A Multilevel Hazards Model for Child Mortality In Nigeria

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 low-mortality 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 better-off 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 UN-World Bank report concludes that the rate of decline in child mortality remains insufficient to achieve Millennium Development Goal 4 (MDG 4)—to reduce under-five mortality rates by two thirds between 1990 and 2015.

In September 2000, the World Community adopted the Millennium Development Goals (MDGs) to re-affirm 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 under-five 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 under-five 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.

We consider a three-state process of infant and child mortality, a non-repeatable 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 time-varying).

Here we look at a situation where there are several children per family. In practical terms, at each month of exposure t, we can define a response variable for each child i belonging to family j in a k community,

Y_ijk= {(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.

Data Organization for Hierarchically Clustered /Multilevel Data

The survival times for a hierarchically clustered sample of individuals are assumed to be conditional on two independent cluster-specific 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, sub-clusters of cluster θ_k. Since the data are hierarchically clustered, all individuals in the j-kth sub-cluster are member of one and only one type-k cluster. Let t_ijk represent the survival times that elapses before the i-th child (level 1) belonging to the j-th family (level 2) in the k-th community (level 3) and let X_ijkrepresents a vector of covariates for the ith member of this cluster. If the random effects are assumed to operate multiplicatively on the baseline hazard, they are interpreted as relative risks and the model is written as

………………….(6)

Where λ₀ (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_ijK=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_ijK=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_ijkand 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 random-effects, 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 Expectation-Miximization (EM) algorithm and we must make assumptions about the probability distributions of the two random-effects

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 weibull-distributed random-effects 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 frailty-effect for a high-risk group and a low-risk group, where for instance, high-risk may refer to a cluster at the 90^th percentile and low-risk 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 ⁴ multivariate hazard model with a single random-effect. If the variance of one of the two random-effects is sufficiently small, the parameter estimates and standard errors in the multilevel model differs little from those estimated using a standard hazard model. Estimating a simpler model is desirable because of the time and computational effort required to estimate the random-effects models, particularly the multilevel model. In addition, as the variance of the random effects becomes difficult-and eventually impossible-to evaluate the numerical integration procedure and certain functions that are part of the estimation procedure.

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 ¹² and ¹³. One can also obtain direct maximization using the Newton-Raphson method which requires the first and second derivatives of the incomplete data log-likelihood function to be evaluated at each iteration using numerical integration techniques. An iteration using the Newton-Raphson method is given by,

………………………………………………………….(11)

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 log-likelihood. However, the matrix of second derivatives that are calculated as part of the Newton- Raphson procedure provides estimates of asymptotic variance. Finally, the Newton-Raphson method generally enjoys rapid convergence.

Finally, the calculation of the first derivative of the incomplete data log-likelihood function at each iteration is more computationally intensive.

From Figure 1 below, we can see that there are more deaths in the rural area than in the urban area. The reason for this increase is the fact that children suffer from social infrastructure and services like (schools, roads, electricity, and health services). Some other identifiable cause of child mortality can also occur from parent unemployment, social deprivation, and endemic conflict, in the rural area than in the urban area.

Figure 1.Data Analysis-Rural Urban Differential

On geo-political zones basis (Figure 2), the North West has the highest mortality rate, followed by the North East, North Central, South South, South West and with South East carrying the least. The reason for these differences is that we have higher proportion of children in the Northern zones who have never received vitamin A supplements and immunization. Also breastfeeding rate in these zones is lower than in the Southern zones, as is the proportion of children who received complementary feeding. Use of un-iodized salt is also more common in the Northern part of Nigeria. Identifiable causes of death in these area are diseases like Malaria, Vaccine preventable diseases particularly measles, pneumonia, diarrhea, and Neonatal Jaundice. Increased in prices of food and poor households became more and more food insecure. The most food insecure households were located in the Northern region where the childhood mortality indices were also high. Children in poor Nigerian households are chronically undernourished, suffering a recurrent lack of access to food of sufficient quality and quantity, good healthcare, and necessary caring practices.

Figure 2.Data Analysis-Distribution by Geopolitical Zones

From the survival graph below (Figure 3), we can see that the hazard rate for Exponential distribution is constant over time. This means that on average 0.5 events will occur per individual at risk per month (during a period in which the hazard remains constant at this value). The survival rate for exponential declines monotonically with time (i.e the risk of a child dying decreases with time)

We can infer that the risk of death occurring among Under-five children decreases with time. This situation is referred to as Negative Duration Dependence.

