Research title on Multilevel Modelling of Time to Death for Under-Five Children: 2016 Ethiopia Demographic and Health Survey, 2019

Background: Reducing child mortality is now a global concern. Globally, Under-ve child mortality rate was decreased by 58% in 2017. In the 2016 EDHS report, under-ve mortality was declined to 60% in Ethiopia in 2016. Methods: The data for the study was obtained from EDHS data conducted in 2016. In the study, we analysed the data using stratied Cox proportional hazard model and multilevel lognormal parametric survival model. Results: From the total of 10,331 under-ve children, 635 (6.1%) deaths had occurred in the 2016 EDHS data. And, the overall probability of survival value was near to 0.92 with the estimated mean survival time was 55.4 months. In the study we found that covariates like birth type of the child, family size, wealth index, frequency of listening radio, place of delivery, place of residence, and geographical region were signicant factors for the death of under-ve children in stratied Cox proportional hazard model. In the multilevel lognormal parametric survival model, we found that the random-intercept effects of variations between region and household levels on the mean survival times of the children were 1.7 and 0.9, respectively. These values indicated that we had enough evidence for the existence of unobserved heterogeneities between regions and households. Conclusion: The covariates like birth type of the child, family size, wealth index, frequency of listening radio, place of delivery, place of residence, and geographical region covariates were signicant factors for under-ve children mortality using stratied Cox proportional hazard regression model. In the random-intercept effects model, the two estimated variances of the random-intercept effects for regional and household levels were 1.7 and 0.9, respectively. The values indicate that we have enough evidence that there were unobserved heterogeneities on the mean survival times of the under-ve children between regions and households levels. Further studies should be conducted to identify the individual, household, and community-level factors associated with infant and child mortality in Ethiopia. and place of delivery were signicant covariates. In the random-intercept effects model, the two estimated variances of the random-intercept effects for regional and household levels are 1.7 and 0.9, respectively. The values indicate that we have enough evidence that there were variations (unobserved heterogeneities) on the mean survival times of the under-ve children between regions and households levels. Further studies should be conducted to identify and compare the signicance of the individual, household, and community-level factors associated with infant and child mortality in Ethiopia.


Background
Reducing child mortality is now a global concern. Globally, Under-ve child mortality rate has decreased by 58%, from an estimated rate of 93 deaths per 1000 live births in 1990 to 39 deaths per 1000 live births in 2017. In Sub-Saharan Africa and South-East Asia still carry the burden of 80 percent of all global under-ve deaths. Those two regions share nearly the same part: 39 percent of all deaths occurred in southern Asia whereas 38 percent occurred in sub Saharan region. The astounding thing is that among 5 countries whose half of the global under-ve deaths three of them is part of sub-Saharan countries; those are Nigeria, Ethiopia, and Congo Democratic Republic (1).
As the 2016 EDHS report, under-ve mortality declined from 166 deaths per 1,000 live births in 2000 to 67 deaths per 1,000 live births in 2016. This represents a 60% decrease in under-ve mortality over a period of 16 years. The report also shows that there were regional disparities problems on under-ve children mortality in Ethiopia. And the mortality rates among children under age ve have been interpreted as there were seven regions have observed that their children death rates were above the average rate (84 deaths per 1, 000 live births). These top 7 regions were Affar, Benishangul-Gumuz, Somali, Dire Dawa, Gambela, SNNP, and Amhara have recorded 125 deaths, 98 deaths, 94 deaths, 93 deaths, 88 deaths in each, and 85 deaths per 1, 000 live births, respectively. But, the lowest deaths rate has been observed in Addis Ababa City; it was 39 deaths per 1,000 live births (CSA and ICF, 2016). And addition to this report, other studies also shown that there was much regional variation on the incidence rate of under-ve mortality in Ethiopia (3)(4)(5).
As we have seen that one study has used the EDHS data of 2016 on under-ve children mortality previously. The study showed that there was an existence of unobserved heterogeneity at the regional level using Cox and Frailty models (3). In their study, they have found ve signi cant factors for the under-ve children deaths using semi-parametric gamma frailty model (used two levels). While in our study we found six additional signi cant factors like number of aged 5 and U5 children, children size at birth, months of breastfeeding, family size, place of delivery, and frequency of following media (like listening of radio) for the under-ve children deaths using multilevel lognormal parametric survival model.
And in our study we have considered the multilevel lognormal random-intercept effects model (considered three levels) and the model found that there were two estimated variances of the randomintercept effects for regional and household levels. These variations indicated that there were existences of unobserved heterogeneities between regions and households on the mean survival times of the underve children. Those all are the gabs between the previous study and our study. The objective of the study was to determine risk factors for the under-ve children deaths using multilevel xed effect parametric survival model and to compare variations of random-intercept effects between region and household levels.

Data and Study Population
The data for this study were taken from the 2016 Ethiopian Demographic and Health Survey.

