The authors have declared that no competing interests exist.
Many researchers have devoted considerable attention to the impact of individuallevel factors on child mortality, but little is known about how family and community characteristics affect health of children. Trend in child mortality as well as its determinants, has long been the subject of academic and policy debates. In spite of this, the problem of child mortality remains as daunting as ever. In fact, advancement in medical sciences and the upsurge in information and telecommunication technology equipment have not significantly reduced child mortality in the country, unlike in the West.
The Multilevel proportional hazards model for data that are hierarchically clustered at three levels was applied to the study of covariates of child mortality in Nigeria. This study merges two parallel developments of statistical tools for data analysis: statistical methods known as hazard models that are used for analyzing eventduration data and statistical methods for analyzing hierarchically clustered data known as multilevel models. These developments have rarely been integrated in research practice and the formalization and estimation of models for hierarchically clustered survival data remain largely uncharted. The model was estimated using the NewtonRaphsons numerical search approach. The model accounts for hierarchical clustering with three random effects or frailty effects. We assume that the random effects are independent and follow the Exponential and Weibull distribution.
The results indicate that biodemographic factors are more important in infancy while socioeconomic factors and household and environmental conditions have a greater effect in childhood. Furthermore, there is significant variation in child mortality risks even after controlling for measured determinants of mortality. Also, factors that fall under family and community level are more significant indicating that child survival is most controlled or determined by family and community factors and variables at the child level is not weighty. This suggests that there may exits unobserved or unobservable factors related to mortality.
In this paper, we present a multilevel hazard models for factors that affects child mortality in Nigeria. These factors are in three categories; Individual level or Child level effects, Family level effects and Community level effects.Using secondary data from the National Demographic Health Survey, estimations were made employing multilevel proportional hazard model, using data that are hierarchical in nature. It is important to consider the realities of hierarchical structure of the data, especially in the Medical and Sociological studies because;
a. Focusing attention on the levels of hierarchy in the population, multilevel analysis enables researchers to understand where (in which level) and how effects are occurring.
b. It provides better estimates for answering simple questions for which singlelevel analysis was once used, and in addition allows for further information about the hierarchical variations to be obtained
c.Carrying out an analysis which does not recognize the existence of clustering creates serious problems. For example, clustering will generally cause standard errors of regression coefficients to be understated.
Hence, this paper investigates two parallel developments of new statistical tools for data analysis: statistical methods known as hazard models that are used for analyzing eventduration data and statistical methods for analyzing hierarchically clustered data known as multilevel models. The objective of the study is (a) To predict the death of a child at time t using Logistic regression model, (b) To compute hazard at time t of a particular child i belonging to family j in a community k from two waiting distributions (Exponential and Weibull), (c) To determine how familylevel factors and communitylevel factors affects child survival, (d) To investigate child mortality at national level through the geographical zones among children in Nigeria.
Most of the earlier studies on Mortality have ignored the hierarchical facts (see for example
Indicators derived from mortality rates provide a good picture of overall population health. These Indicators include infant and child mortality, adult mortality and overall life expectancy at birth. According to the World development indicators database, child mortality rate is the number of deaths of children less than 5 years old per 1000 live birth while Infant Mortality Rate is the number of deaths of children less than 1 year old per 1000 live birth. Despite its significant decline in most parts of the developed world, child mortality remains disturbingly high in most developing countries.
In Nigeria, regional disparities in mortality among children under age 5 have been reported, with higher rates observed in the Northern regions than in the Southern regions. However, these were survey reports rather than empirical studies, and the role of explanatory factors was not investigated. Regional disparities in healthseeking behavior have been reported regarding child immunizations, maternal and child health care utilization, differences in the socioeconomic composition, communicable diseases, childhood nutrition, and malnutrition. Other studies have reported that the Northern regions have higher proportions of home delivery and complications during childbirth, younger age of first marriage, younger age at birth of first child, and lower knowledge and use of contraception compared to the Southern regions. Regional disparities in these parameters are associated with factors at the community level that distinguish these regions from each other. The reduction of these disparities is a key goal of most developing countries’ public health policies, as outlined in the Millennium Development Goals 2015.