Figure 3.Data Analysis and Validation of Result - Exponential Model

For the Weibull distribution in Figure 4, both the hazard rate and the survival rate declines monotonically with time. We can also infer that the risk of under-five mortality decreases as time increases. This will make the United Nations MDGs: to reduce under-five mortality to 2/3 by 2015 achievable.

Figure 4.Data Analysis and Validation of result - Weibull Model

In Table 1 below, we estimated four different models specifications using the Weibull distribution only. Model 1 is a standard hazard model with no correction for clustering. Model II is hazard model with clustering at child level, Model III is hazard model with clustering at Family level, and Model IV is a hazard model with clustering at the community level. The results indicate that there is standard relationship between child mortality risk and the covariates itemized in this sample. From the result obtained, we discovered that nearly all covariates are statistically significant and the estimated effect for the proximate determinant are in close agreement with what we have in open literature and other research studies. Based on the multilevel results, we discovered that mortality variables at the family and community levels are more statistically significant with their p- values less than 0.5, which implies that child survival is most controlled or determined by Family and Community levels and that variables at the Child level is not weighty.

Table 2 shows the result of the conventional parametric hazards model (without random effects) as well as those of two-level parametric hazards model (with random effects). These two levels hazard models contain both fixed and random effects. The fixed effects are in the first part of the table and the random effect in the second part. The fixed effect represents the population mean influences on infant and early child mortality specific to the measured covariates. The child specific random effect captured by the unobserved heterogeneity consists of two components, a measurement error plus the actual variability (heterogeneity) in child specific capacity to survive during the for the five years interval. Some of this variability may be genetic unobserved or unmeasured by the survey. Note that the frailty effects are significantly different from zero. In other words, there are unmeasured child-specific randomly varying risks that affect child survival independently of measured risk factors. Failure to account for such child-specific unmeasured characteristics has several consequences. First, ignoring individual frailty leads to underestimating the standard errors of parameter estimates, creating false impression of precision. An examination of each of the parametric models (exponential, Weibull) under single-level and two-level specifications consistently substantiates the underestimation of all standard errors under the single-level modeling scheme, and confirms the consequences of ignoring random effects in modeling Multilevel Hazard model for Hierarchically clustered data. Second, estimates of the baseline hazard duration pattern might be biased in downward direction (the best way of understanding this is by imagining a process of constant hazard). Third, estimates of covariates may be biased. The comparative results show that while the sign of most parameters are unaffected by randomly varying risk of mortality, their magnitude and level of significance are quite affected when frailty is explicitly modeled.

The result in table 3 below shows the conventional parametric hazards model (without random effects) as well as those of two-level parametric hazards model (with random effects). These two-level hazard models contain both fixed and random effects. The fixed effects are in the first part of the table and the random effects in the second part. The fixed effects represent the population mean influences on infant and early child mortality specific to the measured covariates. The child-specific (or within child) random effect captured by the unobserved heterogeneity consists of two components, a measurement error plus the actual variability (heterogeneity) in the child’s capacity to survive during the month of survival. Some of this variability may be genetic (Stern, 1960; Adams et al., 1990), unobservable or unmeasured by the survey. As seen in Table 2, the general findings are consistent with evidence generated elsewhere: the protective effects of full immunization status, breastfeeding (especially full breastfeeding), possession of modern amenities and increased household income, and the deleterious effects of overcrowding, closely spaced births and low birth weight. Note that the frailty effects are significantly different from zero. In other words, there are unmeasured child-specific randomly varying risks that affect child survival independently of measured risk factors. Failure to account for such child-specific unmeasured characteristics has several consequences. First, ignoring individual frailty leads to underestimating the standard errors of parameter estimates, creating false impression of precision. An examination of each of the parametric models (exponential, Weibull) under single level and two-level specifications consistently substantiates the underestimation of all standard errors under the single-level modeling scheme. Second, estimates of the baseline hazard duration pattern are biased in downward direction (the best way of understanding this is by imagining a process of constant hazard). Third, estimates of covariates may be biased. The comparative results show that while the sign of most parameters is unaffected by randomly varying risk of mortality, their magnitude and level of significance are quite affected when frailty is explicitly modeled.