Outcome variable
The response variable considered in the study was the survival time of a child measured in months from birth until death/censor of children aged less than 60 months. The children who lived start within the reference period are taken into consideration. Children died within the reference period were taken as uncensored cases the children alive in that period were censored cases.

Explanatory variables
The potential explanatory variables were considered in the study were clearly stated ( Table 1). Cox proportional hazard regression model The Cox proportional hazard regression model was used for analyzing survival data. It was used to check the existing association between child mortality and life time variables found to have a signi cant association with child mortality. The Cox proportional hazard model used in this study has the following form: Where: h i (t) is hazard of death for the i th child at time t, h 0 (t) is the baseline hazard at time t for X = 0, β is the vector of unknown coe cients of independent variables (X 1 , X 2 , X 3 ...,X P ), and e β i is the hazard ratio.

Multilevel Mixed Effects Parametric Survival Models
The mixed-effects survival models contain both xed effect and random-intercept effects. The observations in the same cluster are correlated because they share common cluster-level random effects.
In our study, we have used three-level random-intercept effect model with households nested within regions. The two often-used models for adjusting survivor functions for the effects of covariates are the accelerated failure-time model and the proportional hazards model. In this model, the covariates have a multiplicative effect on the hazard function for some baseline hazard function, and it is assumed to be parametric. Thus, the mixed-effects model based on PH model was written as follow: Where: For k = 1, 2,. . ., n sampled under-ve children on j = 1,. . ., m i households nested within i = 1, 2, ..., 11 regions in Ethiopia. The 1xp row vector x ijk contains the covariates for the xed effects, with regression coe cients ( xed effects) β. The two random effects U i (region level) and V j(i) (household level) were used to represent random intercept effect and random coe cients effect. Those random effects u i and v j(i) were realized from a multivariate normal distributions with mean 0, and variances matrixes Σ, respectively.
The ve commonly known parametric survival regression models are exponential, gamma, log-logistic, lognormal, and Weibull (Table 2).

Descriptive statistics for children characteristics'
From the total of 10,331 under-ve children, 635 (6.1%) deaths had occurred in the 2016 EDHS data. And, the average age and standard deviation of the children were 27.2 and 17.9 months respectively. From the sex types of children, the male group has the highest under-ve children deaths of proportion (7.1%) as compare to the female group (5.2%). Children born as a result of multiple births recorded the highest percentage (21.4%) of death compare to those as a result of a single birth (5.7%). Out of 3,867 children, 11.2% had died due to less than two children of aged ve and under in house. This was the highest proportion of death as compare to two and above children of aged ve and under (3.1%) in house. When we compared the birth orders of the children, the 7th and above birth order has the highest under-ve children deaths proportion (7%). From 2,922 under-ve children, 7.5% were died because of below the average size of the children at birth. This proportion was the highest one as compare to the average and above the average sizes of the children at births. The highest death proportion (11.9%) has occurred due to the children had been fed less than six months as compare to those had been fed for six and above months. And, as we have compared the types of under-ve children, the infant group has the highest deaths of proportion (19.3%) whereas the child group has only 1.7% deaths of proportion (Table 3).

The Strati ed Cox Proportional Regression Model
The Strati ed Cox regression model is a modi cation of the Cox regression model by the strati cation of a covariate that does not satisfy the proportional hazards assumption. Covariates that are assumed to satisfy the proportional hazards assumption are included in the model, whereas the predictor being strati ed is not included. Covariates like birth type of the child, family size, wealth index, frequency of listening radio, place of delivery, place of residence, and geographical region were signi cant factors at 5% level of signi cance in strati ed Cox PH model (Table 4). Multilevel mixed-effects parametric models comparison The mixed effects Weibull, exponential, log-normal, and log-logistic parametric regression models are tted to nd factors affecting under-ve child mortality data for the xed effects and a random effects and compared to among. On the basis of AIC criteria log-normal mixed effects model was found a best model to t the data with minimum AIC value (4302.8) and − 2LogL value (Table 5).

The Multilevel Lognormal Parametric Random-intercept Model
In the three-levels mixed effects lognormal parametric model with households nested within regions, the likelihood-ratio test with the chi-square value = 20.4 and p-value = 0.000 < 5% indicated that the model with random-intercept effects model with households nested within regions ts the data better than the xed effect model. In the xed effects model, covariates like sex of child, number of 5 and U5 children, type of birth, size of child, months of breastfeeding, family size, listening radio, place of residence, and place of delivery were signi cant covariates at 5% level of signi cance (Table 6).

