Age distribution, trends, and forecasts of under-5 mortality in 31 sub-Saharan African countries: A modeling study

Background Despite the sharp decline in global under-5 deaths since 1990, uneven progress has been achieved across and within countries. In sub-Saharan Africa (SSA), the Millennium Development Goals (MDGs) for child mortality were met only by a few countries. Valid concerns exist as to whether the region would meet new Sustainable Development Goals (SDGs) for under-5 mortality. We therefore examine further sources of variation by assessing age patterns, trends, and forecasts of mortality rates. Methods and findings Data came from 106 nationally representative Demographic and Health Surveys (DHSs) with full birth histories from 31 SSA countries from 1990 to 2017 (a total of 524 country-years of data). We assessed the distribution of age at death through the following new demographic analyses. First, we used a direct method and full birth histories to estimate under-5 mortality rates (U5MRs) on a monthly basis. Second, we smoothed raw estimates of death rates by age and time by using a two-dimensional P-Spline approach. Third, a variant of the Lee–Carter (LC) model, designed for populations with limited data, was used to fit and forecast age profiles of mortality. We used mortality estimates from the United Nations Inter-agency Group for Child Mortality Estimation (UN IGME) to adjust, validate, and minimize the risk of bias in survival, truncation, and recall in mortality estimation. Our mortality model revealed substantive declines of death rates at every age in most countries but with notable differences in the age patterns over time. U5MRs declined from 3.3% (annual rate of reduction [ARR] 0.1%) in Lesotho to 76.4% (ARR 5.2%) in Malawi, and the pace of decline was faster on average (ARR 3.2%) than that observed for infant (IMRs) (ARR 2.7%) and neonatal (NMRs) (ARR 2.0%) mortality rates. We predict that 5 countries (Kenya, Rwanda, Senegal, Tanzania, and Uganda) are on track to achieve the under-5 sustainable development target by 2030 (25 deaths per 1,000 live births), but only Rwanda and Tanzania would meet both the neonatal (12 deaths per 1,000 live births) and under-5 targets simultaneously. Our predicted NMRs and U5MRs were in line with those estimated by the UN IGME by 2030 and 2050 (they overlapped in 27/31 countries for NMRs and 22 for U5MRs) and by the Institute for Health Metrics and Evaluation (IHME) by 2030 (26/31 and 23/31, respectively). This study has a number of limitations, including poor data quality issues that reflected bias in the report of births and deaths, preventing reliable estimates and predictions from a few countries. Conclusions To our knowledge, this study is the first to combine full birth histories and mortality estimates from external reliable sources to model age patterns of under-5 mortality across time in SSA. We demonstrate that countries with a rapid pace of mortality reduction (ARR ≥ 3.2%) across ages would be more likely to achieve the SDG mortality targets. However, the lower pace of neonatal mortality reduction would prevent most countries from achieving those targets: 2 countries would reach them by 2030, 13 between 2030 and 2050, and 13 after 2050.

where n a x stands for the average person-years lived between x and x + n by those dying in that interval. We use this formula to estimate probabilities of dying in month intervals by setting n = 1/12. We also assume that deaths are distributed uniformly across the month age range; that is, on average, persons dying in the month interval do so half-way through the interval and then n a [x] = (1/12) * (1/2) = 1/24. After these assumptions, we get: Notice that we modified the notation to indicate that age is measured in month intervals: now [x] stands for age in months, and m [x] and q [x] represent monthly death rates and probabilities of dying, respectively.

Two-dimensional P-Spline smoothing
We used a two-dimensional P-Spline smoothing and generalized linear model (GLM) to smooth our calibrated mortality profiles over ages and years, assuming that the number of deaths at a given rate are Poisson-distributed [3]. That is, if  [3], the number of deaths and the number of exposures are arranged in m × n matrices D and E, with rows indexed by age and columns indexed by year, respectively -in the one dimensional case (age dimension), we have a vector of death counts (d), exposures (e), and mortality hazards (µ). The P-Splines consist of a combination of B-Spline basis with roughness penalization (or regularization) on the basis coefficients [4,5], with equally-space B-Splines used as regression basis and adjusted to our Poisson data as follows: log (E(y)) = log (e) + log (µ) = log (e) + Bα, in which E(y) = e · µ (as y ∼ P oi(e · µ)). Eq (S3) represents a GLM with B-Splines as regressors and a log link function of the poisson death counts. With P-Splines, this model is adjusted using an iteratively reweighted least squares (IRWLS) algorithm, but the solution includes a penalization matrix P that controls the tradeoff between smoothness and model accuracy (tuning of 1 or 2 smoothing parameters is performed during the optimization process). This linear prediction model is adjusted using an IRWLS algorithm, which yields the following estimates for α: wherez = y−eμ eμ + Bα is a working dependent variable withμ andα denote current approximations to the solution, and W is a diagonal matrix of weights ( W = diag(eμ)) [4]. The term P in Eq (S4) is defined as P = λD T k D k and represents the main characteristic of the P-Splines model, which is an extension of the standard solution for fitting GLM.
Although the same model specification in Eq (S3) and estimation approach can be applied to both one-and two-dimensional data (age and time dimensions), a generalized linear array model (GLAM) [4] is used to adjust the model in two-dimensional settings as the problem may become computationally intractable with large age and time intervals. More details of this procedure are reported by Camarda [3], who developed the R package MortalitySmooth, and specifically tailored to model mortality data in one-and two-dimensional settings with P-Splines.

Lee-Carter forecast in populations with limited data
For age [x] and year t, the Lee-Carter (LC) model that we fit has the form, where the first 2 terms on the right are estimated in a singular-value decomposition step, and the last term is an error term whose variance is estimated as described by Li and colleagues [6]. The term a [x] represents the age distribution of the latest observed month for each country, k t tracks mortality changes over time, b [x] determines how much the age group [x] mortality changes with a unit change in k t , and e [x]t represents age-period disturbances not captured by the model. We measured the goodness-of-fit of the LC model as the percentage of the variance explained of the mortality profile (m [x]t -after the adjustment to match UN IGME estimates) by the first principal component of the singular-value decomposition, which we compute as: where ε The value of k t in Eq (S5) is adjusted in a second stage to fit the reported values of the observed life expectancy at birth from the observed period [7]. To forecast the k t values into the future, the LC model uses a random walk with drift model, as follows: where c is a drift term that represents the linear trend component in the change of k t , and e t σ represents random fluctuations in this linear change [6]. The drift term is estimated using the following expression: where u 0 , u 1 , ..., u T , represent times with gaps, and the error term in Eq (S7) is estimated as follows:σ Both the underlying variation ofĉ andσ resulted from incomplete information is considered in the projected values of Eq (S7) and the corresponding mortality projections obtained from Eq (S5). To get a deviation from the linear change of k t , more than two years of data are necessary for the LC model to provide uncertainty forecasts -by getting positive values ofσ 2 in Eq (S9). More details on the estimation of the forecasting values and uncertainty can be found in [6]. In our study, we refer to this modified LC approach as Li-Lee-Tuljapurkar model (LLT).

Annual Reduction Rates
In this study, we measure mortality change using the Average Annual Reduction Rate (ARR), which considers the decline of mortality during a period separated by n years (r t ,r t+n ): where r t and r t+n are mortality rates at time t and t + n, respectively, and n is the number of years between t and t + n.