A Survival Framework for Estimating Child Mortality Rates using Multiple Data Types

TL;DR

This paper introduces a Bayesian survival framework to estimate child mortality up to age five by integrating diverse national data sources—FBH microdata from DHS/MICS, VR death counts, and pre-processed mortality-rate estimates—into a single temporal model. It formalizes two parametric survival forms, the $\log$-logistic and the $\text{piecewise-exponential}$, with monotonicity constraints realized through parameter transformations and a temporal random-walk structure, and it implements computation via Template Model Builder. The joint likelihood combines FBH, VR, and pre-processed data (including SBHs) with appropriate interval censoring, overdispersion, and HIV-missing-mothers adjustments, enabling coherent inference and full survival curves (not just three summaries) for each country-year. Across four diverse countries (Kenya, Brazil, Estonia, Syrian Arab Republic), the method yields estimates broadly aligned with UN IGME benchmarks while providing richer age-specific information and explicit uncertainty, highlighting the framework’s potential to unify and improve child mortality estimation at the national level.

Abstract

Child mortality is an important population health indicator. However, many countries lack high-quality vital registration to measure child mortality rates precisely and reliably over time. Research endeavors such as those by the United Nations Inter-agency Group for Child Mortality Estimation (UN IGME) and the Global Burden of Disease (GBD) study leverage statistical models and available data to estimate child survival summaries including neonatal, infant, and under-five mortality rates. UN IGME fits separate models for each age group and the GBD uses a multi-step modeling process. We propose a Bayesian survival framework to estimate temporal trends in the probability of survival as a function of age, up to the fifth birthday, with a single model. Our framework integrates all data types that are used by UN IGME: household surveys, vital registration, and other pre-processed mortality rates. We demonstrate that our framework is applicable to any country using log-logistic and piecewise-exponential survival functions, and discuss findings for four example countries with diverse data profiles: Kenya, Brazil, Estonia, and Syrian Arab Republic. Our model produces estimates of the three survival summaries that are in broad agreement with both the data and the UN IGME estimates, but in addition gives the complete survival curve.

Figures (22)

• Figure 1: Countries by data type included by UN IGME. VR = Vital registration; DHS = Demographic and Health Survey; MICS = Multiple Indicator Cluster Survey; FBH = Full birth history.
• Figure 2: Under-five mortality rate (U5MR), infant mortality rate (IMR), and neonatal mortality rate (NMR) from (a) survey data using pseudo-likelihood estimation and from (b) vital registration data using maximum likelihood estimation, according to the log-logistic survival and piecewise-exponential models. Comparison for survey data is against the discrete hazards model and comparison for vital registration is against the life table method used by UN IGME to convert counts to probabilities of death. The red diagonal line is a line of equivalence.
• Figure 3: Posterior median and 90% intervals for deaths per 1000 live births before each month of age, up to the fifth birthday, for Kenya, according to the (a) log-logistic and (b) piecewise-exponential survival models. Deaths per 1000 live births is equal to $1000 \times \Pr ( A \leq a )=1000\times [1-S( a | \bm{\theta})]$ where $A$ is the random variable for time at death in months. Estimates of the NMR, IMR, and U5MR from UN IGME are indicated with points and error bars at 1, 12, and 60 months, respectively, representing 90% intervals. The figure also includes (c) the probability of death by $a$ months of age conditional on death by 60 months of age, and (d) the same thing but with a log base 2 transformation applied to age, again with posterior median and 90% intervals. Results for all panels are presented for the years 1960, 1970, 1980, 1990, 2000, 2010, and 2020.
• Figure 4: Posterior median and 90% credible intervals for estimates of the U5MR, IMR, and NMR in Kenya according to the log-logistic and piecewise-exponential survival models. The UN IGME 2024 estimates and 90% intervals are also included for reference. Input data are plotted with point estimates and 95% error bars. For DHS full birth histories, the input data presented come from the discrete hazards model. DHS = Demographic and Health Survey; MICS = Multiple Indicator Cluster Survey; SBH = summary birth history; WFS = World Fertility Survey; MIS = Malaria Indicator Survey; U5MR = under-five mortality rate; IMR = infant mortality rate; NMR = neonatal mortality rate.
• Figure 5: Posterior median and 90% credible intervals for estimates of the U5MR, IMR, and NMR in Brazil according to the log-logistic and piecewise-exponential survival models. The UN IGME 2024 estimates and 90% intervals are also included for reference. Input data are plotted with point estimates and 95% error bars. For DHS full birth histories, the input data presented come from the discrete hazards model. DHS = Demographic and Health Survey; SBH = summary birth history; VR = Vital Registration; FBH = full birth history; U5MR = under-five mortality rate; IMR = infant mortality rate; NMR = neonatal mortality rate.