Discussion
In our study using the Strati ed Cox PH model, the multiple or twin birth type children had more than 5 times higher risk of death compared with single birth type children. Many studies showed that birth type of child was a signi cant determinant factor (8)(9)(10). But, other study from Nigeria had reported the reverse of this result. The odd of having infant/child mortality was 1.87 times greater for children with single births as opposed to those with multiple births (11).
In the study, children from family sizes of 4 to 6 members and above 6 members had 0.77 and 0.69 times low risk of dying before the age of ve as compared to children from family sizes of 1 to 3 members, respectively. This is consistent with a study by (8). But contradict with the study by Gebretsadik carried out in Ethiopia found an inverse relationship between under-ve mortality and family size in 2011 Ethiopia demography and health Survey data (10).
Children from rich wealth index households had 0.68 times lower risk of dying before the age of ve as compared to poor wealth index households. They had decreased the death rate by 32%.This was consistent with the ndings by some previous studies. These households have better housing conditions, better nutrition, and hence they may be able to afford better medical attention and care thus signi cantly enhancing the survival probability of all their children (Bello and Joseph, 2014;O et al., 2016;Yu et al., 2018).
In our study, children were delivered in private health sector had 1.6 times high likely of dying before celebrating their 5th year of birth than children were delivered in public health sector. They had increased the death rate by 60%.But, there was no statistically signi cant difference between children were born in home with compared children were born in public health sector. This nding inconsistent with prior studies by (14)(15)(16).
In the study, children from parents were listening radio for at least once a week had 0.7 times less likely to die than children from parents were not listening radio. They had decreased the death rate by 30%.This study explicitly shows the existence of inconsistency in the distribution of under-ve mortalities with a study (17).
Our study suggests that mortality rate was higher in rural area. This is on the ground that those living in the urban area have access to improved water supply, improve sanitation facilities, unlimited access to healthcare as well as other social and economic services. Thus, the likelihood of under-ve mortality was 1.5 times high among children residing in rural areas as compared with their urban counterpart. This concurs with previous studies in and outside Ethiopia (4,10,16,18).
Our ndings showed that the under-ve children from Afar, Somali and Harari geopolitical regions of Ethiopia were signi cantly associated with the highest likelihood of under-ve death as compared with children from Addis Ababa city. This high risk might be attributed to social improvement in the community, population density, territorial advancement, as well as regional economic resources (5,(19)(20)(21).
In our study using the multilevel mixed-effects using the lognormal parametric model, in the xed-effect model the covariates like child's sex, number of children aged 5 and U5, type of birth, size of child at birth, months of breastfeeding, family size, wealth index, place of delivery, place of residence, and frequency of listening radio were the risk factors for the mean survival times of under-ve children at 5% level of signi cance. Other similar ndings also found that factors such as child's sex, type of birth, total number of children, breastfeeding, size of child at birth, family size, wealth index, place of delivery, type of residence, and frequency of listening radio were found as a signi cant factors (3,16,17,21,22).
In our study, the mean survival time of female children is exp (1.8) = 6.05 times longer than male children with 95% CI is between 2.5 and 13.3. A similar study using the analysis of 28 Demographic and Health Surveys in Sub-Saharan Africa countries, and other study from Uganda showed that female children had reduced risks of dying before 5 years of age compared to male children (16,23).
The mean survival time for multiple birth type of the children was exp (-9.9) = 0.034 times less than single birth type of children with 95% CI was between 0.02 and 0.06. This result was consistent with the studies (5,24).
In the multilevel lognormal parametric model, the two estimated variances of the random-intercept effects in the mean survival times of under-ve children between regions and households are 1.7 and 0.9, respectively. These indicated that we have enough evidence for the existence of unobserved heterogeneities between regions and households. And, there was high variation between regions as compared between households. Thus this study provided that there were unmeasured factors other than these included in our analysis that were caused the clustering of under-ve children mortality in some households and regions.
This random-intercept effect between region level was supported by other similar studies using gamma frailty model which found that the variance of the frailty term (Regional frailty) θ = 0.145 with p-value < 0.05 (3) used 2016 EDHS data, and study using multilevel logistic regression model showed that the regional level variance of the under-ve children mortality was 0.218 (5) used 2011 EDHS data. And, a study from Nigeria using multilevel Cox proportional hazard regression model also founded that there were regional variations in under-ve children mortality based on considering community-level variables (11,16,21).

Conclusions
The under-ve children mortality was signi cantly associated with birth type of the child, family size, wealth index, frequency of listening radio, place of delivery, place of residence, and geographical region covariates using strati ed Cox PH regression model after strati ed some covariates to hold Cox PH assumption. The multilevel mixed-effects log-normal parametric model was found the best model to t the data with the minimum AIC and − 2LogL value as compared with the other multilevel mixed-effects parametric models. In the xed effects model, covariates like sex of child, number of 5 and U5 children, type of birth, size of child, months of breastfeeding, family size, listening radio, place of residence, and place of delivery were signi cant covariates. In the random-intercept effects model, the two estimated variances of the random-intercept effects for regional and household levels are 1.7 and 0.9, respectively.
The values indicate that we have enough evidence that there were variations (unobserved heterogeneities) on the mean survival times of the under-ve children between regions and households levels. Further studies should be conducted to identify and compare the signi cance of the individual, household, and community-level factors associated with infant and child mortality in Ethiopia.  The overall estimate of Kaplan-Meier survivor function curve of under-ve children.