To achieve this goal, determinants of high mortality among disadvantaged people, communities, and regions need to be identified. The physical, ecological, and political structure and impoverished socioeconomic milieu in several countries in SubSaharan Africa account for geographic variations in childhood mortality. One such environment is the regional environment. Poor or polluted environments tend to expose children to diseasecausing agents, predisposing them to high mortality risks. In Nigeria, marked regional disparities in mortality of children under age 5 have been reported, with higher rates observed in the Northern regions than in the Southern regions. Children in lowincome countries face much higher risks of mortality compared to their counterparts in more affluent societies. While the infant mortality rate in 1992 was 79per thousand in India, it was only 26 in Thailand and 13 in South Korea. This disadvantage often arises from the lack of parental resources in societies characterized by credit market imperfections. The problem is further aggravated for larger families with more children as these families need to allocate limited available resources across more consumers. Even in the absence of any strategic behavior by family members, children compete for limited parental care and resources – a notion commonly labeled as ‘sibling rivalry’ in economics
Indeed, the incorporation of communitylevel factors in the analysis of child mortality provides an opportunity to identify the health risks associated with particular social structures and community ecologies, which is a key policy tool for the development of public health interventions. The availability of services and social amenities in communities, or the lack thereof, may positively or negatively influence the health of the residents of communities. Some of these factors include differences in communitylevel development, population density, prevalence of poverty, and availability of maternal and child health care services. These are often interrelated aspects of the regional environment that are important for child health and wellbeing, and may also be relevant in exacerbating or mitigating inequities in resources and population health outcomes across regions.
In recent years, all developing regions have experienced considerable improvements in child survival with Northern Africa and East Asia making the fastest progress. Despite these achievements, however, most developing countries still lag significantly behind the developed world, with an average of 66 deaths per 1,000 live births in 2009, compared to only six in developed regions and also, the annual number of children who die before they reach age five is shrinking, falling to 7.6 million global deaths in 2010 from more than 12 million in 1990 (UNICEF and the World Health Organization).
Across the world, children are at higher risk of dying if they are poor. The most impressive declines in child mortality have occurred in developed countries, and in lowmortality developing countries whose economic situation has improved. In contrast, the declines observed in countries with higher mortality have occurred at a slower rate, stagnated or even reversed. Owing to the overall gains in developing regions, the mortality gap between the developing and developed world has narrowed since 1970. However, because the betteroff countries in developing regions are improving at a fast rate, and many of the poorer populations are losing ground, the disparity between the different developing regions is widening.
Today nearly all child deaths occur in developing countries, almost half of them in Africa. While some African countries have made considerable strides in reducing child mortality, the majority of African children live in countries where the survival gains of the past have been wiped out, largely as a result of the HIV/AIDS epidemic. Some developing countries still lag significantly behind the developed world, with an average of 66 deaths per 1,000 live births in 2009, compared to only six in developed regions and also, the annual number of children who die before they reach age five is shrinking, falling to 7.6 million global deaths in 2010 from more than 12 million in 1990 (UNICEF and the World Health Organization). Against this backdrop, the joint UNWorld Bank report concludes that the rate of decline in child mortality remains insufficient to achieve Millennium Development Goal 4 (MDG 4)—to reduce underfive mortality rates by two thirds between 1990 and 2015.
In September 2000, the World Community adopted the Millennium Development Goals (MDGs) to reaffirm community towards a world in which sustaining development and eliminating poverty would be accorded the highest priority. All the 191 United Nations member states pledge that by the year 2015, all the MDGs consisting of a frame of 8 goals, 18 targets and 48 indicators to track progress towards the goals would be meet. The goals are quantifiable with clear cuts objectives. Goal 4 is to reduce underfive mortality rates by two thirds between 1990 and 2015. Goal 5 focuses on issues relating to maternal mortality and maternal health. Goal 6 is aimed at target in epidemiology of HIV/AIDs, Malaria and other major diseases leading to child mortality and their social consequences. It is expected that these diseases must have halted and their spread reversed by 2015.
It is as a result of the dismal failure of meeting the public good across the developing world that the nations of the world in year 2000 met and agreed on a number of Millennium Development Goals (MDGs). Two of these goals are to reduce underfive and maternal mortality rates. But analysis of recent trends shows that Nigeria is making only marginal progress ^{Mid} in reducing these rates and attaining the MDGs even when most of the causes of these deaths are either preventable or treatable. It is sad to note that in spite of all previous efforts, the Under 5 morbidity/mortality indices have shown only marginal reductions in the last five years, making the MDGs targets by 2015 clearly unachievable using current strategies alone.
