | ||||
Standard deviation | ||||
Intercept | 2104.479 | 2796.166 | ||
Linear | 4.814 | |||
Residual | 4919.429 | 4762.647 | ||
s(Time) | 13.287 | <0.0001 | 13.939 | <0.0001 |
The results in Table 5 show that the fixed effects estimates were consistent in sign but have slight difference in magnitude in both semi-parametric and parametric mixed models (see Table 4), except for the effect of time where both the sign and magnitude of its coefficient estimates were different in the two models and the effect of "time square" was nonsignificant in the semi-parametric mixed models. Further, the interaction of weight with time was not statistically significant in any of the semi-parametric mixed model. Except for the subject-specific random slope variance component, there is a slight decrease in subject-specific random intercept and residual variance components in the semi-parametric model compared to variance components in the linear mixed model M_{4} (see Table 4).
Fit statistics for model M_{5} and M_{4}, FBS level of diabetes patients in JUSH, September 2011 - June 2014
Fit statistics | |||||
---|---|---|---|---|---|
Variance structure | −2 log(Lik) | AIC | BIC | E _{ p} | AIC _{adj} |
M_{5} | |||||
Random intercept | 50538.54 | 50554.54 | 50605.63 | 7.087 | 50545.627 |
Random linear | 50507.09 | 50527.09 | 50590.96 | 7.260 | 50514.350 |
M_{4} | 50583.51 | 50601.51 | 50658.98 |
Model selection
The approximate F-test statistic for testing the above hypotheses, i.e. quadratic form of f(t_{ij}) against a quadratic penalized splines, is 83.63 with p-value <0.0001. This strongly suggests a rejection of the null hypothesis \(H_{0}: \sigma ^{2}_{b} = 0\). Thus, the shape of the function f(t_{ij}) is statistically different from a quadratic trend.
Furthermore, consider the semi-parametric mixed model M_{5} in Eq. (10) with random linear effects variance-covariance structure and the linear mixed model M_{4} in Eq. 9. The fit statistics from fitting these two models are displayed in Table 6. The −2 log(Lik), AIC and BIC values indicate a substantial improvement in the fit of M_{5} compared to M_{4}, implying model with penalized spline representation of FBS level was preferred over its parametric counterpart.
The overall results show that, out of the models evaluated, FBS level of diabetes patients at the JUSH diabetic clinic during the study period best characterized by a penalized spline model with truncated quadratic basis, with subject-specific random intercept and slope effects and with linear function of weight and time, called the final model, M_{6}.
Simultaneous confidence band
The confidence bands become noticeably wider after 27 months of follow-up period, demonstrating the increased variability. This increase may be due to a smaller number of FBS level recordings being observed at the later period of the study or a potential artifact induced by the spline smoothing [32]. In practice spline smoothing creates a challenge in semi-parametric regression settings through the inherent bias from using truncated basis functions. The confidence bands obtained for FBS level does not account for this function bias. However, this bias could be corrected, e.g. using bootstrapping methods [33].
Discussion
This study focused on longitudinal data analysis of fasting sugar level of adult diabetic patients at Jimma University Specialized Hospital diabetic clinic using an application of semi-parametric mixed model. The study revealed that the rate of change in FBS level in diabetic patients, due to the clinic interventions, does not continue as a steady pace but changes with time and weight of patients. Furthermore, it clarified the associations between FBS level and some characteristics of adult diabetic patients that weight of a diabetes patient has a significant negative effect whereas patient gender, age, type of diabetes and family history of diabetes did not have a significant effect on the change of FBS level. The result on gender agrees with the findings of [34] where the gender effect on fasting blood glucose level of type 2 diabetes was statistically nonsignificant.
Under the two variance-covariance structures of subject-specific random effects, the semi-parametric mixed models had better fit than their parametric counterparts. This was likely due to the localized splines which captured more variability in FBS level than the linear mixed models. The methodology used in the analysis has implications for clinical monitoring in regular followup of diabetic patients and to assess the effect of intervention packages, such as healthy diet, physical activity and prescribed medicines, because individualized curve may be obtained to follow patient-specific FBS level trends [31].
The main limitation of the study is the limited information on important predictors such as type of interventions including treatment types and nutritional status of a patient that may have influenced the rate of change in FBS level. Due to lack of data on these potential predictors for most of the patients involved in the study, we were unable to include them in the analyses. Therefore, more public health and epidemiology researches are needed to examine the impact of treatments and interventions on population health in general and in particular, people living with diabetes to avoid its complications over time and to identify new risk factors for diabetes.
Conclusion
In this paper, we demonstrate the use of semiparametric mixed effect model for estimation of the rate of change of fasting blood sugar (FBS) level experienced by patients over time. The proposed method can help a physician in clinical monitoring of diabetic patients and to assess the effect of intervention packages such as healthy diet, physical activity.
Notes
Acknowledgements
We thank the Diabetic Clinic of Jimma University Specialized Hospital, Ethiopia for giving access to the data and the staff members of the clinic for their support in extracting the information from patient’s medical card.
Funding
This study project was awarded funding from College of Natural Sciences Research and Postgraduate Office. The second author’s Jimma University visit, when this study started, was financially supported by the Africa Interaction, Knowledge, Interchange and Collaboration (KIC) funding (UID: 105298) from the National Research Foundation (NRF) and the University of South Africa under the Foreign general research grant.
Availability of data and materials
The data sets analyzed in this study available from the corresponding author on reasonable request.
Authors’ contributions
BBY and WKY contributed to the study concept and design. ZMN participated in the data collection and checked the data. TTA performed the statistical analyses and drafted the manuscript. LKD reviewed the findings of data analyses and was a major contributor in writing the manuscript. All authors approved the final manuscript.
Ethics approval and consent to participate
Ethical approval to conduct the study and human subject research approval for this study was received from Jimma University, College of Natural Sciences, Research Ethics Committee and the medical director of the Hospital. As the study was retrospective, informed consent was not obtained from the study participants, but data were anonymous and kept confidential.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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