Modelling the Spatial Distribution and the Factors Associated with Under-Five Mortality in Nigeria

Abstract

Globally, the risk of a child dying before celebrating their fifth birthday is still high at 5.3 million deaths in 2018 alone. Nigeria is among the few countries that are yet to achieve the Sustainable Development Goal Target of keeping under-5 death to as low as 25 deaths per 1000 live births by 2030. A recent study found that the under-5 mortality rate in Nigeria is still high with 1 in 8 Nigerian children dying before reaching the age of 5. In this study, the effect of a child’s spatial location in Nigeria on their likelihood of dying before age 5 was examined alongside other key covariates. Bayesian geo-additive regression models were fitted to the 2018 Nigeria Demographic and Health Surveys data. Statistical inference was based on the Bayesian paradigm via Markov chain Monte Carlo simulation methods, and models were assessed using the deviance information criterion. Under-five mortality rate varied significantly across spatial locations in Nigeria with Kebbi, Jigawa, Kaduna, Kogi and Gombe states having the highest rates. The likelihood of a child dying before age 5 increased among women with primary education and women aged 38 years and over. Other characteristics associated with high under-5 death are poverty, male child, low birth weight and multiple births. The current study has helped to identify geographical ‘hotspots’ as well as the key factors driving under-5 deaths in Nigeria to inform the effective design and implementation of timely and efficient interventions.

Appendix

In the Bayesian paradigm, the unknown functions ${\text{f}}_{j}$ and parameters ${\upgamma }$ and the variance $\sigma^{2}$ or the precision $\tau^{2} = 1/\sigma^{2}$ are treated as latent and random variables which are to be estimated. In general, Bayesian inference is performed by evaluating the posterior distribution $\pi (\theta |{\varvec{y}},{\varvec{x}})$, which may be approximate as the product of the joint likelihood function, $L\left( {\theta ;{\varvec{y}},{\varvec{x}}} \right)$ and the joint prior distributions $\pi \left( {\varvec{\theta}} \right)$, where ${\varvec{\theta}}$ is a vector of the unknown parameters.

Furthermore, for our purpose, we assign independent diffuse priors to the parameters $\gamma_{j} \propto \,{\text{const}},\, j = 1, \ldots ,r$ of the $r$ fixed effects covariates, $z_{i} = \left( {z_{i1} , z_{i2} , \ldots ,z_{ir} } \right)$. Although, highly dispersed Gaussian priors could still be used. Following Kandala et al. (2019).and to reflect spatial neighbourhood structure, we assigned Markov random fields (MRF) priors to the correlated spatial effect $f_{str} \left( s \right), s = 1, \ldots , S$, (Besag et al. 1991). Note that the MRF prior is the spatial extension of random walk models and is defined by Eq. 7

$$f_{str} \left( s \right) | f_{str} \left( r \right) , r \ne s\sim N\left( {\mathop \sum \limits_{{r\smallint \delta_{s} }} \frac{{ f_{str} \left( r \right)}}{{N_{s} }},\frac{{\tau_{str}^{2} }}{{N_{s} }}} \right)$$

(7)

where $N_{s}$ is the number of adjoining states to state $s$, and $r\delta_{s}$ denotes that region $r$ is a neighbour of region $s$. Hence the (conditional) mean of $f_{str} \left( s \right)$ is the average of functions $f_{str} \left( s \right)$ of the neighbouring regions, where $\tau_{str}^{2}$ is a smooth parameter. On the other hand, we assign zero-mean independent and identically distributed Gaussian priors to the uncorrelated (unstructured) spatial effect $f_{unstr} \left( s \right)$ as equation

$$f_{unstr} \left( s \right) | \tau_{unstr}^{2} \sim N\left( {0,\tau_{unstr}^{2} } \right),$$

(8)

As before, we assign inverse gamma-distributed priors to the smooth parameters such that $p\left( {\tau_{j}^{2} } \right)\sim IG\left( {a_{j} ,b_{j} } \right),$ where $j$ here is a generic subscript representing both $str$ and $unstr$ and where $a$ and $b$ are hyperparameters. Usually, the hyperparameters are chosen to be vague and in our case, we chose $a = b = 0.001.$

Furthermore, to estimate the smooth functions, $f_{1} , \ldots , f_{p}$, we used cubic splines which are twice continuously differentiable piecewise cubic polynomials. However, the spline can be written as a linear combination of B-spline basis functions $B_{m} \left( x \right)$, the Bayesian version of the Penalized–Splines (P-Splines) proposed by Eilers and Marx (1996), such that $f\left( x \right) = \mathop \sum \nolimits_{m = 1}^{l} \beta_{m} B_{m} \left( x \right)$. In our approach, this corresponds to 2nd order random walks given by Eq. 9

$$\beta_{m} = 2\beta_{m - 1} - B_{m - 2} + \mu_{m}$$

(9)

with Gaussian increments $\mu_{m} \sim N\left( {0,\tau^{2} } \right)$ which is estimated from data and where the smoothness parameter $\tau$ is also estimated from the data.

For Bayesian inference, samples ${\varvec{\theta}} = \left( {\left\{ {\varvec{f}} \right\},{\varvec{f}}_{{{\varvec{unstr}}}} ,{\varvec{f}}_{{{\varvec{str}}}} ,{\varvec{\tau}}_{{{\varvec{unstr}}}} ,{\varvec{\tau}}_{{{\varvec{str}}}} } \right)$ are drawn from the posterior distribution of the latent parameters $\pi ({\varvec{\theta}}|{\varvec{x}},{\varvec{y}})$ using Markov chain Monte Carlo (MCMC) simulation (Gilks and Wild 2006). For our purpose, we used iteratively weighted least square (IWLS) proposal (Klein et al. 2015; Kneib and Fahrmeir 2006).

The models were then fitted in R statistical programming software version 3.6.1 using R2BayesX (Umlauf et al. 2015), the R interface BayesX, a popular statistical software for fitting various classes of generalized additive mixed models (Belitz et al. 2011). For our study, 20,000 samples were simulated from the posterior distributions. Then, after a burn-in period of 4000 iterations which was discarded on the assumption that the 4000 initial chains may not have converged at the stationary distribution, we summarized the posterior after selecting only every 10^th of the remaining 16,000 samples. This is also called thinning. Both the burn-in and thinning are used to ensure that the posterior samples are approximately independent.

Using sensitivity analysis, we investigated the appropriateness of the MRF priors by fitting the spatial model using Gaussian Random Field (GRF) priors. However, we found no evidence of a better fit with the GRF. Besides, the sparseness introduced by the neighbourhood structure of the MRF of a particular computational advantage and greatly reduces computational costs.