The focus of this analysis is mortality among children in Nigeria. Since the response variable is child mortality, we utilize event history procedures to examine the effect of various factors on the risk of death of children under the age of five. A fundamental discriminating criterion between various event history models, is the distribution that the timing function is assumed to follow
When appropriate eventduration data are available and coupled with relevant multilevel data, survival models provide the best strategy for analyzing processes of qualitative changes in states (transitions) and their multilevel determinants in terms of fixed and random parameter variations across individuals and groups. A weakness of this approach is that time is used as another explanatory variable without its recognition as the domain in which qualitative changes in states takes place in a dynamic way within specific contexts.
Let there be N state an individual can occupy at any moment of time in a given context. Suppose that there are three levels (i, j and k) of hierarchicallyclustered data for a sample of individuals (e.g. a sample of children nested within families and families nested within area of residence or communities). Let t_{ijk} be the survival time that elapses before the
Where f_{lm} (.), is the density function of times to transition from state l to state m, and h^{lm} (.) is the associated hazard function.
The joint density function of the (N1) latent transition times is given by
In a threelevel frame work considered above, let ε_{ijk}, ϑ_{jk}, θ_{k} be the random coefficients at the child level, the family level and the community level respectively. Ignoring the multistate situation, if the random effects are assumed to operate multiplicatively on the baseline hazard, they are interpreted as relative risks and the general multilevel hazard model can be written as follows:
(3)
Where
Where Z_{ijk} is a 1 x
Following
(4)
Where ϖ is the unknown duration dependence?
This general formulation allows ε_{ijk}, ϑ_{jk}, θ_{k} to be function of time. By exponentiating the term in brackets, equation (4) ensures that the hazard function is positive as required since it is a conditional density function. From the multilevel survival formulation in (4), the survivor function at time
..............................(5)
and the likelihood is more generally:
We consider a threestate process of infant and child mortality, a nonrepeatable event. The two states that a child can occupy during the time of interview are ‘alive’ and ‘dead’. Note that a child can exit the alive state only once after a given length of exposure to the risk of death. There are thirteen factors to measure the survival status of a child from birth to age five. My focus is on hazard process in which one or more covariates change values between intervals, but are constant within an interval (i.e one or more covariates are timevarying).
Here we look at a situation where there are several children per family. In practical terms, at each month of exposure
Y_{ij}_{k}= {(1: if child i dies before month t, 0: if child i survive beyond month t) Where t is 5 years.
Let P_{ijk} denotes the response probability of the ith child of the jth family in the kth community that survive up to age 5 (i.e P_{ijk}=prob{Y_{ijk}=1})
For a simple illustration, suppose we have four families (mothers) in four different communities, the first mother from the first community has two children, with the first child dying at 5^{th} month, and the second censored at 2 months. The second mother from the second community has one child censored at 2 months, the third mother from the third community has one child censored at 4 months and the fourth mother from the fourth community has one child dying at 4 month.






1  1  1  1  0 
1  1  1  2  0 
1  1  1  3  0 
1  1  1  4  0 
1  1  1  5=death  1 
1  1  2  1  0 
1  1  2  2=censored  0 
(end of survey)  
2  2  1  1  0 
2  2  1  2=censored  0 
(end of survey)  
3  3  1  1  0 
3  3  1  2  0 
3  3  1  3=censored  0 
(end of survey)  
4  4  1  1  0 
4  4  1  2  0 
4  4  1  3  0 
4  4  1  4=death  1 
The survival times for a hierarchically clustered sample of individuals are assumed to be conditional on two independent clusterspecific random effect, θ_{k} and ϑ_{jk}, We assign one θ_{k} random effect to each of the k=1…K clusters, one ϑ_{jk} random effect to each of the j=1…J, subclusters of cluster θ_{k}. Since the data are hierarchically clustered, all individuals in the
………………….(6)
Where λ_{0} (t_{ijk}) represents the baseline hazard, which we assume to be piecewise constant, andexp (α' x_{ijk}) is the risk associated with covariates X_{ijk}
Individuals who have died (Y_{ij}_{K}=1) contribute to the conditional likelihood function, the product of the conditional hazard and the conditional survival functions, while individuals whose survival times are censored (Y_{ij}_{K}=0) contribute to the conditional survival function:
……………(7)
Here ρ_{ijk} (t_{ijk}) _{=}exp[Λ_{ijk}(t_{ijk})] and Λ_{ijk}(t_{ijk})is the integrated hazard corresponding to
ϖ_{ijk} (t_{ijk}) We will omit the arguments when referring to the hazard and the integrated hazard and denote these quantities simply as ϖ_{ijk}and Λ_{ijk}. We can write the ith cluster’s contribution to the conditional likelihood function as,
…………………………..(8)
Note that once we condition on the unobserved randomeffects, the individual observations in the sample are independent.
If the frailty effects θ_{k}, and ϑ_{jk} were observed, and their distributions were known, one could estimates the parameters of these models by miximising eqn (6) with respect to ξ_{k}, η_{jk} and β_{ijk}
where ξ_{k}, and η_{jk} are the parameters of the two mixing distribution and β_{ijk} is a vector consisting of the covariate effects, α_{ijk}, and parameters that specify the baseline hazard. Since these quantities are not observed, we must either integrate them out of the likelihood function or use the ExpectationMiximization (EM) algorithm and we must make assumptions about the probability distributions of the two randomeffects
We assume that the random effects θ_{k}, ϑ_{jk}, and ε_{ijk}, are independent and are Exponentially distributed with density functions,
f_{θ} (θ_{k}) = ξe^{ξθ}_{k}
f_{ϑ} (ϑ_{jk})=ηe^{ηϑ}_{jk} …………..(9)
The variances for the above distribution are ξ^{2} and η^{2} respectively and their covariances will be estimated. The assumption of the random effects being exponential and weibull follows a previous reserch. The estimated parameters describing the shape of the weibulldistributed randomeffects can be interpreted in several different ways. The most direct interpretation is simply as the variance of the frailty distribution, if the variance is zero, then observations from the same group are independent. A larger variance implies greater heterogeneity in frilty across and greater correlation among individuals belonging to the same group. The second way to describe the results is to construct a risk ratio that compares the conditional expectation of the frailtyeffect for a highrisk group and a lowrisk group, where for instance, highrisk may refer to a cluster at the 90^{th} percentile and lowrisk a cluster at the 10^{th} percentile of the frailty distribution.
As the variance of the random effects tend to zero, the observations in each cluster becomes progressively less correlated. Full independence among all observations in the sample is obtained in the limiting case. Under full dependence, the multilevel hazard model simplifies to
The values for θ_{k}, and ϑ_{jk} that appear in the likelihood function are unobserved. One method for obtaining maximum likelihood estimates of ξ_{k}, η_{jk} and β_{ijk} under these circumstances is to apply the Expectation Maximization (EM) Algorithm
Where φ_{n} represents the parameters of the model at the nth iteration; F_{n} is a vector of first derivatives and H_{n} is a matrix of second derivatives, both evaluated at φ_{n} Under the Newton Raphson method, there is need for a second derivative of the incomplete data loglikelihood. However, the matrix of second derivatives that are calculated as part of the Newton Raphson procedure provides estimates of asymptotic variance. Finally, the NewtonRaphson method generally enjoys rapid convergence.
Finally, the calculation of the first derivative of the incomplete data loglikelihood function at each iteration is more computationally intensive.
In this section, the proposed methodology to data collected from three hierarchical levels in child mortality is applied. The performance of alternative methods to estimating parameters of models with data on child mortality that are clustered was also used. This study was designed to determine those factors that affect the survival of children in Nigeria and to also examine the cluster that is being affected. Most commonly, clustering occurs as a result of sample design that focuses on community or village, mothers or households, and on all the children associated with them so as to measure the extent to which risk of death of children belonging to the same household and community are correlated.
From
On geopolitical zones basis (
From the survival graph below (
We can infer that the risk of death occurring among Underfive children decreases with time. This situation is referred to as Negative Duration Dependence.
For the Weibull distribution in
In
Variables  Model I  Model II  Model III  Model IV 
Sex  
Female        
Male  0.210 (0.55)  0.005 (0.49)  0.146 (0.57)  0.28 (0.52) 
Breastfeeding  
Fully Breastfeed  0.019 (0.60)  0.005 (0.60)  0.121 (0.60)  0.233 (0.60) 
Not fully Breastfeed  0.009 (0.50)  0.005 (0.57)  0.108 (0.50)  0.216 (0.58) 
Size at Birth  
Large  0.68 (0.80)  0.38 (0.071)  0.39 (0.21)  0.034 (0.06) 
Small  0.052 (0.50)  0.035 (0.41)  0.038 (0.49)  0.98 (0.46) 
Mothers Education  
No formal Education  0.07 (0.00)  0.384 (0.00)  0.37 (0.00)  0.305 (0.00) 
Formal Education  0.078 (0.00)  0.03 (0.00)  0.661 (0.00)  0.315 (0.00) 
Bed nets Access  
Yes  0.026 (0.00)  0.009 (0.01)  0.46 (0.00)  0.707 (0.03) 
No  0.068 (0.03)  0.028 (0 04)  0.003 (0.00)  0.644 (0.04) 
Occupation  
Skilled Worker  0.01 (0.32)  0.001 (0.03)  0.293 (0.00)  0.755 (0.00) 
Unskilled Worker  0.003 (0.52)  0.003 (0.02)  0.158 (0.00)  0.438 (0.002) 
Religious Faith  
Christianity  0.154 (0.00)  0.056 (0.002)  0.523 (0.00)  0.85 (0.00) 
Islam  0.029 (0.003)  0.009 (0.003)  0.223 (0.00)  0.354 (0.00) 
Region of Birth  
North Central  0.003 (0.00)  0.34 (0.00)  0.454 (0.00)  0.978 (0.00) 
North East  0.59 (0.00)  0.086 (0.00)  0.821 (0.00)  0.855 (0.00) 
North West  0.803 (0.00)  0.044 (0.00)  0.975 (0.00)  0.722 (0.00) 
South East  0.481 (0.00)  0.027 (0.00)  0.646 (0.00)  0.420 (0.00) 
South South  0.020 (0.00)  0.025 (0.00)  0.561 (0.00)  0.645 (0.00) 
South west  0.004 (0.00)  0.03 8 (0.00)  0.097 (0.00)  0.19 (0.00) 
Residence  
Urban  0.02 (0.00)  0.004 (0.001)  0.04 (0.00)  0.25 (0.00) 
Rural  0.04 (0.00)  0.004 (0.006)  0.154 (0.00)  0.303 (0.00) 
Toilet Type  
Flush  0.034 (0.00)  0.021 (0.001)  0.320 (0.00)  0.116 (0.00) 
Pit  0.68 (0.00)  0.052 (0.008)  0.483 (0.00)  0.842 (0.00) 
Type of Water  
Good water  0.547 (0.00)  0.604 (0.00)  0.605 (0.00)  0.194 (0.05) 
Bad water  0.120 (0.00)  0.040 (0.00)  0.098 (0.00)  0.175 (0.00) 
Variance of Random Effects (Unobserved Heterogeneity)  
Child level    0.324 (0.043)     
Family Level      0.013 (0.01)   
Community Level        0.151 (0.03) 
Sample size  2864  2864  2864  2864 
Exponential Hazards  Weibull Hazards  
Singlelevel  Twolevel  Singlelevel  Twolevel  
Variables  modeling  modeling  modeling  modeling 
Sex  
Female         
Male  0.285(0.00)  0.142(0.00)  0.006(0.00)  0.285(0.06) 
Breastfeeding  
No Breast feeding         
Fully Breastfeed  0,036(0.01)  0.018(0.66)  0.121(0.00)  0.233(0.00) 
Not fully Breastfeed  0.019(0.01)  0.009(0.49)  0.108(0.07)  0.0.216(0.24) 
Age at Birth  
Large Size         
Small  0.052(0.00)  0.108(0.00)  0.490(0.00)  0.976(0.00) 
Mothers Education         
No formal Education        
Formal Education  0.079(0.00)  0.160(0.00)  0.661(0.00)  1.315(0.00) 
Bed nets Access         
Yes         
No  0.068(0.04)  0.226(0.24)  0.707(0.38)  0.644(0.42) 
Occupation         
Not Working         
Skilled Worker  0.010(0.32)  0.034(0.11)  0.293(0.00)  0.575(0.00) 
Unskilled Worker  0.003(0.52)  0.033(0.15)  0.158(0.00)  0.755(0.00) 
Religious Faith  
Christianity  0.154(0.00)  0.066(0.00)  1.523(0.00)  0.850(0.00) 
Islam  0.029(0.00)  0.297(0.00)  0.233(0.00)  3.054(0.00) 
Traditionalist         
Region of Birth         
North Central         
North East  0.339(0.00)  0.043(0.01)  0.561(0.00)  0.645(0.00) 
North West  5.003(0.00)  10.013(0.00)  7.454(0.00)  14.855(0.00) 
South East  1.593(0.00)  0.224(0.00)  1.821(0.00)  1.978(0.00) 
South South  0.803(0.00)  0.198(0.00)  0.975(0.00)  1.722(0.00) 
South west  0.482(0.00)  0.199(0.00)  0.646(0.00)  1.420(0.00) 
Residence         
Urban         
Rural  0.020(0.02)  0.042(0.01)  0.097(0.00)  0.190(0.00) 
Toilet Type         
Flush  0.040(0.00)  0.141(0.00)  0.154(0.00)  1.116(0.00) 
Pit  0.034(0.00)  0.079(0.00)  0.320(0.00)  0.303(0.00) 
No facility         
Type of Water         
Good water         
Bad water  0.120(0.00)  0.243(0.00)  1.098(0.00)  2.175(0.00) 
Part B: random Effects         
Child level Unobserved    1.500(0.00)    1.370(0.0) 
Heterogeneity         
Negative loglikelihood  26674.051  26671.187  25972.22  25542.74 
Sample size  2864  2864  2864  2864 
The result in
Variables Singlelevel  Exponential Hazards  Weibull Hazards  
Twolevel Modeling  Singlelevel Modeling  Singlelevel Modeling  Twolevel Modeling  
PART A: Fixed Effects  
Zn (duration)      0.40(0.05)  0.15(0.06) 
Durationdependence term      
Intercept  0.01(0.26)  0.242(0.44)  1.72(0.30)  3.62(0.63) 
SexMale  0.285(0.00)  0.142(0.00)  0.006(0.00)  0.285(0.06) 
Fully Breastfeed  0,036(0.01)  0.018(0.66)  0.121(0.00)  0.233(0.00) 
Not fully Breastfeed  0.019(0.01)  0.009(0.49)  0.108(0.07)  0.0.216(0.24) 
Small  0.052(0.00)  0.108(0.00)  0.490(0.00)  0.976(0.00) 
Formal Education  0.079(0.00)  0.160(0.00)  0.661(0.00)  1.315(0.00) 
No  0.068(0.04)  0.226(0.24)  0.707(0.38)  0.644(0.42) 
Skilled Worker  0.010(0.32)  0.034(0.11)  0.293(0.00)  0.575(0.00) 
Unskilled Worker  0.003(0.52)  0.033(0.15)  0.158(0.00)  0.755(0.00) 
Christianity  0.154(0.00)  0.066(0.00)  1.523(0.00)  0.850(0.00) 
Islam  0.029(0.00)  0.297(0.00)  0.233(0.00)  3.054(0.00) 
North East  0.339(0.00)  0.043(0.01)  0.561(0.00)  0.645(0.00) 
North West  5.003(0.00)  10.013(0.00)  7.454(0.00)  14.855(0.00) 
South East  1.593(0.00)  0.224(0.00)  1.821(0.00)  1.978(0.00) 
South South  0.803(0.00)  0.198(0.00)  0.975(0.00)  1.722(0.00) 
South west  0.482(0.00)  0.199(0.00)  0.646(0.00)  1.420(0.00) 
Rural  0.020(0.02)  0.042(0.01)  0.097(0.00)  0.190(0.00) 
Flush  0.040(0.00)  0.141(0.00)  0.154(0.00)  1.116(0.00) 
Pit  0.034(0.00)  0.079(0.00)  0.320(0.00)  0.303(0.00) 
Bad water  0.120(0.00)  0.243(0.00)  1.098(0.00)  2.175(0.00) 
Part B: random Effects  
Child level Unobserved    1.500(0.00)  1.370(0.0)  
Heterogeneity  
Negative loglikelihood  26674.05  26671.19  25972.22  25542.74 
Sample size  2864  2864  2864  2864 